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Warwick Mathematics Institute Events

Seminar List Entry | Seminars by subject

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    Upcoming Seminars

  • Junior Number Theory on 24 June 2024 at 11:00 in B3.03

    Speaker: Eda Kirimli (University of Bristol)

    Title: Isogeny graphs of abelian surfaces

    Abstract: Isogeny-based cryptography is an active research area in post-quantum cryptography, and its security depends on variants of isogeny problems, namely the problem of finding an explicit isogeny between two abelian varieties. Although much of the research focused on isogenies of elliptic curves so far, it is interesting to understand isogeny graphs in dimension 2 in full generality, and the recent attacks using isogenies in higher dimensions switched the attention to dimension two. Unlike known methods, we use very geometrical and arithmetical objects, called refined Humbert invariants. We will propose possible promising applications of abelian surfaces with these invariants for isogeny-based cryptography.

  • New Perspectives in Computational Group Theory on 24 June 2024 at 11:15 in MS.01

    Speaker: Tim Burness (University of Bristol)

    Title: The regularity number of a finite group and other base-related invariants.

    Abstract: TBA

  • on 24 June 2024 at 11:15 in MS.01

    Speaker: Tim Burness (University of Bristol)

    Title: The regularity number of a finite group and other base-related invariants

    Abstract: Let G be a transitive permutation group on a finite set X, let H be a point stabiliser and recall that the base size of G, denoted b(G,H), is the minimal size of a base for G. Equivalently, the base size is the minimal integer k such that G has a regular orbit on the Cartesian product (G/H)^k. As a natural generalisation, let us say that a k-tuple (H_1,..., H_k) of core-free subgroups of G is regular if G has a regular orbit on G/H_1 x ... x G/H_k. Then the regularity number of G, denoted R(G), is the minimal integer k such that every k-tuple of core-free subgroups of G is regular. More refined versions can be defined by imposing additional conditions on the component subgroups, such as solubility or nilpotency, and this leads to natural generalisations of several widely studied conjectures on bases due to Cameron, Pyber and Vdovin.

    In this talk, I will introduce the main definitions and I will discuss some of the probabilistic and computational methods we use to study the regularity of subgroup tuples. I will conclude by presenting several new results in the almost simple setting, which extend some of my earlier work on bases with Guralnick, O'Brien, Saxl and Wilson.

    This is joint work with Marina Anagnostopoulou-Merkouri.

  • New Perspectives in Computational Group Theory on 24 June 2024 at 12:00 in MS.01

    Speaker: Cheryl Praeger (University of Oregon)

    Title: A collegial friendship across the globe: group actions from Oxford to Warwick and Perth.

    Abstract: TBA

  • on 24 June 2024 at 12:00 in MS.01

    Speaker: Cheryl Praeger (University of Western Australia)

    Title: A collegial friendship across the globe: group actions from Oxford to Warwick and Perth

    Abstract: Derek Holt and I were DPhil students together in Oxford, and our mathematical paths have criss-crossed over the decades. I will reflect on our mathematical interactions and finish by describing one of my current research projects where Derek’s work is proving invaluable. It involves absolutely irreducible actions of finite simple groups in small dimensions, and it underpin new procedures to compute with finite classical groups.

  • New Perspectives in Computational Group Theory on 24 June 2024 at 14:15 in MS.01

    Speaker: William Kantor (University of Oregon)

    Title: Some groups of order n^3

    Abstract: TBA

  • on 24 June 2024 at 14:15 in MS.01

    Speaker: William Kantor (University of Oregon)

    Title: Some groups of order n^3.

    Abstract: I will discuss the relationship between some groups and a standard combinatorial conjecture.

  • Junior Analysis and Probability Seminar on 24 June 2024 at 15:00 in MB0.07

    Speaker: Ziyang Liu (Warwick)

    Title: Moments of Stochastic heat flow at criticality

    Abstract: The critical SHF is constructed in Caravenna, Sun & Zygouras (2019) as a solution to the two dimensional stochastic heat equation with multiplicative chaos in a window around the rescaled critical temperature. We consider the moments of SHF, focusing on a concentrated initial condition. We prove an upper bound which should indicate the actual growth of these moments.

  • New Perspectives in Computational Group Theory on 24 June 2024 at 15:00 in MS.01

    Speaker: Alex Evetts (University of Manchester)

    Title: Twisted conjugacy growth of virtually nilpotent groups

    Abstract: TBA

  • on 24 June 2024 at 15:00 in MS.01

    Speaker: Alex Evetts (University of Manchester)

    Title: Twisted conjugacy growth of virtually nilpotent groups

    Abstract: The conjugacy growth function of a finitely generated group is a variation of the standard growth function, counting the number of conjugacy classes intersecting the n-ball in the Cayley graph. The asymptotic behaviour is not a commensurability invariant in general, but the conjugacy growth of finite extensions can be understood via the twisted conjugacy growth function, counting automorphism-twisted conjugacy classes. I will discuss what is known about the asymptotic and formal power series behaviour of (twisted) conjugacy growth, in particular some relatively recent results for certain groups of polynomial growth (i.e. virtually nilpotent groups).

  • New Perspectives in Computational Group Theory on 24 June 2024 at 16:30 in MS.01

    Speaker: Tara Macalister Brough (University of Nova de Lisboa)

    Title: Preserving self-similarity in free products of automaton semigroups.

    Abstract: TBA

  • New perspectives in Computational Group Theory on 24 June 2024 at 16:30 in MS.01

    Speaker: Tara Macalister Brough (Nova University Lisbon)

    Title: Preserving self-similarity in free products of automaton semigroups

    Abstract: The concept of self-similarity, famous source of interesting examples in group theory, extends naturally to semigroups. A self-similar semigroup can be viewed as the semigroup generated by the states of an automaton (more formally, a deterministic synchronous - but not necessarily finite - transducer) acting on strings over a finite alphabet. If the automaton is finite, we have an automaton semigroup.

    We examine the extent to which the classes of self-similar semigroups and automaton semigroups are closed under the free product operation, substantially improving on earlier work by Brough and Cain.

    This is joint work with Jan Philipp Wächter and Janette Welker.

  • New Perspectives in Computational Group Theory on 25 June 2024 at 11:15 in MS.01

    Speaker: Marston Conder (University of Auckland)

    Title: Some personal and professional perspectives on Derek Holt and his achievements.

    Abstract: TBA

  • New perspectives in Computational Group Theory on 25 June 2024 at 11:15 in MS.01

    Speaker: Marston Conder (University of Auckland)

    Title: Some personal and professional perspectives on Derek Holt and his achievements

    Abstract: Derek was my acting supervisor for the first year of my doctorate at Oxford (while Graham Higman was away for a year on sabbatical). Derek was very very helpful and I owe him a huge debt of gratitude for his quietly enabling me learn a lot of group theory from him. Over the years since then, I have learnt even more (and benefitted) from a wide variety of mathematical contributions by Derek -- made using his extensive knowledge of group theory, linear algebra, combinatorics, representations, cohomology, computational languages and systems and other things. In this talk I'll describe just a few of those contributions, especially concerning finitely-presented groups and algorithms for gaining important information about them, and some significant outcomes of their application.

  • New Perspectives in Computational Group Theory on 25 June 2024 at 12:00 in MS.01

    Speaker: Sarah Rees (University of Newcastle)

    Title: A regular odyssey.

    Abstract: TBA

  • New perspectives in Computational Group Theory on 25 June 2024 at 12:00 in MS.01

    Speaker: Sarah Rees (University of Newcastle)

    Title: A regular odyssey

    Abstract: I'll survey joint work of myself and Derek over the last nearly 40 years, most of this relating to regular languages. I'll start with our early work, together with David Epstein, calculating, verifying and computing with the regular languages that define automatic structures for groups, then briefly discuss other procedures we developed for finitely presented groups. And I'll discuss the regularity of group geodesics and quasigeodesics, and its implications, in hyperbolic groups and beyond.

  • New Perspectives in Computational Group Theory on 25 June 2024 at 14:15 in MS.01

    Speaker: David Craven (University of Birmingham)

    Title: Computing in (large) exceptional groups.

    Abstract: TBA

  • New perspectives in Computational Group Theory on 25 June 2024 at 14:15 in MS.01

    Speaker: David Craven (University of Birmingham)

    Title: Computing in (large) exceptional groups

    Abstract: Computation in classical groups, even of large dimension, is relatively straightforward, because of their description in terms of forms. Deciding if and element or matrix group lies in a given copy, or any copy, of a classical group is just a matter of computing some forms.

    Determining if an element or (say) 56-dimensional matrix group belongs to some copy of E7(q) is an altogether different matter. In this talk we will discuss how to compute in exceptional groups, both large and small, with examples.

  • New Perspectives in Computational Group Theory on 25 June 2024 at 15:00 in MS.01

    Speaker: Charles Leedham-Green (Queen Mary, University of London)

    Title: Computing the intersection of matrix groups over a finite field.

    Abstract: TBA

  • New perspectives in Computational Group Theory on 25 June 2024 at 15:00 in MS.01

    Speaker: Charles Leedham-Green (Queen Mary, University of London)

    Title: Computing the intersection of matrix groups over a finite field

    Abstract: Given two subgroups of GL(d,q) computing their intersection is a fundamentally important problem, and is generally intractable. But progress can be made. This is joint work with Eamonn O’Brien.

  • New Perspectives in Computational Group Theory on 25 June 2024 at 16:30 in MS.01

    Speaker: Eamonn O'Brien (University of Auckland)

    Title: Algorithms for finite groups: Derek's contributions to CGT.

    Abstract: TBA

  • New perspectives in Computational Group Theory on 25 June 2024 at 16:30 in MS.01

    Speaker: Eamonn O'Brien (University of Auckland)

    Title: Algorithms for finite groups: Derek's contributions to CGT

    Abstract: Derek Holt, often in collaboration with John Cannon, has developed many of the algorithms used in Magma to explore the structure and properties of finite groups. We briefly survey some of this work.

  • New Perspectives in Computational Group Theory on 25 June 2024 at 9:00 in MS.01

    Speaker: Susan Hermiller (University of Nebraska)

    Title: Subgroups of the group of dyadic piecewise linear homeomorphisms of the real line.

    Abstract: TBA

  • New Perspectives in Computational Group Theory on 25 June 2024 at 9:45 in MS.01

    Speaker: Gareth Tracey (University of Warwick)

    Title: Elements contained in a unique maximal subgroup of a finite group.

    Abstract: TBA

  • New perspectives in Computational Group Theory on 25 June 2024 at 9:45 in MS.01

    Speaker: Gareth Tracey (University of Warwick)

    Title: Elements contained in a unique maximal subgroup of a finite group

    Abstract: A finite group of prime power order has the property that every non-generating element is contained in a unique maximal subgroup. But what other examples are there of pairs (G,x) with G a finite group and x contained in a unique maximal subgroup of G? In this talk, I will speak about some recent progress on this problem. I will also discuss applications to Baer-Suzuki type problems in finite groups. This is joint work with R. M. Guralnick.

  • New Perspectives in Computational Group Theory on 26 June 2024 at 11:15 in MS.01

    Speaker: Saul Schleimer (University of Warwick)

    Title: Solving the word problem in the mapping class group in quasi-linear time.

    Abstract: TBA

  • New perspectives in Computational Group Theory on 26 June 2024 at 11:15 in MS.01

    Speaker: Saul Schleimer (University of Warwick)

    Title: Solving the word problem in the mapping class group in quasi-linear time.

    Abstract: The word problem for the mapping class group was first posed, and first solved, by Dehn [1922] in his Breslau lectures. His method was rediscovered, and greatly extended, by Thurston [1970-80's]. Mosher [1995] proved that the mapping class group is automatic and so found a quadratic-time algorithm for the word problem. Hamidi-Tehrani [2000] and Dynnikov [2023] gave quadratic-time algorithms using train-tracks.

    We give the first sub-quadratic-time algorithm. We combine the work of Dynnikov with a generalisation of the half-GCD algorithm to obtain an algorithm running in time $O(n \log^3(n))$.

    This is joint work with Mark Bell.

  • New perspectives in Computational Group Theory on 26 June 2024 at 12:00 in MS.01

    Speaker: Colva Roney-Dougal (University of St Andrews)

    Title: Polynomial-time normalisers for primitive groups

    Abstract: The normaliser problem takes as input generators for groups G and H, both contained in a common overgroup, and asks one to return generators for N_H(G).

    We present a polynomial time algorithm solving the probem for most, but not quite all, permutation groups, provided that G is primitive and G \leq H. Our methods are heavily reliant on ideas and results that I either learned from or proved with Derek.

    This is joint work with Jendrik Brachter.

  • New Perspectives in Computational Group Theory on 26 June 2024 at 12:45 in MS.01

    Speaker: Colva Roney-Dougal (University of St Andrews)

    Title: Polynomial-time normalisers for primitive groups.

    Abstract: TBA

  • New Perspectives in Computational Group Theory on 26 June 2024 at 9:00 in MS.01

    Speaker: Yago Antolin (Universidad Complutense de Madrid & ICMAT)

    Title: The falsification by fellow traveller property.

    Abstract: TBA

  • New perspectives in Computational Group Theory on 26 June 2024 at 9:00 in MS.01

    Speaker: Yago Antolin (Universidad Complutense de Madrid)

    Title: The falsification by fellow traveller property

    Abstract: The falsification by fellow traveller property is a property of graphs introduced by Neumann and Shapiro. It captures the essence of the proof of Cannon of the fact that hyperbolic groups have rational growth functions. It also implies the regularity of geodesics of Cayley graphs. Derek Holt has settle this property for several classes of groups (including Garside and some large-type Artin groups) .It is an useful property to find automatic structures on a group. In this talk I plan to give an overview of this property, some applications and some limitations.

  • New Perspectives in Computational Group Theory on 26 June 2024 at 9:45 in MS.01

    Speaker: Bettina Eick (TU Braunschweig)

    Title: Computing with polycyclic groups: new developments and open problems.

    Abstract: TBA

  • New perspectives in Computational Group Theory on 26 June 2024 at 9:45 in MS.01

    Speaker: Bettina Eick (Technische Universität Braunschweig)

    Title: Computing with polycyclic groups: new developments and open problems

    Abstract: This talk gives a brief introduction into the algorithmic theory of polycyclic groups with a view towards practical algorithms. It then considers some new developments: one is the computation of the Frattini subgroup of a polycyclic group, the other is a solution to the isomorphism problem and the computation of genera for finitely generated torsion free nilpotent groups of small Hirsch length. The talk provides an overview on these two new developments and it finishes with some still open problems.

  • Combinatorics on 28 June 2024 at 14:00 in B3.02

    Speaker: Nika Salia (King Fahd University)

    Title: Linear three-uniform hypergraphs with no Berge path of given length

    Abstract: Extensions of Erd\H{o}s-Gallai Theorem for general hypergraphs are well studied.
    In this work, we prove the extension of Erd\H{o}s-Gallai Theorem for linear hypergraphs.
    In particular, we show that the number of hyperedges in an $n$-vertex $3$-uniform linear hypergraph, without a Berge path of length $k$ as a subgraph is at most $\frac{(k-1)}{6}n$ for $k\geq 4$. The bound is sharp for infinitely many $k$ and $n$.
    Joint work with Ervin Győri.

  • Soft Matter Lunches on 01 July 2024 at 12:00 in B1.01

    Speaker: Enej Caf (University of Ljubljana)

    Title: TBA

    Abstract: TBA

  • Past Seminars

  • Combinatorics on 21 June 2024 at 14:00

    Speaker: George Kontogeorgiou (University of Chile)

    Title: Small weakly separating path systems for complete graphs

    Abstract: Let G be a graph and let P be a set of paths of G. We say that P weakly separates G if for every pair of edges of G there exists a path in P that contains exactly one of them. It is a well-known problem to determine the size of the smallest weakly separating path system of a given graph on n vertices. Around a decade ago, Falgas-Ravry, Kittipassorn, Korandi, Letzter and Narayanan conjectured an upper bound of O(n) paths. This was proved last year by Bonamy, Botler, Dross, Naia and Skokan, who further conjectured an upper bound of (1+o(1))n paths.

    Some authors have considered the restriction of this problem to complete graphs. It is known that a weakly separating path system for a complete graph on n vertices must contain at least n-1 paths. Recently, Fernandes, Oliveira Mota and Sanhueza-Matamala proved that (1+o(1))n paths suffice.

    In recent work with Maya Stein, we proved that every complete graph on n vertices has a weakly separating path system of n+2 paths. I will provide a brief view of the history of the problem and a proof sketch.

  • Geometry and Topology on 21 June 2024 at 14:00

    Speaker: Carl-Fredrik Nyberg Brodda (KIAS)

    Title: TBA

    Abstract: TBA

  • Applied Mathematics on 21 June 2024 at 12:00

    Speaker: Lukas Eigentler (Warwick)

    Title: Modelling dryland vegetation patterns

    Abstract: **Disclaimer: I gave the same talk at the SBIDER seminar in January.** Vegetation patterns are a ubiquitous feature of semi-arid regions and are a prime example of a self-organisation principle in ecology. In this talk, I present a broad overview of how I have used mathematical modelling and analysis to develop new insights into vegetation pattern dynamics, for example on how different plant species can coexist despite competition for a single limiting resource, and a possible resolution for contradicting field data on the long-term migration dynamics of vegetation patterns.

  • Analysis on 20 June 2024 at 16:00

    Speaker: Florent Baudier (Texas A&M)

    Title: Metric invariants from curvature-like inequalities

    Abstract: A central theme in the 40-year-old Ribe program is the quest for metric invariants that characterize local properties of Banach spaces. These invariants are usually closely related to the geometry of certain sequences of finite graphs (Hamming cubes, binary trees, diamond graphs...) and provide quantitative bounds on the bi-Lipschitz distortion of those graphs.

    A more recent program, deeply influenced by the late Nigel Kalton, has a similar goal but for asymptotic properties instead. In this talk, we will motivate the (asymptotic) notion of infrasup umbel convexity (introduced in collaboration with Chris Gartland (UC San Diego)) and discuss the value of this invariant for Heisenberg groups. This (asymptotic) invariant is inspired by the profound work of Lee, Mendel, Naor, and Peres on the (local) notion of Markov convexity. If time permits we will discuss the notion of bicone convexity, a new asymptotic invariant, inspired by the work of Eskenazis, Mendel, and Naor on the (local) notion of diamond convexity.

    All these metric invariants share the common feature of being derived from point-configuration inequalities which generalize curvature inequalities.

  • Geometry and Topology on 20 June 2024 at 14:00

    Speaker: Ramon Flores (Universidad de Sevilla)

    Title: Characterizing graph properties via RAAGs.

    Abstract: In the last years, thorough research has been conducted in order to understand graph properties in terms of group properties of the associated right-angled Artin group (RAAG). These properties should be intrinsic, in the sense that they should not depend on a concrete system of generators of the group. In this talk, we will give a general review of the topic, with emphasis on planarity, self-complementarity, and existence of surjections. In particular, we will highlight the crucial role of the cohomology algebra of the group in our approach.

    This is joint work with Delaram Kahrobaei (CUNY New York) and Thomas Koberda (Virginia).

  • Probability Theory on 19 June 2024 at 16:00

    Speaker: Richard Kraaij (TU Delft)

    Title: Well-posedness for Hamilton-Jacobi equations for stochastic control problems: A new view on the classical approach using couplings

    Abstract: Stochastic control problems for controlled Markov processes can be infinitesimally characterized using a second order Hamilton-Jacobi-Bellman (HJB) equation. The classical work of Crandall-Ishii-Lions (1992) establishes how to obtain uniqueness of viscosity solutions for controlled diffusion processes, and a collection of recent works has pushed the estimates to include classes of spatially inhomogeneous controlled Lévy processes. We introduce a new perspective of the classical proof methods in terms of Markovian couplings, enabling the extension of the applicability to a wider range of Lévy type processes and paving the pay for extensions to new contexts.
    Based on joint work (in progress) with Serena Della Corte (Delft), Fabian Fuchs (Bielefeld) and Max Nendel (Bielefeld).

  • Postgraduate on 19 June 2024 at 12:00

    Speaker: Hamdi Dervodeli (University of Warwick)

    Title: Does tropical geometry know how to factor classical polynomials?

    Abstract: The factoring locus of a polynomial is a list of conditions on its coefficients under which the polynomial factors. The aim of this talk is to explore potential connections tropical geometry has with this factoring locus. More generally, we want to know if the reducibility of a variety is detected by the tropicalization of its defining ideal.

  • Junior Analysis and Probability on 17 June 2024 at 15:00

    Speaker: Artemis Aikaterini Vogiatzi (Queen Mary)

    Title: Singularity models for high codimension mean curvature flow

    Abstract: Mean curvature flow is a geometric evolution equation that describes how a submanifold embedded in a higher-dimensional space changes its shape over time. We establish a codimension estimate that enables us to prove at a singular time of the flow, there exists a rescaling that converges to a smooth codimension one limiting flow in Euclidean space, regardless of the original flow's codimension. Under a cylindrical type pinching, we show that this limiting flow is weakly convex and either moves by translation or is a self-shrinker. These estimates allow us to analyse the behaviour of the flow near singularities and establish the existence of the limiting flow. Considering spaces such as the
    {\displaystyle \mathbb {C} }Pn, we go beyond the finite timeframe of the mean curvature flow, by proving that the rescaling converges smoothly to a totally geodesic limit in infinite time. Our approach relies on the preservation of the quadratic pinching condition along the flow and a gradient estimate that controls the mean curvature in regions of high curvature.

  • Colloquium on 14 June 2024 at 16:00

    Speaker: Nick Sheridan (Edinburgh)

    Title: Integrality of mirror maps via homological mirror symmetry

    Abstract: Classical mirror symmetry is a conjectural relationship between curve-counting invariants on one space and Hodge theory on a 'mirror' space. Givental has proved it for a broad class of complete intersections in toric varieties. A key player in the classical mirror symmetry formalism is the so-called 'mirror map', which relates the relevant parameters on the two sides. Since the early days of mirror symmetry it has been observed that the Taylor coefficients of the mirror map seem always to be integers. As we have explicit formulae for mirror maps in terms of solutions to certain hypergeometric equations, this can be considered as a conjecture in pure arithmetic; it has only been proved in limited settings, by direct analysis of the formulae. The existing proofs of classical mirror symmetry shed no light on 'why' this conjecture should hold. We will explain how it is a consequence of an arithmetic refinement of Kontsevich's homological mirror symmetry conjecture. As that has been proved in a broad class of examples, we obtain the first proof of this conjecture in many new cases. Based on joint works with Ganatra-Hanlon-Hicks-Pomerleano and with Bleau.

  • Combinatorics on 14 June 2024 at 14:00

    Speaker: Sean Dewar (University of Bristol)

    Title: Counting realisations of rigid graphs

    Abstract: A graph is d-rigid if, given any generic positioning of its vertices in d-dimensional Euclidean space, there are finitely many other realisations of the graph in d-dimensional Euclidean space (modulo isometries) with the same length edges. Combinatorial characterisations for 1-rigidity (i.e., connectivity) and 2-rigidity (Geiringer-Laman theorem) are known, but it is currently an open problem for d > 2. Interestingly, these combinatorial characterisations remain the same if we switch the geometry we embed into for either spherical or hyperbolic geometries.

    My talk will be a survey of results involving a variation of this problem: given a d-rigid graph with a generic realisation, how many other realisations of the graph exist with the same length edges (now allowing for complex realisations also)? For example, given the 2-rigid graph formed by gluing two triangles at an edge, every generic realisation in the plane has exactly one other edge-length equivalent realisation that is formed by flipping one of the triangles. We can similarly consider the realisation number of a graph with respect to spherical and hyperbolic geometries, with the former having close links with the study of cross-ratio degrees in algebraic geometry.

    I shall prove that for realisation numbers, the geometry does matter, and in fact the flat Euclidean space realisation number is a lower bound for the realisation number in spherical and hyperbolic geometries. I will also cover recent research of myself and others which uncovered a link between the 2-dimensional realisation numbers, tropical geometry and Tutte polynomials. This work is the culmination of multiple projects with Georg Grasegger (RICAM), Oliver Clarke (Edinburgh), Daniel Green-Tripp (Bristol), James Maxwell (Bristol), Anthony Nixon (Lancaster), Yue Ren (Durham) and Ben Smith (Lancaster).

  • Applied Mathematics on 14 June 2024 at 12:00

    Speaker: Sara Jabbari (Birmingham)

    Title: Mathematical tools to aid with the development of novel antimicrobials

    Abstract: The ability of bacteria to become resistant to previously successful antibiotic treatments is an urgent and increasing worldwide problem. Solutions can be sought via a number of methods including, for example, identifying novel antibiotics, re-engineering existing antibiotics or developing alternative treatment methods. The nonlinear interactions involved in infection and treatment render it difficult to predict the success of any of these methods without the use of computational tools in addition to more traditional experimental work. We use mathematical modelling to aid in the development of anti-virulence treatments which, unlike conventional antibiotics that directly target a bacterium's survival, may instead attenuate bacteria and prevent them from being able to cause infection or evade antibiotics. Many of these approaches, however, are only partially successful when tested in infection models. Our group are studying a variety of potential targets, including preventing bacteria from binding to host cells, inhibiting the formation of persister cells (these can tolerate the presence of antibiotics) and blocking efflux pump action (a key mechanism of antimicrobial resistance). I will present results that illustrate how mathematical modelling can suggest ways in which to improve the efficacy of these approaches.

  • Algebraic Geometry on 12 June 2024 at 15:00

    Speaker: Andrés Ibáñez Núñez (Oxford)

    Title: Refined Harder-Narasimhan filtrations in moduli theory

    Abstract: We introduce a notion of refined Harder-Narasimhan filtration, defined abstractly for algebraic stacks satisfying natural conditions. Examples include moduli stacks of objects at the heart of a Bridgeland stability condition, moduli stacks of K-semistable Fano varieties, moduli of principal bundles on a curve, and quotient stacks. We will explain how refined Harder-Narasimhan filtrations are closely related both to stratifications and to the asymptotics of certain analytic flows, relating and expanding work of Kirwan and Haiden-Katzarkov-Kontsevich-Pandit, respectively. In the case of quotient stacks by the action of a torus, the refined Harder-Narasimhan filtration can be computed in terms of convex geometry.

  • Algebraic Topology on 11 June 2024 at 17:00

    Speaker: Jack Davies (University of Bonn)

    Title: Geometric norms on equivariant elliptic cohomology

    Abstract: A notion of multiplicative structure on equivariant cohomology theories will be defined called a ``geometric norm structure''. We will then introduce equivariant elliptic cohomology, as defined by Lurie and Gepner--Meier, and discuss how it naturally comes with a geometric norm structure. We will also touch on a few applications of these structures: constructions of C_p-normed algebra structures and integral models for Behrens' Q(N) spectra. This is joint work-in-progress with William Balderrama and Sil Linskens.

  • MIR@W on 10 June 2024 at 16:00

    Speaker: Alex Penn (University of Surrey/Defra Systems Research Team)

    Title: Modelling in real world policy contexts: participatory systems mapping, actionable complexity and accounting for human behaviour

    Abstract: Participatory systems mapping (PSM) is increasingly popular in government as a way to understand how policy acts within open, complex adaptive systems. PSM methods allow stakeholders to collaboratively develop causal models of their complex system, consisting of a system’s key components, from any domain, and the causal interconnections between them. Then to analyse them to suggest ways in which system complexity impacts policy-relevant questions and to design policy which exploits system structure and dynamics. These can produce actionable insights that can be directly applied in the policy process, such as potential unexpected effects of interventions, trade-offs between outcomes, policy interactions and potential vulnerabilities or policy levers. Crucially however, the anticipated behaviour of system actors forms part of the causal structure and can radically change this structure, when people adapt or do not behave as expected in response to policy interventions. I will discuss PSM work with UK national and regional government highlighting instances in which behaviour plays a key role in how maps are structured and can be used. I aim to use this to open up a discussion on how mathematical modelling of human behaviour could potentially be integrated with complex systems approaches and the realities of the policy making process.

  • MIR@W on 10 June 2024 at 15:30

    Speaker: Ruth McCabe (Imperial College London)

    Title: Communicating uncertainty in modelling to non-technical audiences

    Abstract: While mathematical models of epidemics are widely used to inform public health decision-making, the uncertainty in results are often poorly communicated, in particular to non-technical audiences. In this talk, I will share work from my DPhil thesis surrounding the communication of uncertainty to both decision-makers and the public and discuss how our community might improve the way in which we share our results.

  • Junior Analysis and Probability Seminar on 10 June 2024 at 15:00

    Speaker: Andrew Rout (Rennes)

    Title: Nonlinear dispersive PDEs: The difficulty of periodicity

    Abstract: Solutions to linear dispersive PDEs want to spread out (disperse) in space. Heuristically, working on a periodic domain constrains this dispersion more than working in ℝ^n. As such, it is usually harder to study well-posedness of nonlinear dispersive equations on periodic domains.

    In this pedagogical talk, we discuss some of the techniques from harmonic analysis used to overcome these difficulties. This talk is aimed at non-specialists.

  • MIR@W on 10 June 2024 at 14:30

    Speaker: David Haw (University of Liverpool)

    Title: Human behaviour and transmission modelling: how to calibrate an new generation of integrated models

    Abstract: We build on an integrated economic/epidemiological model by introducing a parametric description of behavioural change in response to quantities measurable in real time during an outbreak. A proof of concept will be given based on data from the first 18 months of COVID-19 mitigation in the UK.

  • DAGGER on 10 June 2024 at 14:00

    Speaker: Laura Breitkopf (University of Bremen)

    Title: Equidistribution of cusp points of Hecke triangle groups

    Abstract: In the framework of infinite ergodic theory, we derive equidistribution
    results for suitable weighted sequences of cusp points of Hecke triangle
    groups encoded by group elements of constant word length with respect to
    a set of natural generators. This is a generalization of the
    corresponding results for the modular group whose cusp points are
    related to the Stern-Brocot sequence that can be constructed from the
    Farey mapping. In this talk, we will introduce generalizations of the
    Farey mapping and discuss how their dynamical properties yield the
    equidistribution results. We rely on advanced results from infinite
    ergodic theory and transfer operator techniques developed for AFN-maps.
    This is joint work with Marc Keßeböhmer and Anke Pohl.

  • MIR@W on 10 June 2024 at 14:00

    Speaker: Alice Milne (Rothamsted Research)

    Title: Modelling land management

    Abstract: European ash (Fraxinus excelsior), one of the most common trees in the UK has been seriously impacted by the fungal disease Ash Dieback (ADB). Additionally, ash trees are now facing a new threat from the potential invasion of the Emerald Ash Borer (Agrilus planipennis, EAB), a pest beetle that is ranked in the top three most dangerous invasive pests by EFSA. Preventing the introduction of such threats is often difficult, therefore, early detection and successful management are key areas where science can deliver. Scientific research tends to focus on single pest species, ignoring possible interactions of multiple pests/diseases. In addition, surveillance and management strategies also often neglect fail to account for the ability and willingness of managers to adopt advice on the control or management of pests and diseases. To address these challenges, we developed a model of the distribution and spread of ADB and the potential introduction and dispersal of EAB in a spatially explicit landscape. We integrated this with a novel model of stakeholder behaviour based on social science research. Stakeholder perspectives and decision-making was were modelled dynamically allowing them to realistically change over time in response to peer views and external influences. The linked models allowed us to identify the impacts of deploying surveillance and management resources under various scenarios to deliver successful strategies for the detection and management of ash treescapes. These results can be applied to other ecologically important systems where social and epidemiological interactions impact the delivery of appropriate management and control of pests and diseases.

  • MIR@W on 10 June 2024 at 11:45

    Speaker: Suzy Moat and Tobias Preis (University of Warwick)

    Title: Quantifying human behaviour with online data

    Abstract: Our everyday usage of the internet leaves volumes of data in its wake. Can we use this data to help us reduce delays and costs in measuring human behaviour, or even to measure behaviour we couldn’t measure before? Here we will outline a number of studies carried out at the Data Science Lab at Warwick Business School, investigating whether online data can help us monitor disease levels or better understand the relationship between the beauty of outdoor environments and our wellbeing. We will discuss some of the challenges in generating estimates of human behaviour from online data when our relationship with the internet continues to evolve at such rapid pace.

  • MIR@W on 10 June 2024 at 11:15

    Speaker: Anne Kandler (Max Planck Institute for Evolutionary Anthropology)

    Title: Inferring social learning processes from empirical data: Which details matter?

    Abstract: Applying theoretical cultural evolution frameworks to real-world data to understand the underlying mechanism that - potentially - produced them is an interesting inferential challenge. Problems may arise due to characteristics of the available data such as its level of aggregation (e.g., population-level or individual-level data), its sparsity or its spatial/temporal resolution. However, problems may also arise due to misspecifications of the mathematical framework used to analyse the data, i.e., due to the omittance of crucial properties of the cultural system that can affect the observed data. Here, we mainly discuss the model misspecification problem. Focusing on the coarse distinction between unbiased and biased social learning, we use models of unbiased social learning coupled with realistic assumptions about demographic properties or the structure of the learning process to show that population-level frequency data generated by these models may not conform to theoretical expectations under unbiased social learning. In more detail, we show that using expectations blind to (i) demographic characteristics of the population (age structure, social structure or changes in population size) or (ii) details of the learning process (learning of packages of cultural variants vs. the learning of single variants) may generate wrong conclusion about the underlying learning process.

  • MIR@W on 10 June 2024 at 10:15

    Speaker: Kavita Vedhara (Cardiff University)

    Title: Behavioural Science: Warts and all

    Abstract: An overview of some of the current challenges with behavioural data to provoke discussion on how behavioural science and mathematical modelling communities may work together more effectively.

  • Colloquium on 07 June 2024 at 16:00

    Speaker: Radu Cimpeanu (Warwick)

    Title: Interfacing with the real world: how to break mathematical models (and put them together again)

    Abstract: In this presentation I will explore the interplay between classical - as well as modern - analytical techniques in applied mathematics and high performance computing. When not completely disjointed (or indeed one completely absent), these two approaches are often deployed with a verification-oriented mindset. By contrast, I will strive to convince you of the power of analytically-informed computational approaches (or indeed computationally-informed analytical methods) that draw elements from mathematical modelling, differential equations, asymptotic and complex analysis, and control theory. Interfaces will be the main theme of the talk not only as the intersection space between different methodologies, but also in view of the problems motivating this work, which include interfacial flow problems such as elucidating the dynamics of droplets, bubbles and liquid films in non-trivial (and highly non-linear!) regimes as the setup for hybrid technique development. Should time allow it, I will also touch upon knowledge transfer aspects within this vast and exciting problem space.

  • Combinatorics on 07 June 2024 at 14:00

    Speaker: Oliver Janzer (University of Cambridge)

    Title: Tight general bounds for the extremal numbers of 0-1 matrices

    Abstract: A zero-one matrix $M$ is said to contain another zero-one matrix $A$ if we can delete some rows and columns of $M$ and replace some $1$-entries with $0$-entries such that the resulting matrix is $A$. The extremal number of $A$, denoted $ex(n,A)$, is the maximum number of $1$-entries that an $n\times n$ zero-one matrix can have without containing $A$. The systematic study of this function for various patterns $A$ goes back to the work of Furedi and Hajnal from 1992, and the field has many connections to other areas of mathematics and theoretical computer science. The problem has been particularly extensively studied for so-called acyclic matrices, but very little is known about the general case (that is, the case where $A$ is not necessarily acyclic). We prove the first asymptotically tight general result by showing that if $A$ has at most $t$ $1$-entries in every row, then $ex(n,A)\leq n^{2-1/t+o(1)}$. This verifies a conjecture of Methuku and Tomon.

    Our result also provides the first tight general bound for the extremal number of vertex-ordered graphs with interval chromatic number $2$, generalizing a celebrated result of Furedi, and Alon, Krivelevich and Sudakov about the (unordered) extremal number of bipartite graphs with maximum degree $t$ in one of the vertex classes.

    Joint work with Barnabas Janzer, Van Magnan and Abhishek Methuku.

  • Applied Mathematics on 07 June 2024 at 12:00

    Speaker: Roberto Alamino (Aston)

    Title: Machine Learning in Statistical Physics & Condensed Matter - Physical Information and Explainability

    Abstract: The talk will provide an overview of some key applications of machine learning (ML) to problems involving condensed matter and statistical physics taking a closer look into two main issues: (i) the role played by physical information in the design of the ML models and (ii) the explainability of the trained parameters and the possibility of extracting meaningful physics from them.

  • Statistical Mechanics on 06 June 2024 at 14:00

    Speaker: Ben Lees (University of Leeds)

    Title: Discrete lattice models as complex spin systems

    Abstract: It has long been known that some important spin systems on lattices, such as the spin-O(N) model, have representations in terms of discrete structures (loops, paths etc) on the lattice. This has been a fruitful connection, however many discrete models on lattices, such as percolation, the dimer model, spatial random permutations, and others, are also of considerable interest. In these cases there was no precise link to spin systems, either because the motivation came from elsewhere, or because these are toy models that have taken on a life of their own. In this talk I will present a spin system where spins take continuous complex values. By choosing the measures appropriately, virtually any discrete system of interest can be obtained as a special case. These spin systems then lend themselves naturally to several of the well developed tools of classical spin systems.
    This is joint work with Volker Betz.

  • Probability Theory on 05 June 2024 at 16:00

    Speaker: Louis-Pierre Arguin (University of Oxford)

    Title: Large Values of the Riemann Zeta Function: A Probabilistic Approach

    Abstract: I will give an account of the recent progress in probability and in number theory to understand the large values of the zeta function on the critical line, especially in short intervals. The problems have interesting connections to statistical mechanics of disordered systems, both in their interpretations and in the techniques of proofs. These connections will be emphasized. This is based in part on joint works with Emma Bailey, and with Paul Bourgade & Maksym Radziwill.

  • Dynamical Systems mini-course on 05 June 2024 at 10:00

    Speaker: Zhiyuan Zhang (Imperial College)

    Title: Exponential mixing of 3D Anosov flows (part III)

    Abstract: TBA

  • Algebraic Topology on 04 June 2024 at 17:00

    Speaker: Daniel Graves (University of Leeds)

    Title: Reflexive homology and beyond

    Abstract: Reflexive homology is a homology theory for involutive algebras. In this talk I will introduce reflexive homology, explain how it fits into the framework of crossed simplicial groups and present some recent results and calculations. Time permitting, I will discuss some recent work (joint with Sarah Whitehouse) where we build an action of a group into these constructions.

  • Dynamical Systems mini-course on 04 June 2024 at 14:00

    Speaker: Zhiyuan Zhang (Imperial College)

    Title: Exponential mixing of 3D Anosov flows (part II)

    Abstract: TBA

  • Junior Analysis and Probability Seminar on 03 June 2024 at 15:00

    Speaker: Zdeněk Mihula (Czech Technical University)

    Title: Different shades of noncompactness of optimal Sobolev embeddings

    Abstract: Let Ω⊆Rn be a bounded domain, p∈[1,n), and p∗=np/(n−p). The optimal Sobolev embedding
    V1,p0(Ω)↪Lp∗(Ω) and its Lorentz improvement, V1,p0(Ω)↪Lp∗,p(Ω), are notoriously known to be noncompact. In this talk, we will discuss their noncompactness from the quantitative point of view. In particular, we will look into how to measure the difference in their noncompactness. We will also discuss critical Sobolev embeddings, which correspond to p=n.

  • Dynamical Systems mini-course on 03 June 2024 at 12:00

    Speaker: Zhiyuan Zhang (Imperial College)

    Title: Exponential mixing of 3D Anosov flows

    Abstract: TBA

  • Colloquium on 31 May 2024 at 16:00

    Speaker: Mark Cannon (Oxford)

    Title: Difference of Convex Functions in Robust and Adaptive Model Predictive Control

    Abstract: This talk will provide an overview of problems and methods in control systems research, with a particular focus on data-driven optimisation-based control in applications where safety is critical. Starting with a brief introduction to the main concepts and challenges, the discussion will motivate recent work on Model Predictive Control (MPC) in this context. We will consider how to use differences of convex (DC) functions to derive convex conditions that allow the future control system performance to be optimized as a sequence of convex sub-problems. Key contributions to theoretical results, including feasibility, convergence, optimality and closed loop stability will be discussed.

    The talk will explain how DC function representations can be computed directly from data and how model estimation can be performed online simultaneously with control to define safe learning-based control algorithms. We describe an application of these ideas to a problem of controlling the transistions of a tiltwing vertical take-off and landing (VTOL) aircraft subject to wind disturbances and model uncertainty. Results of a case study will be discussed involving a VTOL aircraft model with unknown wind gusts and aerodynamics defined by experimental data.

  • Combinatorics on 31 May 2024 at 14:00

    Speaker: Katharina Jochemko (KTH Royal Institute of Technology)

    Title: Weighted Ehrhart theory

    Abstract: In 1962 Ehrhart proved that the number of lattice points in integer dilates of a lattice polytope is given by a polynomial — the Ehrhart polynomial of the polytope. Since then Ehrhart theory has developed into a very active area of research at the intersection of combinatorics, geometry and algebra.
    The Ehrhart polynomial encodes important information about the polytope such as its volume and the dimension. An important tool to study Ehrhart polynomials is the h*-polynomial, a linear transform of the Ehrhart polynomial which is given by the numerator of the generating series. By a famous theorem of Stanley the coefficients of the h*-polynomial are always nonnegative integers. In this talk, we discuss generalizations of this result to weighted lattice point enumeration in rational polytopes where the weight function is given by a polynomial. In particular, we show that Stanley’s Nonnegativity Theorem continues to hold if the weight is a sum of products of linear forms that a nonnegative.
    This is joint work with Esme Bajo, Robert Davis, Jesús De Loera, Alexey Garber, Sofía Garzón Mora and Josephine Yu.

  • Applied Mathematics on 31 May 2024 at 12:00

    Speaker: Giulia Celora (UCL)

    Title: Self-organized patterning in complex fluids

    Abstract: Understanding how multicomponent systems self-organise to control their emergent dynamics across spatial and temporal scales is a fundamental problem with important applications in many areas; from the design of soft materials to the study of developmental biology. In this talk, I will discuss how we can use mathematical modelling to understand the role of microscale physical interactions in the self-organisation of complex fluids. I will illustrate this by presenting two examples. Firstly, I will discuss self-organization in stimuli-responsive polyelectrolyte gels surrounded by an ionic solution; secondly, I will discuss self-organization during collective migration of multicellular communities. Our results reveal hidden connections between these two initially disconnected applications hinting at the existence of general principles controlling self-organisation of both inanimate and living matter.

  • Analysis on 30 May 2024 at 16:00

    Speaker: Ulrich Menne (National Taiwan Normal University)

    Title: A Priori Bounds for Geodesic Diameter

    Abstract: Our starting point is P. Topping's a priori bound for the
    geodesic diameter of compact connected manifolds immersed in Euclidean
    space. In joint work with C. Scharrer, we transfer this result to
    indecomposable varifolds, include surfaces with boundary in the
    treatment, and ensure the applicability to geometric variational
    problems (including several formulations of Plateau's problem).

  • Statistical Mechanics on 30 May 2024 at 14:00

    Speaker: Leonid Pastur (King's College London)

    Title: Entanglement Entropy of Free Disordered Fermions

    Abstract: We consider a macroscopic system of free lattice fermions, and we are interested in the entanglement entropy (EE) of a large block of size L of the system, treating the rest of the system as the macroscopic environment of the block. Entropy is a widely used quantifier of quantum correlations between a block and its environment. We begin with known results (mostly one-dimensional) on the asymptotics form of EE of translation-invariant systems for large L, where for any value of the Fermi energy there are basically two asymptotics known as area law and enhanced (violated ) area law. We then show that in the disordered case and for the Fermi energy belonging to the localized spectrum of a one-body Hamiltonian, the EE obeys the area law for all typical realizations of disorder and any dimension. As for the enhanced area law, it turns out to be possible for some special values of the Fermi energy in the one-dimensional case.

  • Geometry and Topology on 30 May 2024 at 14:00

    Speaker: David Hume (University of Birmingham)

    Title: Coarse embeddings, and yet more ways to avoid them

    Abstract: Coarse embeddings (maps between metric spaces whose distortion can be controlled by some function) occur naturally in various areas of pure mathematics, most notably in topology and algebra. It may therefore come as a surprise to discover that it is not known whether there is a coarse embedding of three-dimensional real hyperbolic space into the direct product of a real hyperbolic plane and a 3-regular tree. One reason for this is that there are very few invariants which behave monotonically with respect to coarse embeddings, and thus could be used to obstruct coarse embeddings.

    In this talk I will discuss some new invariants which combine two very classical invariants – asymptotic dimension and growth – to give different obstructions to coarse embeddings. This is joint work with John Mackay and Romain Tessera.

  • Probability Theory on 29 May 2024 at 16:00

    Speaker: Giovanni Conforti (University of Padova)

    Title: Convergence analysis of score-based diffusion models

    Abstract: Score-based generative models are a new paradigm to sample efficiently from a target distribution. They find plenty of applications in statistical machine learning and beyond. These methods work in two steps. In a first step, an ergodic Markov process is run to convert data into pure noise. Such process is called the forward process. Then, one learns how to sample from the time-reversal of the forward process, called the backward process. To this aim, it is necessary to approximately compute the score function associated to the forward process. In this talk, I will present a convergence analysis of score based diffusion models inspired from ideas coming from optimal transport and stochastic control which takes into account the three main sources of error: initialisation, time-discretisation and score approximation. The presentation is based on joint work with Alain Durmus and Marta Gentiloni Silveri.

  • Algebraic Geometry on 29 May 2024 at 15:00

    Speaker: Patrick Kennedy-Hunt (Sheffield)

    Title: Degenerating Hilbert schemes and the secondary fan

    Abstract: I will start by explaining the secondary fan of a polytope and its connection to a moduli space of tropical curves. Associated to any fan is a toric variety, and I will explain how to interpret the fan associated to a moduli space of tropical curves as a natural moduli space. Time permitting, I will outline how this toric story is a special case of  a theory built to study Hilbert schemes through degeneration techniques.

  • Algebraic Topology on 28 May 2024 at 17:00

    Speaker: Rhiannon Savage (University of Oxford)

    Title: A Representability Theorem for Stacks in Derived Geometry Contexts

    Abstract: The representability theorem for stacks, due to Artin in the underived setting and Lurie in the derived setting, gives conditions under which a stack is representable by an n-geometric stack. In recent work of Ben-Bassat, Kelly, and Kremnizer, a new theory of derived analytic geometry has been proposed as geometry relative to the ∞-category of simplicial commutative Ind-Banach R-modules, for R a Banach ring. In this talk, I will present a representability theorem which holds in a very general context, encompassing both the derived algebraic geometry context of Toën and Vezzosi and these new derived analytic geometry contexts.

  • Junior Number Theory on 28 May 2024 at 11:00

    Speaker: Harmeet Singh (King's College London / LSGNT)

    Title: The Hasse norm principle

    Abstract: For an extension of number fields K/k, we say that the Hasse norm principle holds if an element of k that is a norm everywhere locally is also a norm globally. The study of when this holds has been of interest ever since Hasse proved his famous norm theorem in 1931. In this down-to-earth talk, I'll begin by introducing the Hasse norm principle, discuss some of the instances when it holds, and provide an overview of the main ideas that go into proving these results.

  • Colloquium on 24 May 2024 at 16:00

    Speaker: Henry Wilton (Cambridge)

    Title: Rational curvature invariants of 2-dimensional complexes

    Abstract: I will discuss some new invariants of 2-complexes. They are inspired by recent developments in the theory of one-relator groups, but also have the potential to unify the theories of many well-studied families including small-cancellation presentation complexes, CAT(0) 2-complexes and 3-manifold spines, in addition to the motivating examples of one-relator presentation complexes. The fundamental result is that these invariants are the extrema of explicit linear-programming problems, and in particular are rational, computable and realised. The definitions suggest a conjectural “map” of 2-complexes, which I will attempt to describe.

  • Combinatorics on 24 May 2024 at 14:00

    Speaker: Levente Bodnar (University of Oxford)

    Title: Extremal Problems in Circuit Complexity

    Abstract: Finding correlation bounds between CNFs and various Boolean functions is one of the central questions in low-depth circuit complexity. Each function's complexity is associated with a family of extremal combinatorial problems. This talk focuses on two natural Boolean functions—parity and threshold—the corresponding combinatorial problems, and the progress towards answering them.

  • Applied Mathematics on 24 May 2024 at 12:00

    Speaker: Valeria Giunta (Swansea)

    Title: Bifurcations, pattern formation and multi-stability in non-local models of interacting species

    Abstract: Understanding the mechanisms behind the spatial distribution, self-organisation and aggregation of organisms is a central issue in both ecology and cell biology. Since self-organisation at the population level is the cumulative effect of behaviours at the individual level, it requires a mathematical approach to be elucidated.

    In nature, every individual, be it a cell or an animal, inspects its territory before moving. The process of acquiring information from the environment is typically non-local, i.e. individuals have the ability to inspect a portion of their territory. In recent years, a growing body of empirical research has shown that non-locality is a key aspect of movement processes, while mathematical models incorporating non-local interactions have received increasing attention for their ability to accurately describe how interactions between individuals and their environment can affect their movement, reproduction rate and well-being. In this talk, I will present a study of a class of advection-diffusion equations that model population movements generated by non-local species interactions. Using a combination of analytical and numerical tools, I will show that these models support a wide variety of spatio-temporal patterns that are able to reproduce segregation, aggregation and time-periodic behaviours commonly observed in real systems. I will also show the existence of parameter regions where multiple stable solutions coexist and hysteresis phenomena.

    Overall, I will describe various methods for analysing bifurcations and pattern formation properties of these models, which represent an essential mathematical tool for addressing fundamental questions about the endless aggregation phenomena observed in nature.

  • Analysis on 23 May 2024 at 16:00

    Speaker: Alessandro Carlotto (Trento)

    Title: Attaching faces of positive scalar curvature manifolds with corners

    Abstract: Motivated by a conjecture due to Gromov, we prove a novel desingularization theorem that allows to smoothly attach two given manifolds with corners by suitably gluing a pair of isometric faces, with control on both the scalar curvature of the resulting space and the mean curvature of its boundary.
    Various significant applications and related open problems will also be presented. This lecture is based on joint work with Chao Li (NYU).

  • Geometry and Topology on 23 May 2024 at 14:00

    Speaker: Henry Segerman (Oklahoma State University)

    Title: Avoiding inessential edges

    Abstract: Results of Matveev, Piergallini, and Amendola show that any
    two triangulations of a three-manifold with the same number of
    vertices are related to each other by a sequence of local
    combinatorial moves (namely, 2-3 and 3-2 moves). For some applications
    however, we need our triangulations to have certain properties, for
    example that all edges are essential. (An edge is inessential if both
    ends are incident to a single vertex, into which the edge can be
    homotoped.) We show that if the universal cover of the manifold has
    infinitely many boundary components, then the set of essential ideal
    triangulations is connected under 2-3, 3-2, 0-2, and 2-0 moves. Our
    results have applications in veering triangulations and in quantum
    invariants such as the 1-loop invariant. This is joint work with Tejas
    Kalelkar and Saul Schleimer.

  • Statistical Mechanics on 23 May 2024 at 14:00

    Speaker: Costanza Benassi (Northumbria University)

    Title: The Symmetric Matrix Ensemble: integrable lattices, orthogonal polynomials and hydrodynamic chains

    Abstract: Random matrix models arise in relation to a great variety of problems in mathematics and physics, and constitute a paradigm for the modelling of complex phenomena. We focus on the Symmetric Matrix Ensemble (SME) and its relationship with integrable lattices and with a class of skew-orthogonal polynomials. We exploit these properties of the system in order to characterise and investigate the thermodynamic limit of the relevant order parameters and show that in this limit they satisfy a novel integrable chain of PDEs as functions of the coupling constants. This provides results from C.B. M. Dell'Atti, A. Moro (L. Math. Phys., 2021), and C.B. M. Dell'Atti, A. Moro (in preparation).

  • Algebraic Topology on 21 May 2024 at 16:00

    Speaker: Cheuk Yu Mak (University of Southampton)

    Title: Loop group action on symplectic cohomology

    Abstract: For a compact Lie group G, its massless Coulomb branch algebra is the G-equivariant Borel-Moore homology of its based loop space. This algebra is the same as the algebra of regular functions on the BFM space. In this talk, we will explain how this algebra acts on the equivariant symplectic cohomology of Hamiltonian G-manifolds when the symplectic manifolds are open and convex. This is a generalization of the closed case where symplectic cohomology is replaced with quantum cohomology. Following Teleman, we also explain how it relates to the Coulomb branch algebra of cotangent-type representations. This is joint work with Eduardo González and Dan Pomerleano.

  • Junior Number Theory on 20 May 2024 at 11:00

    Speaker: Zachary Feng (University of Oxford)

    Title: Eigenvarieties and p-adic propagation of automorphy

    Abstract: Functoriality is a key feature in Langlands’ conjectured relationship between automorphic representations and Galois representations; it predicts that certain Galois representations are automorphic, i.e. should come from automorphic representations. We discuss the idea of p-adic propagation of automorphy, which seeks to establish the automorphy of everything in a “neighbourhood” given the automorphy of something in that neighbourhood. The “neighbourhoods” that we consider will be the irreducible components of a p-adic analytic space called the eigenvariety, which parameterizes p-adic automorphic representations. This technique was introduced by Newton and Thorne in their proof of symmetric power functoriality, and can be adapted to investigate similar problems.

  • Colloquium on 17 May 2024 at 16:00

    Speaker: Tim Burness (Bristol)

    Title: Simple points, fixed point ratios and applications

    Abstract: Let G be a group acting on a finite set X. The fixed point ratio (FPR) of an element g in G is simply the proportion of points in X fixed by g. Calculating, or bounding, FPRs has been a central problem in permutation group theory for many decades, finding numerous applications. In this talk, I will survey some of the main FPR results in the special case where G is a simple group, which is an area where there has been several major advances in recent years. I will also highlight the diverse range of applications, which includes powerful new results on bases for permutation groups, the connectivity of generating graphs and the commuting probability of finite groups.

  • Applied Mathematics on 17 May 2024 at 12:00

    Speaker: Yuyang Zhou (Edinburgh Napier)

    Title: Fully Probabilistic Control Algorithm Design for a class of Complex Stochastic Systems

    Abstract: Complex dynamical systems have garnered significant interest in the realms of control and engineering, as they offer a cohesive and natural framework for the mathematical modeling of diverse real-world systems, such as communication networks, power grids, and chemical processes. The inherent characteristics of these systems, such as high dimensionality, intricate structures, complex models with multiple modes of switching, and substantial uncertainties, pose notable challenges for system analysis, estimation, and particularly for control.

    This talk will introduce a novel decentralised probabilistic control framework designed for complex stochastic systems. It will offer a fresh research outlook dedicated to refining the control of such systems. In this talk, we will delineate the core principles and applications of the framework, showcasing its contribution to the development of innovative control strategies for complex stochastic systems.

  • Analysis on 16 May 2024 at 16:00

    Speaker: Linhan Li (Edinburgh)

    Title: Uniform rectifiability, smooth distance, and Green function

    Abstract: Uniformly rectifiable sets are often viewed as good sets for singular integrals, potential theory, and geometric measure theory, which are sets having big overlaps with Lipschitz images on every scale. In this talk, I will survey several equivalent characterizations of uniform rectifiability of an Ahlfors regular set, including a characterization in terms of oscillation of the Green function for a class of elliptic operators on the complement of the set. This Green function characterization, in contrast to some earlier results in analysis and PDE, still holds for sets of co-dimension higher than 1.
    This is based on joint projects with Joseph Feneuil and Svitlana Mayboroda (DOI: 10.1007/s00208-023-02715-6, DOI: 10.1016/j.aim.2023.109220).

  • Probability Theory on 15 May 2024 at 16:00

    Speaker: Brune Massoulié (Dauphine University)

    Title: Cutoff for the transience time of the Facilitated Exclusion Process

    Abstract: The facilitated exclusion process (FEP) is a particle system, where particles evolve on a discrete lattice, making random jumps while obeying local constraints. Because of these constraints, the process will almost surely become blocked (frozen), or reach an absorbing set of configurations after a finite time. This process can thus be seen as a toy model for the liquid-solid transition. In our work, we study the timescale after which these events occur, by introducing a new representation of this process. We therefore study a nonreversible variant of the simple symmetric exclusion process (SSEP), where particles can be destroyed by traps. This allows to make precise estimates on the transience time of both the FEP and the SSEP with traps, we also establish cutoff for the mixing time of the latter, and this opens the way for estimating precisely the mixing time of the FEP.

  • Combinatorics on 15 May 2024 at 15:00

    Speaker: Robert Hancock (University of Oxford)

    Title: Typical Ramsey properties of the primes and abelian groups

    Abstract: Given a matrix $A$ with integer entries, a subset $S$ of an abelian group and $r\in\mathbb N$, we say that $S$ is $(A,r)$-Rado if any $r$-colouring of $S$ yields a monochromatic solution to the system of equations $Ax=0$. A classical result of Rado characterises all those matrices $A$ such that $\mathbb N$ is $(A,r)$-Rado for all $r \in \mathbb N$. R\"odl and Ruci\'nski, and Friedgut, R\"odl and Schacht proved a random version of Rado’s theorem where one considers a random subset of $[n]:=\{1,\dots,n\}$.

    In this paper, we investigate the analogous random Ramsey problem in the more general setting of abelian groups. Given a sequence $(S_n)_{n\in\mathbb N}$ of finite subsets of abelian groups, let $S_{n,p}$ be a random subset of $S_n$ obtained by including each element of $S_n$ independently with probability $p$. We are interested in determining the probability threshold for $S_{n,p}$ being $(A,r)$-Rado.

    Our main result is a general black box for hypergraphs which we use to tackle problems of this type. Using this tool in conjunction with a series of supersaturation results, we determine the probability threshold for a number of different cases. A consequence of the Green--Tao theorem is the van der Waerden theorem for the primes: every finite colouring of the primes contains arbitrarily long monochromatic arithmetic progressions. Using our machinery, we obtain a random version of this result. We also prove a novel supersaturation result for $[n]^d$ and use it to prove an integer lattice generalisation of the random version of Rado's theorem.

    This is joint work with Andrea Freschi and Andrew Treglown (both University of Birmingham).

  • Ergodic Theory and Dynamical Systems on 15 May 2024 at 14:00

    Speaker: Ian Morris (QMUL)

    Title: Exceptional projections of self-affine sets

    Abstract: If a subset X of Rd is projected onto a linear subspace then the Hausdorff dimension of its image is trivially bounded above by the rank of the projection and by the dimension of the set X itself. When the Hausdorff dimension of the image is smaller than both of these values the projection is called an exceptional projection for the set X. By the classical theorem of Marstrand, the set of exceptional projections of a Borel set always has Lebegue measure zero when considered as a subset of the relevant Grassmannian. I will describe some results from an ongoing systematic study of the exceptional projections of self-affine sets, including an example of a strongly irreducible self-affine set whose set of exceptional projections includes a nontrivial subvariety of the Grassmannian. This is joint work with Çağrı Sert.

  • Postgraduate on 15 May 2024 at 12:00

    Speaker: Hamdi Dervodeli (University of Warwick)

    Title: Does tropical geometry know how to factor classical polynomials?

    Abstract: The factoring locus of a polynomial is a list of conditions on its coefficients under which the polynomial factors. The aim of this talk is to explore potential connections tropical geometry has with this factoring locus. More generally, we want to know if the reducibility of a variety is detected by the tropicalization of its defining ideal.

  • Algebraic Topology on 14 May 2024 at 17:00

    Speaker: Christian Dahlhausen (Heidelberg University)

    Title: Towards A^1-homotopy theory of rigid analytic spaces

    Abstract: In this talk, I will report about work in progress with Can Yaylali (Darmstadt/Orsay) towards A^1-homotopy theory of rigid analytic spaces. In the beginning, I will recall the motivating algebraic theory such as Morel-Voevodsky's seminal work on unstable A^1-homotopy of schemes, Voevodsky's stable version, and Ayoub's proof of a six-functor formalism for that. We seek to study a rigid analytic analogue using the rigid affine line A^1 as an interval. For this purpose, I will give some background on rigid analytic spaces where there are (at least) two canonical interval objects for doing homotopy theory, the closed unit disc B^1 and the rigid affine line A^1. The B^1-homotopy category has already been defined and studied by Ayoub and a full six-functor formalism was established by Ayoub-Gallauer-Vezzani. One drawback of the B^1-invariant theory is that analytic K-theory for rigid analytic spaces (as defined and studied by Kerz-Saito-Tamme) is not representable since it is not B^1-invariant. Thus we aim for an A^1-invariant version with coefficients in any presentable category. For the stable theory, we can prove the existence of a partial six-functor formalism for analytifications of schemes and algebraic morphisms between them by using the results of Ayoub's thesis. Furthermore, using coefficients in condensed categories, we render analytic K-theory representable in the unstable category and identify it with Z x BGL, in analogy to the case of schemes.

  • Junior Analysis and Probability Seminar on 13 May 2024 at 15:00

    Speaker: Hrit Roy (Edinburgh)

    Title: Bochner–Riesz means associated with rough convex domains in R2

    Abstract: We consider generalized Bochner–Riesz operators associated with planar convex domains. We construct domains which yield improved Lp bounds for the Bochner–Riesz over previous results of Seeger–Ziesler and Cladek. The constructions are based on the existence of large Bm sets due to Bose–Chowla, and large Λ(p) sets due to Bourgain.

  • Junior Number Theory on 13 May 2024 at 11:00

    Speaker: Bijay Bhatta (University of Manchester)

    Title: Height bounds on Lattices with skew-Hermitian forms over Type IV algebras

    Abstract: In this talk, we will discuss about the certain cases of Zilber-Pink conjecture on unlikely intersections in moduli space $\mathcal{A}_g$ of principally polarised abelian varieties of dimension g. In particular, we will talk about the case (mainly proving the effective height bound) when the associated endomorphism algebra is of Type IV (Albert types). This extends the work of Daw and Orr who proved Type I and II cases assuming Galois bounds.

  • Colloquium on 10 May 2024 at 16:00

    Speaker: June Barrow-Green (Open University)

    Title: Ronald Ross and Hilda Hudson: A collaboration on the mathematical theory of epidemics

    Abstract: In 1916 the physician Ronald Ross published the first of three papers on the mathematical study of epidemiology or, as he called it, ‘pathometry’. The second and third of these papers appeared the following year co-authored with the mathematician Hilda Hudson.   At the time Hudson, who had ranked equivalent to the 7th wrangler in the 1903 Cambridge Mathematical Tripos, was known for her work on Cremona Transformations. So how and why did Hudson, a geometer, end up collaborating with Ross on the theory of epidemics? And what role did she play? In my talk I shall discuss the nature and extent of their collaboration, as well as the genesis, content, and significance of their work.

  • Women in Mathematical Sciences Day on 10 May 2024 at 16:00

    Speaker: June Barrow-Green (Open University)

    Title: Ronald Ross and Hilda Hudson: A collaboration on the mathematical theory of epidemics

    Abstract: In 1916 the physician Ronald Ross published the first of three papers on the mathematical study of epidemiology or, as he called it, ‘pathometry’. The second and third of these papers appeared the following year co-authored with the mathematician Hilda Hudson.   At the time Hudson, who had ranked equivalent to the 7th wrangler in the 1903 Cambridge Mathematical Tripos, was known for her work on Cremona Transformations. So how and why did Hudson, a geometer, end up collaborating with Ross on the theory of epidemics? And what role did she play? In my talk, I shall discuss the nature and extent of their collaboration, as well as the genesis, content, and significance of their work.

  • Women in Mathematical Sciences Day on 10 May 2024 at 14:00

    Speaker: Tina Torkaman (University of Chicago)

    Title: TBA

    Abstract: TBA

  • Applied Mathematics on 10 May 2024 at 12:00

    Speaker: Katarzyna Macieszczak (Warwick)

    Title: Metastability in Open Quantum Systems

    Abstract: In this talk, I will be discussing the phenomenon of timescale separation, known as metastability, for the dynamics of Markovian open quantum systems. It arises, among others, in proximity to first-order dissipative phase transitions, and leads to the existence of a time regime in which system states appear to be stationary before their eventual relaxation towards a true stationary state. I will present necessary conditions for the metastability to occur and how, as a result, the long-lived states can be found in numerical simulations of master equations. I will also explain how those states can be characterised in terms of classical and quantum degrees of freedom, discussing their signatures in continuous measurement records and quantum trajectories, which are relevant for experimental realisations and stochastic simulations of those systems.

  • Women in Mathematical Sciences Day on 10 May 2024 at 12:00

    Speaker: Mina Dalirrooyfard (Morgan Stanley)

    Title: Fine-grained complexity: why we are stuck at certain running times

    Abstract: We measure the running time of problems as a function of the size of their input. On many problems such as finding the diameter of the graphs, we can achieve polynomial running time, and on many problems, such as various graph partitioning problems, the best algorithms run in exponential time. Through traditional complexity theory, for many problems, it is known whether the problem is polynomially solvable or it is NP-complete, hence unlikely to have a polynomial time algorithm. However, for problems with known polynomial-time algorithms, in many applications, it is important to know whether there might exist faster algorithms for the problem. For example, finding the diameter of the graph can be done in quadratic time via a simple algorithm. However, can one find a linear time algorithm for it? If not, why?
    These types of questions are answered by “fine-grained complexity theory”. I will give an introduction to what fine-grained complexity is, what it has achieved and what techniques are mostly used. I will also introduce average-case fine-grained complexity, which aims to find the hardness of problems where the input comes from a random distribution, which is the more realistic use case of many problems. Finally, I will talk a bit about my research after grad school.

  • Analysis on 09 May 2024 at 16:00

    Speaker: Jakub Takac (Warwick)

    Title: A proof of the flat chain conjecture of Ambrosio and Kirchheim

    Abstract: The theory of currents was developed by Federer and Fleming in 1960 in order to show the existence for the Plateau problem (finding a minimal surface with prescribed boundary) for arbitrary dimension and co-dimension in Euclidean space. Pioneering work of Ambrosio and Kirchheim from 2000 developed a theory of metric currentsthat holds in any complete metric space. This theory includes proofs of Federer and Fleming's closure and boundary rectifiability theorems for integral currents that hold in any complete metric space, from which they solved the Plateau problem in compact metric spaces and various Banach spaces. However, a natural question remained open: how do the Federer-Fleming and Ambrosio-Kirchheim theories relate in Euclidean space? Ambrosio and Kirchheim naturally conjectured that metric currents in Euclidean space correspond precisely to Federer-Fleming flat chains. This is known as the flat chain conjecture. In this talk, I will introduce both notions of the currents and present the proof of the conjecture, in the level of detail that time permits.

  • Geometry and Topology on 09 May 2024 at 14:00

    Speaker: Rohini Ramadas (University of Warwick)

    Title: Thurston theory in complex dynamics: a tropical perspective

    Abstract: A rational function in one complex variable defines a branched covering from Riemann sphere CP^1 to itself. In the 1980s, William Thurston proved a theorem addressing the question: which branched coverings of the topological sphere S^2 are (suitably equivalent to) rational functions on CP^1? Thurston’s theorem is still central in one-variable complex and arithmetic dynamics.

    Tropical geometry is a field in which polyhedral geometry and combinatorics are used to describe degenerations in algebraic geometry. There are connections with geometric group theory; for example, Culler-Vogtmann Outer Space is closely related to the space of tropical curves.

    I will introduce Thurston’s theorem and describe a connection with tropical geometry.

  • Probability Theory on 08 May 2024 at 16:00

    Speaker: Francesca Cottini (University of Luxembourg)

    Title: (Ir)reducible central limit theorems for polynomial chaos via spectral graph theory

    Abstract: In this talk, we will first present a central limit theorem (CLT), based only on second moment assumptions, for polynomial chaos, i.e. multilinear polynomials of independent random variables. This reducible CLT (as it can be reduced to the application of a classical result by Feller-Lindeberg) finds its main application in the theory of 2d directed polymers and the related 2d Stochastic Heat Equation.
    Among the class of polynomial chaos satisfying a universal CLT (according to Nourdin, Peccati, Reinart and their celebrated Fourth Moment Theorem), we will exhibit a sufficient condition for a polynomial chaos to be irreducible, i.e. it does not satisfy our reducible CLT via second moment only. The novelty of our approach and findings lies in the interpretation we offer solely based on the spectral and connectivity properties of relevant associated graphs and hypergraphs.
    The talk is based on an ongoing project with Francesco Caravenna (Milano-Bicocca) and Giovanni Peccati (Luxembourg).

  • Postgraduate on 08 May 2024 at 12:00

    Speaker: Lucas Araujo Bonomo (University of Warwick)

    Title: Acoustic Liners for Turbofan Aircraft Engines: Duct Acoustics, Experimental Methods and What on earth a Mechanical Engineer is doing in WMI

    Abstract: Noise stands as a key factor in aircraft regulation. In the case of modern turbofan aircraft, the fan emerges as a primary source of noise. Due to its distinctive acoustic signature, characterised by a profusion of tones spanning a broadband spectrum, fan noise incurs significant penalties in aircraft noise metrics, given its pronounced annoyance to humans. Acoustic liners are passive devices installed on the interior walls of turbofan nacelles to mitigate fan noise. This presentation aims to introduce the mathematical problem inherent in the engineering challenge of modelling and optimizing such devices. The talk covers the fundamentals of Duct Acoustics and the current methodologies for modelling these devices into numerical models of turbofan engines. Additionally, I talk about the experimental techniques utilised to characterise liners under realistic conditions. Finally, the current debate on the academic liner community is introduced. By the end of this talk, I hope you will understand what an experimentalist mechanical engineer is doing in a maths department.

  • Ergodic Theory and Dynamical Systems on 07 May 2024 at 14:00

    Speaker: Sven Sandfeldt (KTH)

    Title: Centralizer rigidity for partially hyperbolic diffeomorphisms on nilmanifolds

    Abstract: The smooth centralizer of a diffeomorphism f is the set of smooth maps g that commute with f. When f is an Anosov diffeomorphism on a nilmanifold, then a result by Rodriguez Hertz and Wang shows that: If the centralizer of f is higher rank, then f is smoothly conjugated to an affine map. In particular, for Anosov diffeomorphisms f, the only mechanism giving f a large centralizer is a smooth conjugacy to an algebraic model. In this talk, I will speak about rigidity of partially hyperbolic diffeomorphisms on nilmanifolds with large centralizer, extending the case for Anosov diffeomorphisms.

  • Junior Number Theory on 06 May 2024 at 11:00

    Speaker: Beatriz Barbero Lucas (University College Dublin)

    Title: Obtaining new quantum codes from Generalized Monomial-Cartesian Codes

    Abstract: Quantum computers are a great tool to attack some intractable problems for classical computers, such as the prime factorization problem and the discrete logarithm problem. However, quantum computer implementations have higher error rates than classical computers, making reliability a challenge. That is where Quantum Error correction codes come into play.
    In the first part of this talk I will give an introduction to error correcting codes, in particular to evaluation codes, in order to later understand the good properties of the new quantum codes that we obtained from the Generalized Momonial-Cartesian codes that we have proposed.
    This talk is based on the paper with F. Hernando, H. Martín-Cruz and G. McGuire.

  • Colloquium on 03 May 2024 at 16:00

    Speaker: Patrick Farrell (Oxford)

    Title: Computing multiple solutions of nonlinear problems

    Abstract: Many nonlinear problems exhibit multiple solutions—think of an umbrella inverted by the wind, a light switch, or a child's seesaw. Computing the distinct solutions of a nonlinear problem as its parameters are varied is a central task in applied mathematics and engineering. The solutions are captured in a bifurcation diagram, plotting (some functional of) the solutions as a function of the parameters. In this talk I will present a new algorithm, deflated continuation, for this task.

    Deflated continuation has three advantages. First, it is capable of computing disconnected bifurcation diagrams; previous algorithms only aimed to compute that part of the bifurcation diagram continuously connected to the initial data. Second, its implementation is very simple: it only requires a minor modification to an existing Newton-based solver. Third, it can scale to very large discretisations if a good preconditioner is available; no auxiliary problems must be solved.

    We will present applications to hyperelastic structures, liquid crystals, and Bose-Einstein condensates, among others.

  • Combinatorics on 03 May 2024 at 14:00

    Speaker: Martin Winter (University of Warwick)

    Title: Kalai's 3^d conjecture for coordinate-symmetric polytopes

    Abstract: Polyhedral combinatorics is an active area of research with many intriguing open questions. In particular our understanding of the combinatorics and geometry of centrally symmetric polytopes is still severely limited. This is best illustrated by two elementary yet widely open conjectures due to Kalai and Mahler: Kalai's 3^d conjecture asserts that the d-cube has the minimal number of faces among all centrally symmetric d-polytopes (that is, 3^d many); Mahler's conjecture asserts that the d-cube has the minimal Mahler volume (= volume times volume of dual) among all centrally symmetric d-polytopes. Both conjectures claim the same family of minimizers (the Hanner polytopes) and generally show many parallels, though their connection is not well understood.

    A polytope is coordinate-symmetric (or unconditional) if it is invariant under reflection on coordinate hyperplanes.

    While Mahler's conjecture was spectacularly solved for this class of polytopes already in 1987, the same special case was still open for Kalai's conjecture. In joint work with Raman Sanyal, we found two short and elegant proofs for Kalai's conjecture for coordinate-symmetric (actually, locally coordinate-symmetric) polytopes, that I will present in this talk.

  • Applied Mathematics on 03 May 2024 at 12:00

    Speaker: Joe Webber & Julio Hurtado (University of Warwick)

    Title: A linear-elastic-nonlinear-swelling model for hydrogels & Continual Learning in Deep Learning and how this impact my research in CAMaCS

    Abstract: (a) Joseph Webber - A linear-elastic-nonlinear-swelling model for hydrogels
    Hydrogels are an important example of poroelastic materials, formed from hydrophilic polymer chains surrounded by adsorbed water molecules. Key to many of their uses is the fact that they can swell to hundreds of times their initial size upon imbibition of water, but at any swelling state they behave as solid, soft, elastic media. Modelling such gels can be challenging: large-strain elasticity arising from swelling and drying must be incorporated alongside the interaction between water and polymer molecules and the transport of water through the pore spaces. This has, in the past, led to many complicated nonlinear approaches based on deriving a free energy density from a bottom-up understanding of the molecular-scale dynamics. In this talk, I will summarise a new model for the dynamics of hydrogels that treats them as instantaneously incompressible linear-elastic materials, whilst allowing for nonlinearities in the isotropic strains corresponding to swelling. This approach is not only analytically tractable, but also able to describe the gel using only three (macroscopic) swelling-state-dependent material parameters. I will discuss the new predictions that such a model can make and show how the aforementioned material parameters for any gel could be easily determined experimentally.

    (b) Julio Hurtado - Continual Learning in Deep Learning and how this impact my research in CAMaCS
    Deep Learning models rely on a fixed and limited dataset to learn. This strategy works fine as long as the test data remains similar to the train distribution. However, when the test set changes, we must retrain the model by adjusting its weight to learn the latest distribution. One naive solution is to train the model with the new distribution, which modifies the weight of the model but overwrites previous knowledge and forgets previous distributions. Continual Learning is a technique that aims to train a model continuously over a stream of changing datasets, aiding with the mitigation of forgetting previous tasks. In this talk, I will discuss how we can indirectly tackle the forgetting of previous data by addressing the issue of generalisation, and how this is guiding my research in CAMaCS.

  • Algebraic Geometry on 01 May 2024 at 16:00

    Speaker: Lucie Devey (Edinburgh)

    Title: Toric vector bundles, Parliament of polytopes and Stability

    Abstract: Given any toric vector bundle, we may construct its parliament of polytopes. This is a generalization of the moment polytope of a toric line bundle. It contains a huge amount of information about the original bundle: notably on its global sections and its positivity. In this talk, we explain how to determine algorithmically if a fixed toric vector bundle is semi-stable or not (with respect to any polarisation), we illustrate it on its parliament of polytopes. This is a first step in getting a classification of toric vector bundles.

  • Probability Theory on 01 May 2024 at 16:00

    Speaker: William FitzGerald (University of Manchester)

    Title: Ordered random walks and the Airy line ensemble

    Abstract: Consider a collection of one-dimensional random walks which are conditioned to remain in the same order for all time. This system has been named ordered random walks, non-intersecting/non-colliding random walks or vicious walkers.
    Ordered random walks have many connections including with last passage percolation, interacting particle systems and random matrices. I will describe some of these connections and the behaviour as the number of walkers grows to infinity. This is based on joint work with Denis Denisov and Vitali Wachtel.

  • Postgraduate on 01 May 2024 at 12:00

    Speaker: William O'Regan (University of Warwick)

    Title: On the Erdős distance problem and fractal variants.

    Abstract: The Erdős distance problem asks us to find the number of distinct distances that can be found between points in the plane. This is more or less resolved by Guth and Katz: n points generate about n distinct distances. In this talk I will, time allowing, introduce this problem, give some simple arguments that give some reasonable upper and lower bounds, discuss ideas of the proof of Guth and Katz, and give fractal variants of the problem. An A-level in mathematics, or equivalent, will be required to follow this talk.

  • Algebraic Topology on 30 April 2024 at 16:00

    Speaker: Stephen Theriault (University of Southampton)

    Title: Homotopy theoretic properties of Poincaré Duality complexes

    Abstract: Let M be a simply-connected closed Poincaré Duality complex of dimension n. The goal is to gain insight into the homotopy theory of M by determining properties of its based loop space. An approach will be outlined that has had some success for particular families of Poincaré Duality complexes.
    We will go on to discuss a new development. Rationally, it is known that if X is the (n-1)-skeleton of M then the based loops on M retracts off the based loops on X, provided the rational cohomology ring of M is not generated by a single element. We will show that under certain conditions this is true integrally. Families for which the integral statement holds include moment-angle manifolds and quasi-toric manifolds.

  • Ergodic Theory and Dynamical Systems on 30 April 2024 at 14:00

    Speaker: Tom Kempton (University of Manchester)

    Title: The dynamics of the Fibonacci Partition Function

    Abstract: The Fibonacci partition function R(n) counts the number of ways of representing a natural number n as the sum of distinct Fibonacci numbers. For example, R(6)=2 since 6=5+1 and 6=3+2+1. An explicit formula for R(n) was recently given by Chow and Slattery. In this talk we express R(n) interms of ergodic sums over an irrational rotation, which allows us to prove lots of statements about the local structure of R(n).

  • Junior Number Theory on 29 April 2024 at 11:00

    Speaker: Benjamin Bedert (University of Oxford)

    Title: On Unique Sums in Abelian Groups

    Abstract: In this talk, we will study the problem in additive combinatorics of determining for a finite Abelian group $G$ the size of its smallest subset $A\subset G$ that has no unique sum, meaning that for every two $a_1,a_2\in A$ we can write $a_1+a_2=a’_1+a’_2$ for different $a’_1,a’_2\in A$. We begin by using classical rectification methods to obtain the previous best lower bounds of the form $|A|\gg \log p(G)$, which stood for 50 years. Our main aim is to outline the proof of a recent improvement and discuss some of its key notions such as additive span, additive dimension and the density increment method. This talk is based on Bedert, B. On Unique Sums in Abelian Groups. Combinatorica (2023).

  • Colloquium on 26 April 2024 at 16:00

    Speaker: J Nathan Kutz (University of Washington)

    Title: Data-driven model discovery and physics-informed learning

    Abstract: A major challenge in the study of dynamical systems is that of model discovery: turning data into reduced order models that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data. We introduce a number of data-driven strategies for discovering nonlinear multiscale dynamical systems and their embeddings from data. We consider two canonical cases: (i) systems for which we have full measurements of the governing variables, and (ii) systems for which we have incomplete measurements. For systems with full state measurements, we show that the recent sparse identification of nonlinear dynamical systems (SINDy) method can discover governing equations with relatively little data and introduce a sampling method that allows SINDy to scale efficiently to problems with multiple time scales, noise and parametric dependencies. For systems with incomplete observations, we show that the Hankel alternative view of Koopman (HAVOK) method, based on time-delay embedding coordinates and the dynamic mode decomposition, can be used to obtain a linear models and Koopman invariant measurement systems that nearly perfectly captures the dynamics of nonlinear quasiperiodic systems. Neural networks are used in targeted ways to aid in the model reduction process. Together, these approaches provide a suite of mathematical strategies for reducing the data required to discover and model nonlinear multiscale systems.

  • Combinatorics on 26 April 2024 at 14:00

    Speaker: Asier Calbet-Ripodas (Queen Mary University of London)

    Title: The asymptotic behaviour of $sat(n,\mathcal{F})$

    Abstract: Given a family $\mathcal{F}$ of graphs, we say that a graph $G$ is \emph{$\mathcal{F}$-saturated} if it is maximally $\mathcal{F}$-free, meaning $G$ does not contain a graph in~$\mathcal{F}$ but adding any new edge to $G$ creates a graph in~$\mathcal{F}$. We then define $sat(n,\mathcal{F})$ to be the minimum number of edges in an $\mathcal{F}$-saturated graph on $n$ vertices. In 1986, K\'aszonyi and Tuza showed that $sat(n,\mathcal{F})=O(n)$ for all families $\mathcal{F}$ and Tuza conjectured that for singleton families $sat(n,\mathcal{F})/n$ converges. Tuza's Conjecture remains wide open. In this talk, I will discuss recent results about the asymptotic behaviour of $sat(n,\mathcal{F})$, mostly in the sparse regime $sat(n,\mathcal{F}) \leq n+o(n)$, in each of the cases when $\mathcal{F}$ is a singleton, when $\mathcal{F}$ is finite and when $\mathcal{F}$ is possibly infinite. Joint work with Andrea Freschi.

  • Analysis on 25 April 2024 at 16:00

    Speaker: Mattia Magnabosco (Oxford)

    Title: Failure of the curvature-dimension condition in sub-Finsler manifolds

    Abstract: The Lott–Sturm–Villani curvature-dimension condition CD(K,N) provides a synthetic notion for a metric measure space to have curvature bounded from below by K and dimension bounded from above by N. It has been recently proved that this condition does not hold in any sub-Riemannian manifold equipped with a positive smooth measure, for every choice of the parameters K and N. In this talk, we investigate the validity of the analogous result for sub-Finsler manifolds, providing two results in this direction. On the one hand, we show that the CD condition fails in sub-Finsler manifolds equipped with a smooth strongly convex norm and with a positive smooth measure. On the other hand, we prove that, on the sub-Finsler Heisenberg group, the same result holds for every reference norm. Additionally, we show that the validity of the measure contraction property MCP(K,N) on the sub-Finsler Heisenberg group depends on the regularity of the reference norm.

  • Geometry and Topology on 25 April 2024 at 14:00

    Speaker: Nikolai Prochorov (Marseille)

    Title: Thurston theory for critically fixed branched covering maps

    Abstract: In the 1980’s, William Thurston obtained his celebrated characterization of rational mappings. This result laid the foundation of such a field as Thurston's theory of holomorphic maps, which has been actively developing in the last few decades. One of the most important problems in this area is the questions about characterization, which is understanding when a topological map is equivalent (in a certain dynamical sense) to a holomorphic one, and classification, which is an enumeration of all possible topological models of holomorphic maps from a given class.

    In my talk, I am going to focus on the characterization and classification problems for the family of postcritically finite branched coverings, i.e., branched coverings of the 2-dimensional sphere S^2 with all critical points being fixed. Maps of this family can be defined by combinatorial models based on planar embedded graphs, and it provides an elegant answer to the classification problem for this family. Further, I plan to explain how to understand whether a given critically fixed branched cover is equivalent to a critically fixed rational map of the Riemann sphere and provide an algorithm of combinatorial nature that allows us to answer this question. Finally, if time permits, I will briefly mention the connections between Thurston's theory, Teichmüller spaces and Mapping Class Groups of marked spheres.
    This is a joint work with Mikhail Hlushchanka.

  • Combinatorics on 25 April 2024 at 14:00

    Speaker: Jun Gao (Institute for Basic Science (IBS))

    Title: Generalized Ramsey--Tur\'an density for cliques

    Abstract: We study the generalized Ramsey--Tur\'an function $\mathrm{RT}(n,K_s,K_t,o(n))$, which is the maximum possible number of copies of $K_s$ in an $n$-vertex $K_t$-free graph with independence number $o(n)$. The case when $s=2$ was settled by Erd{\H{o}}s, S{\'o}s, Bollob{\'a}s, Hajnal, and Szemer\'{e}di in the 1980s. We combinatorially resolve the general case for all $s\ge 3$, showing that the (asymptotic) extremal graphs for this problem have simple (bounded) structures. In particular, it implies that the extremal structures follow a periodic pattern when $t$ is much larger than $s$. Our results disprove a conjecture of Balogh, Liu, and Sharifzadeh and show that a relaxed version does hold. This is a joint work with Suyun Jiang, Hong Liu and Maya Sankar.

  • Probability Theory on 24 April 2024 at 16:00

    Speaker: Sarah-Jean Meyer (University of Oxford)

    Title: The FBSDE approach to sine-Gordon up to $6\pi$.

    Abstract: Title:

    Abstract: I will present a stochastic analysis of the sine-Gordon Euclidean quantum field $(\cos(\beta\phi))_2$ on the full space up to the second threshold, i.e. for $\beta^2<6\pi$. The basis of our method is a stochastic quantisation equation given by a forward-backward stochastic differential equation (FBSDE) for a decomposition $(X_t)_(t\geq0)$ of the interacting Euclidean field $X_\infty$ along a scale parameter $t\geq0$ using an approximate version of the renormalisation flow equation. The FBSDE produces a scale-by-scale coupling of the interacting field with the Gaussian free field without cut-offs and describes the optimiser of a stochastic control problem for Euclidean QFTs introduced by Barashkov and Gubinelli. I will first explain the general set-up for the FBSDE approach. In the case of the sine-Gordon model, I will mention some applications of the FBSDE to illustrate that it can be used effectively to obtain results about large deviations, integrability, decay of correlations for local observables, singularity with respect to the free field, Osterwalder-Schrader axioms and other properties. This is joint work with Massimiliano Gubinelli.

  • Algebraic Geometry on 24 April 2024 at 15:00

    Speaker: Barbara Fantechi (SISSA)

    Title: Deformations of semi-smooth varieties and the boundary of the moduli space of Godeaux surfaces

    Abstract: A variety X is semismooth if étale locally it is isomorphic to a product of a pinch point (x^2y-z^2) with some affine space; equivalently, its normalization is smooth and X is obtained by gluing a smooth divisor to itself via an involution with fixed points in codimension 1. In joint work with Marco Franciosi and Rita Pardini, we calculate the sheaves T^1_X and T_X in terms of the normalization and the gluing, and use this to show that all semi-smooth non normal stable Godeaux surfaces are smoothable, and nonsingular points of the moduli space.

  • Algebraic Topology on 23 April 2024 at 17:00

    Speaker: Thomas Wasserman (University of Oxford)

    Title: Cohen-Macaulay simplicial complexes and duality groups

    Abstract: Duality groups are groups that admit a Poincaré-duality-like relationship between their cohomology and their homology twisted by a module known as the dualising module. Many interesting groups are (virtually) duality groups, like free groups and their (outer) automorphisms, mapping class groups, and the general linear groups over the integers. Knowing duality is useful for homology computations, and the dualising module often has a nice interpretation. In this talk I will discuss work with Ric Wade where we explore the relationship between groups acting on simplicial complexes that are locally Cohen-Macaulay (meaning that their local homology is concentrated in a single degree and free) and groups having duality, and the implications this has for the dualising module.

  • Junior Number Theory on 22 April 2024 at 11:00

    Speaker: Lillybelle Cowland Kellock (University College London)

    Title: A generalisation of Tate’s algorithm for hyperelliptic curves

    Abstract: Tate's algorithm tells us that, for an elliptic curve $E$ over a discretely valued field $K$ with residue characteristic $\geq 5$, the dual graph of the special fibre of the minimal regular model of $E$ over $K^{\textup{unr}}$ can be read off from the valuation of $j(E)$ and $\Delta_E$. This is really important for calculating Tamagawa numbers of elliptic curves, which are involved in the refined Birch and Swinnerton-Dyer conjecture formula. For a hyperelliptic curve $C/K$, we can ask if we can give a similar algorithm that gives important data related to the curve and its Jacobian from polynomials in the coefficients of a Weierstrass equation for $C/K$. This talk will be split between being an introduction to cluster pictures of hyperelliptic curves, from which the important data can be gathered, and a presentation of how the cluster picture can be recovered from polynomials in the coefficients of a Weierstrass equation.

  • Combinatorics on 15 March 2024 at 14:00

    Speaker: Natalie Behague (University of Warwick)

    Title: The rainbow saturation number

    Abstract: The saturation number of a graph is a famous and well-studied counterpoint to the Turán number, and the rainbow saturation number is a generalisation of the saturation number to the setting of coloured graphs. Specifically, for a given graph $F$, an edge-coloured graph is $F$-rainbow saturated if it does not contain a rainbow copy of $F$, but the addition of any non-edge in any colour creates a rainbow copy of $F$. The rainbow saturation number of $F$ is the minimum number of edges in an $F$-rainbow saturated graph on $n$ vertices. Girão, Lewis, and Popielarz conjectured that, like the saturation number, for all $F$ the rainbow saturation number is linear in $n$. I will present our attractive and elementary proof of this conjecture, and finish with a discussion of related results and open questions.

    This is joint work with Tom Johnston, Shoham Letzter, Natasha Morrison and Shannon Ogden.

  • Mathematics Teaching and Learning on 14 March 2024 at 16:00

    Speaker: Lara Alcock (Loughborough)

    Title: Conditional Inference in Mathematics Students

    Abstract: Conditional inference – inference from statements of the form ‘if A then B’ – has long been studied in cognitive psychology. It has been studied only a little in mathematics education, despite its relevance for understanding and constructing mathematical arguments. In this talk, I will describe three studies of conditional inference in mathematics students. Study 1 used comparative judgement to score mathematical conditionals for believability, then investigated whether undergraduates accept more inferences from more believable conditionals. Study 2 investigated patterns of inference across mathematical, abstract and everyday content. For both of these studies, cluster analyses revealed educationally relevant individual differences. Study 3 checked whether believability effects are better understood in terms of easiness. I will discuss implications for theoretical understanding of mathematical expertise, and for teaching and learning.

  • Analysis on 14 March 2024 at 16:00

    Speaker: Alexander Lytchak (KIT)

    Title: Convex subsets in generic Riemannian manifolds

    Abstract: In the talk I would like to discuss some statements and questions about convex subsets and convex hulls in generic Riemannian manifolds of dimension at least 3. The statements, obtained jointly with Anton Petrunin, are elementary but somewhat surprising for the Euclidean intuition. For instance, the convex hull of any finite non-collinear set turns out to be either the whole manifold or non-closed.

  • Geometry and Topology on 14 March 2024 at 14:00

    Speaker: Davide Spriano (University of Oxford)

    Title: Uniquely geodesic groups.

    Abstract: A group is uniquely geodesic (aka geodetic) if it admits a locally finite Cayley graphs where any two vertices can be connected by a unique shortest path. Despite this being a very natural geometric property, an algebraic characterization of uniquely geodetic groups has been elusive for quite some time, even for simple questions such as “are uniquely geodesic groups finitely presented”? With Elder, Gardam, Piggot and Townsend we provide the first algebraic classification of uniquely geodesic groups.

  • Algebraic Geometry on 13 March 2024 at 15:00

    Speaker: Thomas Gauthier (Université Paris-Saclay)

    Title: Sparsity of Postcritically finite maps in higher dimension

    Abstract: In this talk, we focus on connections between arithmetic and holomorphic dynamics. The first goal of the talks is to present several problems in arithmetic dynamics of endomorphisms of projective spaces, all inspired from classical problems in arithmetic geometry. The second goal is to explain how these problems are related to the notions of bifurcation currents and measures in complex dynamics. I will start with several motivations for the problem we study. If time allows, I will sketch a proof strategy to solve two problems at the same time. This is a joint work with Johan Taflin and Gabriel Vigny.

  • Combinatorics on 08 March 2024 at 14:00

    Speaker: Alexandra Wesolek (Technische Universität Berlin)

    Title: Reconfiguration of plane trees in convex geometric graphs

    Abstract: A reconfiguration graph consists of solutions to a problem, with an edge between two solutions if one solution can be obtained from the other by what is called a flip. This talk focuses on reconfiguration graphs where the vertices i.e. solutions are non-crossing spanning trees of a set of n points in the plane (edges of the tree are drawn as straight lines and none of the edges cross each other). A flip consists of exchanging an edge in a non-crossing spanning tree with another one so that the resulting graph is also a non-crossing spanning tree. Avis and Fukuda proved in 1996 that there always exists a flip sequence of length at most 2n-4 between any pair of non-crossing spanning trees. In a joint work with Nicolas Bousquet, Lucas De Meyer and Théo Pierron, we show that for a convex set of n points there always exists a flip sequence of length at most c n where c is approximately 1.95. This improves the upper bound for convex point sets by a linear factor for the first time in about 30 years.

  • Analysis on 07 March 2024 at 16:00

    Speaker: Emanuele Caputo (Warwick)

    Title: Geometric characterizations of metric measure spaces satisfying the Poincaré inequality

    Abstract: In the setting of doubling metric measure spaces, we characterize in geometric terms those that satisfy a Poincaré inequality. The goal of the presentation is to give an overview of previous results, as the characterization of such spaces in terms of the existence of a thick family of curves connecting points. Then, we present another characterization obtained in collaboration with N. Cavallucci (EPFL). We define the concept of separating sets and we prove that the 1-Poincaré inequality is equivalent to the fact that separating sets have a big energy in a suitable sense.

  • Geometry and Topology on 07 March 2024 at 14:00

    Speaker: Marco Linton (University of Oxford)

    Title: The coherence of one-relator groups.

    Abstract: (Joint work with Andrei Jaikin-Zapirain.) A group is said to be coherent if all of its finitely generated subgroups are finitely presented. In this talk I will sketch a proof of Baumslag’s conjecture that all one-relator groups are coherent, discussing connections with the non-positive immersions property and the vanishing of the second L^2 Betti number.

  • Algebraic Geometry on 06 March 2024 at 15:00

    Speaker: Xenia de la Ossa (Oxford)

    Title: On the arithmetic of families Calabi-Yau manifolds

    Abstract: In this seminar I will discuss what I know (and don’t know) about the arithmetic of Calabi-Yau 3-folds. The main goal is to explore whether there are questions of common interest in this context to physicists, number theorists and geometers. The main quantities of interest in the arithmetic context are the numbers of points of the manifold considered as a variety over a finite field. We are interested in the computation of these numbers and their dependence on the moduli of the variety. The surprise for a physicist is that the numbers of points over a finite field are also given by expression that involve the periods of a manifold. The number of points are encoded in the local zeta function, about which much is known in virtue of the Weil conjectures. I will discuss interesting topics related to the zeta function and the appearance of modularity for one parameter families of Calabi-Yau manifolds. I will report on an example for which the quartic numerator of the zeta function factorises into two quadrics at special values of the parameter which satisfy an algebraic equation with coefficients in Q (so independent of any particular prime), and for which the underlying manifold is smooth. We note that these factorisations are due to a splitting of the Hodge structure and that these special values of the parameter are rank two attractor points in the sense of black hole solutions of type IIB supergravity. Modular groups and modular forms arise in relation to these attractor points. To our knowledge, the rank two attractor points that were found by the application of these number theoretic techniques, provided the first explicit examples of such points for Calabi-Yau manifolds of full SU(3) holonomy. The work presented is based on joint research with Philip Candelas, Mohamed Elmi and Duco van Straten and, time permitting, further work with Philip Candelas, Pyry Kuusela and Joseph McGovern. I will not be assuming familiarity with type II string theory.

  • Combinatorics on 01 March 2024 at 14:00

    Speaker: Joanna Lada (LSE)

    Title: Hamilton decompositions of regular tripartite tournaments

    Abstract: Given a Hamiltonian graph $G$, it is natural to ask whether its edge set admits a full decomposition into Hamilton cycles. In the setting of regular oriented graphs, this has been confirmed for sufficiently large regular tournaments (K\"{u}hn, Osthus (2013)), bipartite tournaments (Granet (2022)), and $k$-partite tournaments when $k\geq 4$ (K\"{u}hn, Osthus (2013)).

    For regular tripartite tournaments, a complete decomposition is not always possible; the known counterexample, $\mathcal{T}_\Delta$, consists of the blowup of $C_3$ with a single triangle reversed. However, it is reasonable to think that all such non-decomposable regular tripartite tournaments fall within a specifiable class, and perhaps are even all isomorphic to $\mathcal{T}_\Delta$.

    In this talk, we make progress towards this question by proving the approximate result; that is, given a regular tripartite tournament, one can find a decomposition of its edge set into edge-disjoint Hamilton cycles covering all but $o(n^2)$ edges.

    Joint work with Francesco Di Braccio, Viresh Patel, Yanitsa Pehova, and Jozef Skokan.

  • Geometry and Topology on 29 February 2024 at 14:00

    Speaker: Joe MacManus (University of Oxford)

    Title: Groups quasi-isometric to planar graphs

    Abstract: A classic and important theorem originating in work of Mess states that a f.g. group is quasi-isometric to a complete Riemannian plane if and only if it is a virtual surface group. Another related result obtained by Maillot states that a f.g. group is virtually free if and only if it is quasi-isometric to a complete planar simply connected Riemannian surface with non-compact geodesic boundary. These results illustrate the general philosophy that planarity is a very `rigid' property amongst f.g. groups.

    In this talk I will build on the above and sketch how to characterise those f.g. groups which are quasi-isometric to planar graphs. Such groups are virtually free products of free and surface groups, and thus virtually admit a planar Cayley graph. The main technical step is proving that such a group is accessible, in the sense of Dunwoody and Wall. This is achieved through a careful study of the dynamics of quasi-actions on planar graphs.

  • Junior Number Theory on 26 February 2024 at 11:00

    Speaker: Seth Hardy (University of Warwick)

    Title: Exponential sums with random multiplicative coefficients

    Abstract: The study of exponential sums with multiplicative coefficients is classical in analytic number theory. For example, understanding exponential sums with coefficients given by the Liouville function would offer profound insights into the distribution of primes in arithmetic progressions. Unfortunately, our current understanding of these sums is far from what we expect to be the truth. In this talk, we will explore an alternative approach: considering exponential sums with random multiplicative coefficients. We will introduce the relevant theory and discuss recent progress in proving conjecturally sharp lower bounds for the size of a large proportion of these exponential sums.

  • Colloquium on 23 February 2024 at 16:00

    Speaker: Kenneth Falconer (St Andrews)

    Title: Fractals and intermediate dimensions

    Abstract: We will give a short overview of aspects of fractals such as iterated function systems and self-similar and self-affine sets and introduce Hausdorff and box-counting dimensions. In particular we will look at sets where these dimensions differ. We will then show how Hausdorff and box-counting dimensions can be regarded as particular cases of a spectrum of `intermediate’ dimensions’ and discuss properties and examples of intermediate dimensions.

  • Combinatorics on 23 February 2024 at 14:00

    Speaker: William Turner (University of Birmingham)

    Title: Simultaneously embedding sequences of graphs

    Abstract: Graph embeddability is a well-studied problem in topological combinatorics. Perhaps the most famous result in this field is Kuratowski’s Theorem (1930), which characterises when a graph is planar in terms of forbidden substructures. In recent years, progress has been made characterising different types of embeddability for combinatorial objects of higher dimension. In this talk, we will present a natural way to embed “temporal sequences”, sequences (G1,...,Gn) of graphs where each consecutive pair is related via the minor relation. For a restricted case, we will build up a characterisation in terms of “forbidden sequential minors”. To do so, we will need to develop a theory of abstract duals for temporal sequences, analogous to Whiteny’s Criterion (1932). Finally, we will discuss some wider complexity results and open problems.

  • Analysis on 22 February 2024 at 16:00

    Speaker: Mikhail Karpukhin (UCL)

    Title: New embedded minimal surfaces in 3-sphere and 3-ball via eigenvalue optimisation

    Abstract: The study of optimal upper bounds for Laplace eigenvalues on closed surfaces under area constraint is a classical problem of spectral geometry. It is particularly interesting due to the fact that optimal metrics (if they exist) correspond to branched minimal surfaces in n-dimensional sphere. In general, determining whether such metrics exist, whether the corresponding maps are embeddings, and determining the dimension of the sphere are very challenging problems, where very few results are known. In the present talk we will discuss how one can use group action to resolve these issues and, as a result, construct many new examples of embedded minimal surfaces in the 3-sphere. The same considerations can be applied to the Steklov eigenvalue problem. As a consequence, we completely resolve the realisation problem for free boundary minimal surfaces in the unit 3-ball: we show that any compact orientable surface with boundary can be embedded in the 3-ball as a free boundary minimal surface. Based on a joint work with R. Kusner, P. McGrath and D. Stern.

  • Statistical Mechanics on 22 February 2024 at 14:00

    Speaker: Olga Iziumtseva (University of Nottingham)

    Title: Asymptotic and geometric properties of Volterra Gaussian processes

    Abstract: In this talk we discuss properties of Gaussian processes with representation int_0^t c(t,s) dG(s), where G is a continuous Gaussian martingale, and c is a square integrable Volterra kernel. Volterra Gaussian processes described in terms of a stochastic integral with respect to a Wiener process were first introduced by P. Lévy in 1956 as the canonical Volterra representation for a given Gaussian process, and continue to be an active area of research. In this talk we establish the law of iterated logarithm for a one-dimensional Volterra Gaussian process, we discuss the existence of local time in dimensions d>1, and construct the Rosen renormalized self-intersection local time for a planar Volterra Gaussian process. Joint work with Wasiur Khudabukhsh.

  • Probability Theory on 21 February 2024 at 16:00

    Speaker: Sabine Jansen (LMU Munich)

    Title: Cluster expansions & Kirkwood-Salsburg: 2-body vs. multi-body interactions

    Abstract: Cluster expansions give power series expansions in statistical mechanics, for example, for correlation functions (factorial moment densities) or the pressure of a classical gas at low density in a grand-canonical Gibbs measure (Gibbs point process). The series captures corrections towards the ideal gas (Poisson point process). I will present an abstract if and only if convergence condition for non-negative pair potentials based on the Kirkwood-Salsburg equations (joint work with Leonid Kolesnikov) and present some challenges and ideas for multi-body interactions.

  • Junior Number Theory on 19 February 2024 at 11:00

    Speaker: Khalid Younis (University of Warwick)

    Title: The distribution of smooth numbers

    Abstract: A number is said to be y-smooth if all of its prime factors are at most y. In much the same way as one studies primes, one can ask how many smooth numbers there are less than a large quantity x, whether they are spread evenly among arithmetic progressions, or how they are distributed in short intervals. In this talk, we will address some of these questions, with a focus on recent work on short intervals. In doing so, we will explore the connection with zeros of the Riemann zeta function.

  • Colloquium on 16 February 2024 at 16:00

    Speaker: Viveka Erlandsson (Bristol)

    Title: Counting curves à la Mirzakhani

    Abstract: While it is a classical result that the number of closed geodesics of bounded length on a hyperbolic surface grow exponentially in their length, the situation is very different when one considers certain subsets of closed geodesics. An important breakthrough in this direction was Maryam Mirzakhani’s work counting simple geodesics, proving they grow (asymptotically) polynomially in the length. Since then there has been a lot of interest in this type of questions and I will describe some recent development and generalizations of Mirzakhani’s work.

  • Combinatorics on 16 February 2024 at 14:00

    Speaker: Asier Calbet Ripodas (Queen Mary University of London)

    Title: The asymptotic behaviour of $sat(n,\mathcal{F})$

    Abstract: Given a family $\mathcal{F}$ of graphs, we say that a graph $G$ is \emph{$\mathcal{F}$-saturated} if it is maximally $\mathcal{F}$-free, meaning $G$ does not contain a graph in~$\mathcal{F}$ but adding any new edge to $G$ creates a graph in~$\mathcal{F}$. We then define $sat(n,\mathcal{F})$ to be the minimum number of edges in an $\mathcal{F}$-saturated graph on $n$ vertices. In 1986, K\'aszonyi and Tuza showed that $sat(n,\mathcal{F})=O(n)$ for all families $\mathcal{F}$ and Tuza conjectured that for singleton families $sat(n,\mathcal{F})/n$ converges. Tuza's Conjecture remains wide open. In this talk, I will discuss recent results about the asymptotic behaviour of $sat(n,\mathcal{F})$, mostly in the sparse regime $sat(n,\mathcal{F}) \leq n+o(n)$, in each of the cases when $\mathcal{F}$ is a singleton, when $\mathcal{F}$ is finite and when $\mathcal{F}$ is possibly infinite. Joint work with Andrea Freschi.

  • Statistical Mechanics on 15 February 2024 at 14:00

    Speaker: Andreas Koller (University of Warwick)

    Title: Scaling limit of gradient models on $\Z^d$ with non-convex energy

    Abstract: Random fields of gradients are a class of model systems arising in the study of random interfaces, random geometry, field theory and elasticity theory. The models we consider are characterised by an imposed boundary tilt and the free energy (called surface tension in the context of random interface models) as a function of tilt. Of interest are, in particular, whether the surface tension is strictly convex and whether the large-scale behaviour of the model remains that of the massless free field (Gaussian universality class). Where the Hamiltonian (energy) of the system is determined by a strictly convex potential, good progress has been made on these questions over the last two decades. For models with non-convex energy fewer results are known. Open problems include the conjecture that, in any regime of the parameters such that the scaling limit is Gaussian, its covariance (diffusion) matrix should be given by the Hessian of surface tension as a function of tilt. I will survey some recent advances in this direction using renormalisation group arguments and describe our result confirming the conjectured behaviour of the scaling limit for a class of non-convex potentials in the regime of low temperatures and small tilt. This is based on joint work with Stefan Adams.

  • Probability Theory on 14 February 2024 at 16:00

    Speaker: Peter Morfe (Max Planck Institut-Leipzig)

    Title: Anomalous Diffusion for a Passive Tracer Advected by the Curl of the 2D GFF

    Abstract: I will describe recent work on anomalous diffusion asymptotics for diffusions advected by turbulent velocity fields. Precisely, the model of interest involves a passive tracer subjected to Brownian diffusion and advection by the curl of the Gaussian free field (or divergence-free white noise). Recent work of Cannizzaro, Haunschmidt-Sibitz, and Toninelli (2022) established that the mean-square displacement is borderline superdiffusive with a logarithmic correction, confirming earlier predictions of the physics literature and a conjecture of Toth and Valko (2011). In joint work with Chatzigeorgiou, Otto, and Wang, we give an alternative proof built around ideas from stochastic homogenization, with a slightly stronger conclusion.

  • Algebra on 12 February 2024 at 17:00

    Speaker: Rudradip Biswas (University of Warwick)

    Title: Algebraic conditions for discrete groups to admit finite dimensional models for their classifying space of proper actions

    Abstract: Amongst people caring about cohomology questions for infinite groups, having a neat checkable algebraic property that, when imposed on a group G, implies the existence of a finite dimensional model for \underline{E}G has been a stimulating question for more than two decades. In this talk, I will try to highlight my contributions to this question. I will make sure to provide all the necessary definitions. My hope is the talk will be accessible to anyone with a basic knowledge of group cohomology. A part of the material is already in published form - "Injective generation of derived categories and other applications of cohomological invariants of infinite groups." Comm. Algebra 50 (2022), no. 10, 4460-4480.

  • Junior Number Theory on 12 February 2024 at 11:00

    Speaker: Jackie Voros (University of Bristol)

    Title: On the average least negative Hecke eigenvalue

    Abstract: In this talk we discuss the first sign change of Fourier coefficients of newforms, or equivalently Hecke eigenvalues. We will see this to be an analogue of the least quadratic non-residue problem, of which the average was investigated by Erdős in 1961. In fact, we will see that the average least negative prime Hecke eigenvalue holds the same (finite) value as the average least quadratic non-residue, under GRH. This is mainly due to the fact that Hecke eigenvalues at primes are equidistributed with respect to the Sato-Tate measure, a consequence of the Sato-Tate conjecture that was proven in 2011. We further explore the so-called vertical Sato-Tate conjecture to show the average least Hecke eigenvalue has a finite value unconditionally.

  • Combinatorics on 09 February 2024 at 14:00

    Speaker: Debsoumya Chakraborti (University of Warwick)

    Title: Bandwidth theorem for graph transversals

    Abstract: The bandwidth of a graph is the minimum $b$ such that there is a labeling of the vertex set of $H$ by the numbers $1,2,\ldots,n$ with $|i-j|\le b$ for every edge $ij$ of $H$. The celebrated bandwidth theorem states that if $H$ is an $n$-vertex graph with chromatic number $r$, bounded maximum degree, and bandwidth $o(n)$, then every $n$-vertex graph with minimum degree at least $\left(1-\frac{1}{r}+o(1)\right)n$ contains a copy of $H$. This minimum degree is best possible up to the $o(1)$ term.

    Recently, there have been systematic studies extending spanning subgraph problems to the setting of transversals over a collection of graphs. Given a graph-collection $\mathcal{G}=\{G_1,\dots, G_h\}$ with the same vertex set $V$, an $h$-edge graph $H$ on the vertex set $V$ is a $\mathcal{G}$-transversal if there exists a bijection $\lambda : E(H) \rightarrow \{1,\ldots,h\}$ such that $e\in E(G_{\lambda(e)})$ for each $e\in E(H)$. The minimum degree condition $\delta(\mathcal{G})=\min\{ \delta(G_i)\}$ for finding a spanning $\mathcal{G}$-transversal $H$ have been actively studied when $H$ is a Hamilton cycle, an $F$-factor, a tree with maximum degree $o(n/\log n)$, and power of Hamilton cycles, etc.

    Generalizing both these results and the original bandwidth theorem, we obtain asymptotically sharp minimum degree conditions for graphs with bounded degree, sublinear bandwidth, and given chromatic number, proving the bandwidth theorem for graph transversals. This talk is based on joint work with Seonghyuk Im, Jaehoon Kim, and Hong Liu.

  • Analysis on 08 February 2024 at 16:00

    Speaker: Joshua Daniels-Holgate (Hebrew University of Jerusalem)

    Title: Mean curvature flow from conical singularities

    Abstract: We discuss some regularity results for mean curvature flow from smooth hypersurfaces with conical singularities. We then discuss how to use these results to tackle two conjectures of Ilmanen.

  • Probability Theory on 07 February 2024 at 16:00

    Speaker: Perla Sousi (University of Cambridge)

    Title: Phase transition for the late points of random walk

    Abstract: Let X be a simple random walk in \mathbb{Z}_n^d with d\geq 3 and let t_{\rm{cov}} be the expected time it takes for X to visit all vertices of the torus. In joint work with Prévost and Rodriguez we study the set \mathcal{L}_\alpha of points that have not been visited by time \alpha t_{\rm{cov}} and prove that it exhibits a phase transition: there exists \alpha_* so that for all \alpha>\alpha_* and all \epsilon>0 there exists a coupling between \mathcal{L}_\alpha and two i.i.d. Bernoulli sets \mathcal{B}^{\pm} on the torus with parameters n^{-(a\pm\epsilon)d}with the property that \mathcal{B}^-\subseteq \mathcal{L}_\alpha\subseteq \mathcal{B}^+ with probability tending to 1 as n\to\infty. When \alpha\leq \alpha_*, we prove that there is no such coupling.

  • Algebra on 05 February 2024 at 17:00

    Speaker: Beth Romano (Kings College London)

    Title: Constructing graded Lie algebras

    Abstract: I'll talk about a construction that starts with a Heisenberg group of a certain kind and produces a graded Lie algebra. While purely algebraic, this construction is motivated by questions in number theory about rational points on families of curves, a setting where graded Lie algebras have proven to give valuable information. In addition to the number-theoretic applications, the construction has some nice applications to Lie theory, giving a way to lift certain Weyl group elements. I won't assume any background knowledge about graded Lie algebras or algebraic curves, and I'll give examples throughout the talk.

  • Junior Number Theory on 05 February 2024 at 11:00

    Speaker: Yan Yau Cheng (University of Edinburgh)

    Title: Arithmetic Chern Simons Theory

    Abstract: Mazur first observed in the 60s a deep analogy between the embedding of a knot in a 3-manifold and primes in a number field. Witten showed that knot invariants can be obtained by computations from quantum field theory. Using ideas from this analogy, Minhyong Kim and his collaborators developed the study of arithmetic field theories. This talk will be an introduction to Arithmetic Field Theories, in particular focusing on Arithmetic Chern-Simons Theory.

  • Combinatorics on 02 February 2024 at 14:00

    Speaker: Gal Kronenberg (University of Oxford)

    Title: Partitioning graphs into isomorphic linear forests

    Abstract: The linear arboricity of a graph G, denoted by la(G), is the minimum number of edge-disjoint linear forests (i.e. collections of disjoint paths) in G whose union is all the edges of G. It is known that the linear arboricity of every cubic graph is 2. In 1987 Wormald conjectured that every cubic graph with even number of edges, can be partitioned such that the two parts are isomorphic linear forests. This is known to hold for Jeager graphs and for some further classes of cubic graphs (see, Bermond-Fouquet-Habib-Peroche, Wormald, Jackson-Wormald, Fouquet-Thuillier-Vanherpe-Wojda). In this talk we will present a proof of Wormald's conjecture for all large connected cubic graphs.

    This is joint work with Shoham Letzter, Alexey Pokrovskiy, and Liana Yepremyan.

  • Analysis on 01 February 2024 at 16:00

    Speaker: Costante Bellettini (UCL)

    Title: Analysis of Stable Minimal Hypersurfaces: Curvature Estimates and Sheeting

    Abstract: We consider properly immersed two-sided stable minimal hypersurfaces of dimension n. We illustrate the validity of curvature estimates for n \leq 6 (and associated Bernstein-type properties). For n \geq 7 we illustrate sheeting results around "flat points". The proof relies on PDE analysis. The results extend respectively the Schoen-Simon-Yau estimates (obtained for n \leq 5) and the Schoen-Simon sheeting theorem (valid for embeddings).

  • Geometry and Topology on 01 February 2024 at 14:00

    Speaker: Samuel Shepherd (Vanderbilt University)

    Title: One-ended halfspaces in group splittings

    Abstract: I will introduce the notion of halfspaces in group splittings and discuss the problem of when these halfspaces are one-ended. I will also discuss connections to JSJ splittings of groups, and to determining whether groups are simply connected at infinity. This is joint work with Michael Mihalik.

  • Algebra on 29 January 2024 at 17:00

    Speaker: Hong Yi Huang (University of Bristol)

    Title: Bases for permutation groups

    Abstract: Let G < Sym(Omega) be a permutation group on a finite set Omega. A base for G is a subset of Omega with trivial pointwise stabiliser, and the base size of G, denoted b(G), is the minimal size of a base for G. This classical concept has been studied since the early years of permutation group theory in the nineteenth century, finding a wide range of applications. Recall that G is called primitive if it is transitive and its point stabiliser is a maximal subgroup. Primitive groups can be viewed as the basic building blocks of all finite permutation groups, and much work has been done in recent years in bounding or determining the base sizes of primitive groups. In this talk, I will report on recent progress of this study. In particular, I will give the first family of primitive groups arising in the O'Nan-Scott theorem for which the exact base size has been computed in all cases.

  • Junior Number Theory on 29 January 2024 at 11:00

    Speaker: Sebastian Monnet (King's College London / LSGNT)

    Title: Nonabelian number fields with prescribed norms

    Abstract: Let α be a rational number and let Σ be a family of number fields. For each number field K in Σ, either α is a norm of K, or it is not. We might ask for what proportion of K in Σ that is the case. We will see that this is a natural question to ask, and that it is extremely hard in general. For an abelian group A, the case Σ = {A-extensions} was solved by Frei, Loughran, and Newton. We will discuss new results for the simplest class of nonabelian extensions: so-called "generic" number fields of a given degree.

  • Combinatorics on 26 January 2024 at 11:00

    Speaker: Peter van Hintum (University of Oxford)

    Title: Additive Structure, the Brunn-Minkowski inequality and beyond.

    Abstract: We’ll explore additive structure in continuous and discrete settings, i.e. the question of what structure subsets A and B of a group (in particular R^k and Z) must display if their Minkowski sum A+B is small. In the continuous world, A and B must look similar and almost convex as described by stability results for the Brunn-Minkowski inequality. In the discrete world, A and B must additionally posses a lattice-like structure as described by Freiman-Ruzsa type results. In this talk, we’ll consider both perspectives and the connection between them.

    This talk is based on various papers joint with Alessio Figalli, Peter Keevash, and Marius Tiba among others.

  • Analysis on 25 January 2024 at 16:00

    Speaker: Giada Franz (MIT)

    Title: Topological control for min-max free boundary minimal surfaces

    Abstract: A free boundary minimal surface (FBMS) in a three-dimensional Riemannian manifold is a critical point of the area functional with respect to variations that constrain its boundary to the boundary of the ambient manifold. A very natural question is the one of constructing FBMS (in a given ambient manifold) of a given topological type.
    In this talk, we will focus on one of the methods that have been employed so far to tackle this problem, that is Simon-Smith variant of Almgren-Pitts min-max theory.
    We will see how this method allows us to control the topology (i.e. genus and number of boundary components) of the resulting surface, and we will present several applications.

  • Geometry and Topology on 25 January 2024 at 14:00

    Speaker: Francesco Fournier-Facio (University of Cambridge)

    Title: Infinite simple characteristic quotients

    Abstract: The rank of a finitely generated group is the minimal size of a generating set. Several questions that received a lot of attention around 50 years ago ask about the rank of finitely generated groups, and how this relates to the rank of their direct powers. In this context, Wiegold asked about the existence of infinite simple characteristic quotients of free groups. I will review this framework, present several open questions – old and new – and present a solution to Wiegold’s problem.

    Joint with Rémi Coulon

  • Probability Theory on 24 January 2024 at 16:00

    Speaker: Sandro Franceschi (Polytechnique Paris)

    Title: Reflected Brownian motion in a cone: a study of the transient case

    Abstract: One of the classic problems in the literature devoted to reflected Brownian motion in a two-dimensional cone is the study of its stationary distribution in the recurrent case. On the other hand, we will focus in this talk on the transient case to study the Green's functions of this process and their asymptotics. This will naturally lead us to consider the Martin boundary of the process which allows us to determine the harmonic functions satisfying oblique Neumann conditions on the edges. For some models, we will illustrate this by studying the probability of escape of the process along an axis or its probability of absorption at the origin.

    To establish our results, we use analytical methods historically developed in probability and combinatorics to study random walks in the quadrant. We establish functional equations satisfied by the Laplace transforms of Green's functions and the probabilities of escape or absorption. Thanks to the theory of boundary value problems (of Riemann and Carleman) it is possible to determine explicit formulas for these transforms involving hypergeometric functions. The saddle point method and transfer lemmas enable us to obtain asymptotic results and establish the Martin boundary.

  • Junior Number Theory on 22 January 2024 at 11:00

    Speaker: Cedric Pilatte (University of Oxford)

    Title: Graph eigenvalues and the logarithmic Chowla conjecture in degree 2

    Abstract: The Liouville function \lambda(n) is defined to be +1 if n is a product of an even number of primes, and -1 otherwise. The statistical behaviour of \lambda is intimately connected to the distribution of prime numbers. In many aspects, the Liouville function is expected to behave like a random sequence of +1 and -1. For example, the two-point Chowla conjecture predicts that the average of \lambda(n)\lambda(n+1) over n < x tends to zero as x goes to infinity. In this talk, I will discuss quantitative bounds for a logarithmic version of this problem.

  • Colloquium on 19 January 2024 at 16:00

    Speaker: John MacKay (Bristol)

    Title: Group actions on L^1 spaces

    Abstract: An important (and classical) way to study groups is through their possible affine isometric actions on Hilbert spaces: this leads to the study of Kazhdan's Property (T) and related notions which have useful applications both in group theory and outside, e.g. in dynamics. But it is natural to consider actions on other Banach spaces too, for example, there have been breakthroughs recently by Oppenheim and de Laat-de la Salle for actions on L^p spaces (1

  • Combinatorics on 19 January 2024 at 14:00

    Speaker: Katherine Staden (Open University)

    Title: Colourful Hamilton cycles

    Abstract: A classical question in graph theory is to find sufficient conditions which guarantee that a graph G contains a Hamilton cycle. A colourful variant of this problem has graphs G_1, ..., G_s on the same n-vertex set, where we think of each graph as having a different colour, and we want to find a Hamilton cycle coloured with 1, ..., s whose colouring is from some given list of allowed colourings. I will discuss a selection of results on this problem.

    This is joint work with Candy Bowtell, Patrick Morris and Yani Pehova, and with Yangyang Cheng.

  • Statistical Mechanics on 18 January 2024 at 14:00

    Speaker: Josephine Evans (University of Warwick)

    Title: Non-Equilibrium steady states in BGK models for gas dynamics

    Abstract: This talk is based on a joint work with Angeliki Menegaki from Imperial. We consider the BGK kinetic model for a dilute gas coupled to a thermostat modelling gain and loss of kinetic energy for the particles. I will discuss how to show existence and linear stability for a model of this type and also why stability results are difficult in this area and how such results are related to one way of deriving Fourier's law from microscopic processes.

  • Geometry and Topology on 18 January 2024 at 14:00

    Speaker: Ian Leary (University of Southampton)

    Title: Residual finiteness of generalized Bestvina-Brady groups

    Abstract: (joint with Vladimir Vankov)
    I discovered/created generalized Bestvina-Brady groups to give an uncountable family
    of groups with surprising homological properties. In this talk, I will introduce the
    groups and describe joint work with Vladimir Vankov addressing the following questions:
    when are they virtually torsion-free?
    when are they residually finite?
    This leads naturally to a third question:
    when do they virtually embed in right-angled Artin groups?
    There are nice conjectural answers to all three questions, which we have proved in
    some cases.

  • Probability Theory on 17 January 2024 at 16:00

    Speaker: Tamara Grava (University of Bristol)

    Title: A soliton gas for the focusing nonlinear Schrödinger equation

    Abstract: We consider the focusing nonlinear Schrödinger equation on the line. This equation admits a N soliton solution parametrised by 2N complex spectral data. In recent years many significant numerical simulations that study large sets of random solitons have been realised. We consider a random configuration of N solitons with respect to a probability distributions on the spectral data. We then show that the N soliton solution, after suitable rescalings converges to a normal distribution as N goes to infinity.

  • Junior Analysis and Probability Seminar on 17 January 2024 at 14:00

    Speaker: Simon Gabriel (Münster)

    Title: Rooted trees and singular SPDEs

    Abstract: In this talk, we give an introduction on how rooted trees are used to encode iterated stochastic integrals, appearing for example in the study of singular stochastic PDEs. To this end, we first discuss what makes an SPDE singular and a possible approach on how to “solve” them in a certain (subcritical) regime. Here a notation using rooted trees comes in handy. Time permitting, we briefly discuss the (more ore less) open problem of treating so called critical singular SPDEs.

    The talk is aimed at a non-specialist audience.

  • Junior Number Theory on 15 January 2024 at 11:00

    Speaker: Sven Cats (University of Cambridge)

    Title: Higher descent on elliptic curves

    Abstract: Let $E$ be an elliptic curve over a number field $K$ and $n \geq 2$ an integer. We recall that elements of the $n$-Selmer group of $E/K$ can be written as certain \emph{$n$-coverings} of $E/K$. Writing the elements in this way is called conducting an \emph{explicit $n$-descent}. One of the applications of explicit $n$-descent is in finding generators of large height for $E(K)$ and from this point of view one would like to be able to take $n$ as large as possible. Already for $n \geq 5$ it is computationally challenging to apply the known algorithms for explicit $n$-descent. We discuss two ways around this: Improving a $p$-isogeny descent to a $p$-descent and combining $n$- and $(n+1)$-descents to $n(n+1)$-descent.

  • Colloquium on 12 January 2024 at 16:00

    Speaker: Chiara Saffirio (Basel)

    Title: From microscopic to macroscopic scales: many interacting quantum particles and their semiclassical approximation.

    Abstract: Systems of interacting particles describing notable physical phenomena, such as Bose-Einstein condensation, superconductivity or superfluidity, exhibit a daunting complexity. This complexity renders the exact many-body theory computationally non-approachable, even for physicists conducting computer experiments and simulations. Therefore, an approximate description using effective macroscopic models is highly useful, and the rigorous study of the regime of validity of such approximations is of primary importance in mathematical physics.

    In this talk we will focus on the dynamics of systems made of quantum particles in the mean-field regime, where weakly interacting particles exhibit a collective behavior approximated by an averaged potential in convolution form. We study time scales where the semiclassical description becomes relevant. Through a novel technique based on weak-strong stability principles for partial differential equations, we show that the many-body dynamics is well approximated by a Vlasov equation describing the evolution of the effective probability density of particles on the phase space.

  • Combinatorics on 12 January 2024 at 14:00

    Speaker: Jane Tan (University of Oxford)

    Title: Reconstructing cube complexes from boundary distances

    Abstract: Given a quadrangulation of a disc, suppose we know all the pairwise distances (measured by the graph metric) between vertices on the boundary of the disc. Somewhat surprisingly, a result of Haslegrave states that this is enough information to recover the whole interior structure of the quadrangulation provided all internal vertex degrees are at least 4. In this talk, we look at a generalisation of this result to 3 dimensions: it is possible to reconstruct 3D cube complexes that are homeomorphic to a ball from the pairwise distances between all points on the boundary sphere provided certain curvature conditions hold. This is joint work with Haslegrave, Scott and Tamitegama. We’ll also survey some related results from before and after our work.

  • Analysis on 11 January 2024 at 16:00

    Speaker: Ilaria Mondello (Paris-East Créteil)

    Title: Gromov-Hausdorff limits of manifolds with a Kato bound on the Ricci curvature

    Abstract: The goal of this talk is to present some recent results on manifolds for which the negative part of the Ricci curvature satisfies a Kato bound, inspired by Kato potentials in the Euclidean space. This condition is implied for instance by a lower Ricci curvature bound, or an integral Ricci bound in the spirit of Gallot and Petersen-Wei. We will explain some analytic consequences of a Kato bound, and how we used them to study the structure of Gromov-Hausdorff limits of manifolds satisfying this kind of bounds. This talk is based on a joint work with G. Carron and D. Tewodrose.

  • Geometry and Topology on 11 January 2024 at 14:00

    Speaker: Richard Wade (University of Oxford)

    Title: Quasi-flats in the Aut free factor complex

    Abstract: We will describe families of quasi-flats in the "$Aut(F_n)$ version" of the free factor complex. This shows that, unlike its more popular "Outer" cousin, the Aut free factor complex is not hyperbolic. The flats are reasonably simple to describe and are shown to be q.i. embedded via the construction of a coarse Lipschitz retraction. This leaves many open problems about the coarse geometry of this space, and I hope to talk about a few of them. This is joint work with Mladen Bestvina and Martin Bridson.

  • Algebraic Topology on 09 January 2024 at 16:00

    Speaker: Doosung Park (University of Wuppertal)

    Title: Syntomic cohomology and real topological cyclic homology

    Abstract: In this talk, I will show that real topological cyclic homology admits a complete exhaustive filtration whose graded pieces are equivariant suspensions of syntomic cohomology. Combined with the announced results of Antieau-Krause-Nikolaus and Harpaz-Nikolaus-Shah, this would lead to the computation of the equivariant slices of the real K-theory of Z/p^n after a certain suspension. The key ingredients of the proof are a real refinement of the Hochschild-Kostant-Rosenberg filtration and the computation of real topological Hochschild homology of perfectoid rings in my joint work with Hornbostel.

  • Junior Number Theory on 08 January 2024 at 11:00

    Speaker: Kenji Terao (University of Warwick)

    Title: Isolated points on modular curves

    Abstract: As is well known, Faltings's theorem settles the question of determining when a curve, defined over a number field, has infinitely many rational points. However, Faltings's work can also be used to understand when such a curve has infinitely many higher degree points, a study which gives rise to the notion of isolated points. In this talk, we will study some techniques for finding isolated points on curves, and see how they can be applied to the more structured world of modular curves.

  • Algebraic Topology on 12 December 2023 at 16:00

    Speaker: Anna Marie Bohmann (Vanderbilt University)

    Title: Scissors congruence and the K-theory of covers

    Abstract: Scissors congruence, the subject of Hilbert's Third Problem, asks for invariants of polytopes under cutting and pasting operations. One such invariant is obvious: two polytopes that are scissors congruent must have the same volume, but Dehn showed in 1901 that volume is not a complete invariant. Trying to understand these invariants leads to the notion of the scissors congruence group of polytopes, first defined the 1970s. Elegant recent work of Zakharevich allows us to view this as the zeroth level of a series of higher scissors congruence groups.
    In this talk, I'll discuss some of the classical story of scissors congruence and then describe a way to build the higher scissors congruence groups via K-theory of covers, a new framework for such constructions. We'll also see how to relate coinvariants and K-theory to produce concrete nontrivial elements in the higher scissors congruence groups. This work is joint with Gerhardt, Malkiewich, Merling and Zakharevich.

  • Ergodic Theory and Dynamical Systems on 12 December 2023 at 14:00

    Speaker: Christian Wolf (CUNY)

    Title: Ergodic theory on coded shifts spaces

    Abstract: In this talk we present results about ergodic-theoretic properties of
    coded shift spaces. A coded shift space is defined as a closure of all bi-infinite
    concatenations of words from a fixed countable generating set. We derive
    sufficient conditions for the uniqueness of measures of maximal entropy and
    equilibrium states of H\"{o}lder continuous potentials based on the partition of the coded
    shift into its sequential set (sequences that are concatenations of generating words)
    and its residual set (sequences added under the closure). We also discuss
    flexibility results for the entropy on the sequential and residual set. Finally, we present
    a local structure theorem for intrinsically ergodic coded shift spaces which shows
    that our results apply to a larger class of coded shift spaces compared to previous works
    by Climenhaga, Climenhaga and Thompson, and Pavlov. The results presented
    in this talk are joint work with Tamara Kucherenko and Martin Schmoll.

  • Colloquium on 08 December 2023 at 16:00

    Speaker: John Gibbon (Imperial)

    Title: Regularity and multifractality in passive and active turbulent Navier-Stokes-like flows

    Abstract: I will begin with a survey of the regularity properties of the incompressible Navier-Stokes equations (NSEs) – one of the Millenium Clay Prize problems ­ – including the weak solution properties of Leray (1934). I will contrast these with the results that we would like to prove to gain full regularity but have not yet done so. Then I will move on to a brief description of the Multifractal Model (MFM), developed by Parisi and Frisch (1985) to describe homogeneous turbulence. I will show that there exists an intriguing correspondence between the NSEs and the MFM. Finally, I will consider the incompressible Toner-Tu equations (ITT) that describe flocking phenomena in active turbulence. They enjoy many similar properties to those possessed by the NSEs, so many results can be lifted over.

  • Combinatorics on 08 December 2023 at 14:00

    Speaker: Ryan Martin (Iowa State)

    Title: Counting cycles in planar graphs

    Abstract: TBA

  • Statistical Mechanics on 07 December 2023 at 16:00

    Speaker: Christian Korff (University of Glasgow)

    Title: Exactly solvable lattice models, symmetric functions and vertex operators

    Abstract: The ring of symmetric functions plays a central role in representation theory. It connects with exactly solvable lattice models of statistical mechanics and quantum many-body systems by observing that the eigenfunctions of the transfer matrices or Hamiltonian (the Bethe wave functions) are symmetric polynomials. For periodic boundary conditions so-called cylindric symmetric functions emerge whose product (and co-product) expansions lead to 2D topological quantum field theories. For infinite lattices and with suitable boundary conditions at infinity, one can use the transfer matrices of exactly solvable lattice models to obtain combinatorial formulae for vertex operators of symmetric functions. This links the area of statistical lattice models and quantum spin-chains (via the boson-fermion correspondence) with integrable hierarchies of PDEs such as the Kadomtsev-Petiashvili equation where it is known that particular solutions, tau-functions, are given by symmetric functions.

  • Analysis on 07 December 2023 at 16:00

    Speaker: Alix Deruelle (Paris-Saclay)

    Title: Ancient Solutions to the Ricci Flow Coming Out of Spherical Orbifolds

    Abstract: Given a 4-dimensional Einstein orbifold that cannot be desingularized by smooth Einstein metrics, we investigate the existence of an ancient solution to the Ricci flow coming out of such a singular space. In this talk, we will focus on singularities modeled on a cone over RP3 that are desingularized by gluing Eguchi-Hanson metrics to get a first approximation of the flow. We show that a parabolic version of the corresponding obstructed gluing problem has a smooth solution: the bubbles are shown to grow exponentially in time, a phenomenon that is intimately connected to the instability of such orbifolds.
    This is joint work with Tristan Ozuch.

  • Geometry and Topology on 07 December 2023 at 14:00

    Speaker: Sam Hughes (University of Oxford)

    Title: Centralisers and classifying spaces for Out(F_N)

    Abstract: In this talk I will outline reduction theory for mapping classes and explain various attempts to construct similar machinery for elements of Out(F_N). I will then present a new reduction theory for studying centralisers of elements in IA_3(N), the finite index level 3 congruence subgroup of Out(F_N). Using this I will explain an application to the classifying space for virtually cyclic subgroups, a space notable for its appearance in the Farrell--Jones Conjecture. Based on joint work with Yassine Guerch and Luis Jorge Sánchez Saldaña.

  • Probability Theory on 06 December 2023 at 16:00

    Speaker: Ilya Chevyrev (University of Edinburgh)

    Title: Decorated path spaces with applications to fast-slow systems

    Abstract: In this talk, I will present a space of decorated paths that allows one to keep track of oscillations of paths that happens in infinitesimal time. Despite its simple definition as a naive completion of the Skorokhod space, this notion is fruitful in the study of ordinary differential equations with jumps, generalising the framework of Marcus, and applies in situations where classical Skorokhod topologies are too restrictive. As an application, I will show how homogenisation theorems of superdiffusive fast-slow systems, including billiards with flat cusps, can be stated and proved in this framework. Based on a joint work with Alexey Korepanov and Ian Melbourne.

  • Algebraic Geometry on 06 December 2023 at 15:00

    Speaker: Rudradip Biswas (Warwick)

    Title: There is only one equivalence class of bounded t structures in the derived bounded category of coherent sheaves on a noetherian finite dimensional scheme.

    Abstract: In the study of triangulated categories, bounded t-structures have always attracted a lot of attention. I will talk about one of my new papers, joint with Hongxing Chen, Kabeer Manali Rahul, Chris Parker, and Junhua Zheng, where we show that all bounded t structures on the derived bounded category of coherent sheaves on a noetherian finite dimensional scheme are equivalent to each other. Some major groundbreaking work in this area recently came from Amnon Neeman. Our paper generalizes his results because he could only prove this equivalence result when the scheme was separated and quasi excellent or had a dualising complex.

  • Algebraic Topology on 05 December 2023 at 16:00

    Speaker: Irakli Patchkoria (University of Aberdeen)

    Title: Posets of finite abelian subgroups and Morava K-theory

    Abstract: A result of K. Brown says that for a nice enough discrete group G, the orbifold Euler characteristic of G and the equivariant Euler characteristic of the poset of its non-trivial finite subgroups have the same fractional parts. We will present an analogous result for the poset of non-trivial finite abelian subgroups for which we will use Morava K-theory. After presenting some computations, we will discuss potential applications in number theory analogous to Brown’s results on denominators of special values of zeta functions.

  • Ergodic Theory and Dynamical Systems on 05 December 2023 at 14:00

    Speaker: Weikun He (Institute of Mathematics, Beijing)

    Title: Dimension theory of groups of circle diffeomorphisms.

    Abstract: n this talk, we consider the action of a finitely generated group on the circle by analytic diffeomorphisms. We will discuss some results concerning the dimensions of objects arising from this action. More precisely, we will present connections among the dimension of minimal subsets, that of stationary measures, entropy of random walks, Lyapunov exponents and critical exponents. These can be viewed as generalizations of well-known results in the situation of PSL(2,R) acting on the circle.

  • Algebra on 04 December 2023 at 17:00

    Speaker: Tim Burness (University of Bristol)

    Title: Topological generation of algebraic groups

    Abstract: Let G be an algebraic group over an algebraically closed field and recall that a subset of G is a topological generating set if it generates a dense subgroup. In this talk, I will report on recent work with Spencer Gerhardt and Bob Guralnick on the topological generation of simple algebraic groups by elements in specified conjugacy classes. I will also present an application concerning the random generation of finite simple groups of Lie type.

  • Junior Analysis and Probability Seminar on 04 December 2023 at 16:00

    Speaker: Tom Sales (Warwick)

    Title: The Cahn–Hilliard equation on an evolving surface

    Abstract: In recent years there has been interest on partial differential equations (PDEs) posed on domains which evolve in time, and in particular evolving surfaces. Applications for these systems can be found, for example, in the study of lipid biomembranes. In this talk we consider the Cahn–Hilliard equation on an evolving surface and discuss the corresponding analysis and numerical analysis. This includes a framework for PDEs on evolving domains, and techniques for the discretisation of PDEs on evolving surfaces via the evolving surface finite element method (ESFEM). Assuming a smooth potential function, we outline the main proofs for the well-posedness of the Cahn–Hilliard equation, as well as optimal order error bounds for a numerical scheme using backward-Euler time discretisation and isoparametric ESFEM.

  • Junior Number Theory on 04 December 2023 at 11:00

    Speaker: Harvey Yau (University of Cambridge)

    Title: An introduction to Brauer-Manin obstruction

    Abstract: To study the rational points on a variety, one useful tool is to study it over a completion of the rationals, and in many cases this suffices to prove there are no rational points. However, sometimes this method is insufficient to prove the nonexistence of rational points, and many such examples have been found over the years. The Brauer-Manin obstruction provides a general explanation for these examples, and was first described by Y. Manin. This talk will give an introduction to the topic and construct some explicit examples of the obstruction on curves and surfaces.

  • Combinatorics on 01 December 2023 at 14:00

    Speaker: Kyriakos Katsamaktsis (UCL)

    Title: Ascending subgraph decomposition

    Abstract: TBA

  • Analysis on 30 November 2023 at 16:00

    Speaker: Guido De Philippis (NYU Courant)

    Title: Monge Ampere equation and unique continuation for differential inclusions

    Abstract: The Monge Ampere equation is a prototypical non linear equations arising in several questions concerning Geometry, Optimal Design, Optimal transport etc etc. I will review some of the applications and some of the known results, in particular concerning Sobolev regularity of the solutions.
    I will then show how these results have an equivalent formulation in terms of a unique continuation property for solution of differential inclusions and use this link to reprove the Sobolve regularity result for planar solutions f the MA equation obtained by Figalli Savin and myself in 2013. This is a joint work with Andre Guerra and Richard Tione.

  • Statistical Mechanics on 30 November 2023 at 16:00

    Speaker: Baptiste Cerclé (University of Paris Saclay)

    Title: Integrability in Toda Conformal Field Theories and Whittaker functions

    Abstract: Toda conformal field theories form a family of two-dimensional quantum field theories that enjoy, in addition to conformal invariance, an enhanced level of symmetry.
    Initially introduced in the physics literature, they admit a mathematical definition based on two key probabilistic objects: Gaussian Free Fields and Gaussian Multiplicative Chaos.

    The aim of this talk is twofold: first we will explain how the probabilistic definition of Toda theories allows to provide integrability results for such models and, as a consequence, for Gaussian Multiplicative Chaos measures. To this end we will sketch the proof of the Fateev-Litvinov formula for a family of three-point correlation functions.
    We will then detail a connection between class one Whittaker functions and certain quantities key in the study of Toda theories: the reflection coefficients. The derivation of these reflection coefficients relies on a new Brownian path decomposition, generalizing Williams celebrated result, that we will also present.

  • Mathematics Teaching and Learning on 30 November 2023 at 16:00

    Speaker: Barry Griffiths (University of Central Florida)

    Title: American Trends in Teaching and Researching Mathematics: A Glimpse of the Future?

    Abstract: In this talk I will look at how teaching and research in the United States is being affected by technology, the drive to educate an increasing number of students, the business of higher education, and the global academic community. I will draw parallels with the situation in the UK and discuss how these issues might lead to changes in the academic culture.

  • Geometry and Topology on 30 November 2023 at 14:00

    Speaker: Cameron Rudd (MPIM Bonn)

    Title: Stretch laminations and hyperbolic Dehn surgery

    Abstract: Given a hyperbolic manifold M and a homotopy class of maps from M to the circle, there is an associated geodesic "stretch" lamination encoding at which points in M the Lipschitz constant of any map in the homotopy class must be large. Recently, Farre-Landesberg-Minsky related these laminations to horocycle orbit closures in infinite cyclic covers and when M is a surface, they analyzed the possible structure of these laminations. I will discuss the case where M is a 3-manifold and give the first 3-dimensional examples where these laminations can be identified. The argument uses the Thurston norm and tools from quantitative Dehn surgery.

  • Probability Theory on 29 November 2023 at 16:00

    Speaker: Marielle Simon (University of Lyon)

    Title: A few scaling limits results for the facilitated exclusion process in 1d

    Abstract: The aim of this talk is to present some recent results which have been obtained for the facilitated exclusion process in one dimension. This stochastic lattice gas is subject to strong kinetic constraints which create a continuous phase transition to an absorbing state at a critical value of the particle density. If the microscopic dynamics is symmetric, its macroscopic behavior, under periodic boundary conditions and diffusive time scaling, is ruled by a non-linear PDE belonging to free boundary problems (or Stefan problems). One of the ingredients is to show that the system typically reaches an ergodic component in subdiffusive time.
    The asymmetric case can also be fully treated: in this case, considered on the infinite line, the empirical density converges to the unique entropy solution to a hyperbolic Stefan problem. All these results rely, to various extent, on a mapping argument with a zero-range process, which completely fails in dimension higher than 1.
    Based on joint works with O. Blondel, C. Erignoux, M. Sasada and L. Zhao.

  • Algebraic Topology on 28 November 2023 at 16:00

    Speaker: Eric Finster (University of Birmingham)

    Title: A Topos Theoretic View of Goodwillie Calculus

    Abstract: I will describe a framework for understanding the unstable version of
    Goodwillie’s calculus of functors from a topos-theoretic perspective
    which builds on an analogy between higher topos theory and commutative
    algebra. In particular, I will describe how both Goodwilile’s original
    ”homotopy calculus” as well as the ”orthogonal calculus” of Michael
    Weiss can be understood in this framework. Along the way, we will see
    emerge a picture of the topos of n-excisive functors as classifying
    ”n-nilpotent” objects. This is joint with with M. Anel, G. Biedermann
    and A. Joyal.

  • Junior Analysis and Probability Seminar on 27 November 2023 at 15:00

    Speaker: Sotirios Kotitsas (Warwick)

    Title: The KPZ equation in dimensions d ≥ 2: a survey and recent results

    Abstract: The KPZ equation:
    ∂h/∂t(t,x) =1/2∆h(t,x) + β|∇h(t,x)|^2 + ξ(t,x)
    where ξ is a random forcing term is one of the most important stochastic PDEs in mathematical physics. It is conjectured to encode the fluctuations of many natural models of randomly growing interfaces and it has been the study of intense research in the past decade. Due to its nonlinear nature it is hard to make sense of the equation directly and this has only been achieved in dimension d= 1. In this talk we will explain why the KPZ equation in d ≥ 2 is fundamentally different and we will survey some known results regarding its fluctuations and its connections to the theory of random polymers. Time permitting we will talk about some new work in progress in d = 2.

  • Junior Number Theory on 27 November 2023 at 11:00

    Speaker: Benjamin Bedert (University of Oxford)

    Title: On Unique Sums in Abelian Groups

    Abstract: In this talk, we will study the old problem in additive combinatorics of determining for a finite Abelian group $G$ the size of its smallest subset $A\subset G$ that has no unique sum, meaning that for every two $a_1,a_2\in A$ we can write $a_1+a_2=a’_1+a’_2$ for different $a’_1,a’_2\in A$. We begin by using classical rectification methods to obtain the previous best lower bounds of the form $|A|\gg \log p(G)$. Our main aim is to outline the proof of a recent improvement and discuss some of its key notions such as additive dimension and the density increment method. This talk is based on Bedert, B. On Unique Sums in Abelian Groups. Combinatorica (2023).

  • Colloquium on 24 November 2023 at 16:00

    Speaker: Aretha Teckentrup (Edinburgh)

    Title: Deep Gaussian process priors in infinite-dimensional inverse problems

    Abstract: Deep Gaussian processes have proved remarkably successful as a tool for various statistical inference tasks. This success relates in part to the flexibility of these processes and their ability to capture complex, non-stationary behaviours. In this talk, we introduce deep Gaussian processes as prior distributions in infinite-dimensional inverse problems, and demonstrate their superiority in example applications including computational imaging and regression. We will discuss recent algorithmic developments for efficient sampling, as well as recent theoretical results which give crucial insight into the behaviour of the methodology.

  • Combinatorics on 24 November 2023 at 14:00

    Speaker: Andrea Freschi (Birmingham)

    Title: Discrepancy in edge-coloured and oriented graphs

    Abstract: TBA

  • Analysis on 23 November 2023 at 16:00

    Speaker: Thomas Körber (University of Vienna)

    Title: Schoen's Conjecture for Limits of Isoperimetric Surfaces

    Abstract: R. Schoen has conjectured that an asymptotically flat Riemannian n-manifold (M,g) with non-negative scalar curvature is isometric to Euclidean space if it admits a non-compact area-minimizing hypersurface. This has been confirmed by O. Chodosh and M. Eichmair in the case where n=3. In this talk, I will present recent work with M. Eichmair where we confirm this conjecture in the case where 3

  • Geometry and Topology on 23 November 2023 at 14:00

    Speaker: Jeffrey Giansiracusa (University of Durham)

    Title: Topology of the matroid Grassmannian

    Abstract: The matroid Grassmannian is the moduli space of oriented matroids; this is an important combinatorial analogue of the ordinary oriented real Grassmannian. Thirty years ago MacPherson showed us that understanding the homotopy type of this space can have significant implications in manifold topology, such as providing combinatorial formulae for the Pontrjagin classes. In some easy cases, the matroid Grassmannian is homotopy equivalent to the oriented real Grassmannian, but in most cases we have no idea whether or not they are equivalent. This question is known as MacPherson's conjecture. I'll show that one of the important homotopical structures of the oriented Grassmannians has an analogue on the matroid Grassmannian: the direct sum monoidal product (which gives rise to topological K-theory) is E-infinity.

  • Ergodic Theory Meeting on 22 November 2023 at 16:45

    Speaker: Henna Koivusalo (Bristol)

    Title: Shrinking targets on self-affine sets

    Abstract: The classical shrinking target problem concerns the following set-up: Given a dynamical system (T, X) and a sequence of targets (B_n) of X, we investigate the size of the set of points x of X for which T^n(x) hits the target B_n for infinitely many n. In this talk I will discuss shrinking target problems in the context of iterated function systems, where `size' is studied from the perspective of dimension. I will give an overview of the topic, with the aim to, by the end, cover an upcoming result on geometric shrinking targets on Przytycki-Urbanski-type affine iterated function systems. Analysing this particular model requires heavy use of the theory of Bernoulli convolutions.

    This talk is based on a work joint with Thomas Jordan.

  • Other on 22 November 2023 at 16:00

    Speaker: Christophoros Panagiotis (University of Bath)

    Title: Quantitative sub-ballisticity of self-avoiding walk on the hexagonal lattice

    Abstract: In this talk, we will consider the self-avoiding walk on the hexagonal lattice, which is one of the few lattices whose connective constant can be computed explicitly. This was proved by Duminil-Copin and Smirnov in 2012 when they introduced the parafermionic observable. In this talk, we will use the observable to show that, with high probability, a self-avoiding walk of length n does not exit a ball of radius n/logn. This improves on an earlier result of Duminil-Copin and Hammond, who obtained a non-quantitative o(n) bound. Along the way, we show that at criticality, the partition function of bridges of height T decays polynomially fast to 0. Joint work with Dmitrii Krachun.

  • Ergodic Theory Meeting on 22 November 2023 at 15:30

    Speaker: Tim Austin (Warwick)

    Title: A dynamical proof of the Shmerkin—Wu theorem

    Abstract: Let a
    A few years ago, Shmerkin and Wu independently gave two different proofs of Furstenberg's conjecture. In this talk I will sketch a more recent third proof that builds on some of Furstenberg's original results. In addition to those, the main ingredients are a version of the Shannon—McMillan—Breiman theorem relative to a factor and some standard calculations with entropy and Hausdorff dimension.

  • Algebraic Geometry on 22 November 2023 at 15:00

    Speaker: Sara Veneziale (Imperial)

    Title: Machine learning and the classification of Fano varieties

    Abstract: In this talk, I will describe recent work in the application of AI to explore questions in algebraic geometry, specifically in the context of the classification of Fano varieties. We ask two questions. Does the regularized quantum period know the dimension of a toric Fano variety? Is there a condition on the GIT weights that determines whether a toric Fano has at worst terminal singularities? We approach these problems using a combination of machine learning techniques and rigorous mathematical proofs. I will show how answering these questions allows us to produce very interesting sketches of the landscape of weighted projective spaces and toric Fanos of Picard rank two. This is joint work with Tom Coates and Al Kasprzyk.

  • Ergodic Theory Meeting on 22 November 2023 at 14:00

    Speaker: Terry Soo (UCL)

    Title: Independent, but not identically distributed coin-flips

    Abstract: In joint work with Zemer Kosloff, we will discuss the dynamical properties of a seemingly innocuous perturbation of a sequence of independent and identically distributed (iid) coin-flips to one that is no longer stationary. In the stationary case, Ornstein proved that iid systems are completely classified up to isomorphism by their Shannon entropy. We will find that in the nonstationary case, the usual entropy theory no longer applies, but we will recover an explicit version of the Sinai factor theorem that allows us to generate iid randomness from a nonstationary source.

  • Algebraic Topology on 21 November 2023 at 16:00

    Speaker: Emel Yavuz (Queen's University Belfast)

    Title: C_2-Equivariant Orthogonal Calculus

    Abstract: Orthogonal homotopy calculus is the branch of functor calculus involving the study of functors from the category of finite dimensional real vector spaces to the category of pointed topological spaces. Using it, one can construct a Taylor tower of approximations to such functors, consisting of polynomial functors, and the layers of the tower are characterised by orthogonal spectra, making them much easier to compute.
    A natural question is; what happens when the functors come with a group action? Such functors are of great interest, as they arise naturally within algebraic topology, for example the functor V \mapsto BO(V) where V is a G-representation. After an introduction to orthogonal calculus, I will discuss the main constructions and theorems of a C_2-equivariant orthogonal calculus, that works for functors from finite dimensional C_2-inner product spaces to C_2-spaces.

  • Ergodic Theory and Dynamical Systems on 21 November 2023 at 14:00

    Speaker: Li Dongchen (Imperial College London)

    Title: Persistence of heterodimensional cycles

    Abstract: TBA

  • Partial Differential Equations and their Applications on 21 November 2023 at 12:00

    Speaker: Mattia Zanella (Univ. Pavia)

    Title: TBA

    Abstract: TBA

  • Colloquium on 17 November 2023 at 16:00

    Speaker: Jon Chapman (Oxford)

    Title: Asymptotics beyond all orders: the devil's invention?

    Abstract: "Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever."
    — N. H. Abel.

    The lecture will introduce the concept of an asymptotic series, showing how useful divergent series can be, despite Abel's reservations. We will then discuss Stokes' phenomenon, whereby the coefficients in the series appear to change discontinuously. We will show how understanding Stokes' phenomenon is the key which allows us to determine the qualitative and quantitative behaviour of the solution in many practical problems. Examples will be drawn from the areas of surface waves on fluids, crystal growth, dislocation dynamics, and Hele-Shaw flow.

  • Mathematics Teaching and Learning on 16 November 2023 at 16:00

    Speaker: Martyn Parker (Warwick Statistics)

    Title: QAA subject benchmark statements for Mathematics, Statistics and Operational Research

    Abstract: In this talk I will discuss the Subject Benchmarking Statements (SBS) for Mathematical Sciences and Operational Research (MSOR). Subject Benchmark Statements describe the nature of study and the academic standards expected of graduates in MSOR. They show what graduates might reasonably be expected to know, do and understand at the end of their studies.
    The benchmarking statements are creating and updated by members of the MSOR community under the guidance of the QAA (Quality Assurance Agency for Higher Education). The most recent MSOR updates took place over a period of approximately 1 year and were published in 2023.

    I will discuss the interface between the HE regulator, the Office for Students (OfS), and the SBS statements. In particular, how the current OfS regulatory framework and requirements interact with SBS updates.

  • Statistical Mechanics on 16 November 2023 at 16:00

    Speaker: Francesco Mezzadri (University of Bristol)

    Title: A model for complex $\beta$ ensembles of random matrices

    Abstract: We introduce the first random matrix model of a complex $\beta$-ensemble. The matrices are tridiagonal and can be thought of as the non-Hermitian analogue of the Hermite $\beta$-ensembles discovered by Dumitriu and Edelman (2002). This is work in collaboration with Henry Taylor.

  • Analysis on 16 November 2023 at 16:00

    Speaker: Carlo Gasparetto (Pisa)

    Title: A Viscosity and Monotonicity Approach to Epsilon-Regularity

    Abstract: Allard’s theorem states that a minimal surface that is close enough to a plane coincides with the graph of a smooth function which enjoys suitable a-priori estimates. In this talk I show how to prove this result and its parabolic counterpart by exploiting viscosity techniques and a weighted monotonicity formula. Based on a joint work with G. De Philippis and F. Schulze.

  • Geometry and Topology on 16 November 2023 at 14:00

    Speaker: Rob Kropholler (Warwick)

    Title: The landscape of Dehn functions

    Abstract: -

  • Probability Theory on 15 November 2023 at 16:00

    Speaker: Luisa Andreis (Politecnico di Milano)

    Title: Spatial coagulation processes: large deviations and phase transitions

    Abstract: We consider a spatial Markovian particle system with pairwise coagulation: after independent exponential random times, particle pairs merge into a single particle, and their masses are summed. We derive an explicit formula for the joint distribution of the particle configuration at a given fixed time, which involves the binary trees describing the history of how each of the particles has been formed via coagulations. While usually these processes are studied with the help of PDE (generalisation of the well-known Smoluchowski equation), our approach comes from statistical mechanics. The description is indeed in terms of a reference process, a Poisson point process of point group distributions, where each of the histories is an independent tree, and the non-coagulation between any two of them induces an exponential pair-interaction. Based on this formula, we can give a (conditional) large-deviation principle for the joint distribution of the particle histories in the limit of many particles with explicit identification of the rate function. We characterise its minimizer(s) and give criteria for the occurrence of a gelation phase transition, i.e., a loss of mass in the limiting configuration. This talk is based on an ongoing joint work with W. König, H. Langhammer and R.I.A. Patterson (WIAS Berlin).

  • Ergodic Theory and Dynamical Systems on 14 November 2023 at 14:00

    Speaker: Matteo Tanzi (Kings College London)

    Title: Uniformly Expanding Coupled Maps: Self-Consistent Transfer Operators and Propagation of Chaos

    Abstract: TBA

  • Algebra on 13 November 2023 at 17:00

    Speaker: Peiran Wu (University of St Andrews)

    Title: Irredundant bases for the symmetric and alternating groups

    Abstract: An irredundant base of a group G acting faithfully on a finite set Γ is a sequence of points in Γ that produces a strictly descending chain of pointwise stabiliser subgroups in G, terminating at the trivial subgroup. I will give an overview of known results about the irredundant base size, before focusing on the case where G is the symmetric or alternating group of degree n with a non-standard primitive action. It was proved in 2011 that an irredundant base of size 2 exists for such an action in all but finitely many cases. I will speak about the recent work by me and my supervisor, where we have shown that the maximum size of an irredundant base for the action is O(√n) and in most cases O((log n)^2). These upper bounds are also best possible in their respective cases, and I will present some interesting examples constructed to prove their optimality.

  • Junior Number Theory on 13 November 2023 at 11:00

    Speaker: Amelia Livingston (University College London)

    Title: The Langlands correspondence for algebraic tori

    Abstract: This talk is an introduction to the easiest case of the Langlands correspondence. The correspondence "for $GL_1$" reduces to class field theory, and using elementary techniques from group cohomology, Langlands extended this from $GL_1$ to any algebraic torus. This setting involves no analysis, and provides a friendly first look at a couple of the objects involved in more general cases of the Langlands program.

  • Colloquium on 10 November 2023 at 16:00

    Speaker: Rob Hollingworth, Tom Montenegro-Johnson, Randa Herzallah (Warwick)

    Title: Impact - what it is, how it's done, and why it's good for you

    Abstract: Impact is about how academics reach out to the wider world. This can arise through working with industry, local or national agencies, or through public understanding and involvement. The success of the Maths Institute in the next REF assessment will be critically dependent on both specific Impact Case Studies and the general role of impact within the department.

    This three-part talk will explain what Impact is and what it means to the Maths Institute, it will inform about how you can get involved with Impact activities and what this means for you, and it will give one (or two depending on time) examples of Impact Case studies. The colloquium will specifically address topics of relevance for those at the start of their own impact journey (or who may not even know how impactful their activities could be!).

  • Combinatorics on 10 November 2023 at 14:00

    Speaker: Tom Johnston (Bristol)

    Title: Shotgun assembly of random graphs

    Abstract: TBA

  • Statistical Mechanics on 09 November 2023 at 16:00

    Speaker: Laurent Thomann (Université de Lorraine)

    Title: Almost sure scattering for the one dimensional nonlinear Schrödinger equation

    Abstract: We exhibit measure on the space of initial data for which we describe the non trivial evolution by the linear Schrödinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. Actually, we give precise (and optimal) bounds on the Radon-Nikodym derivatives of these measures with respect to each other and we characterise their L^p regularity. We deduce from this precise description the global well-posedness of the equation for p>1 and scattering for p>3. This is joint work with Nicolas Burq.

  • Geometry and Topology on 09 November 2023 at 14:00

    Speaker: Monika Kudlinska (University of Oxford)

    Title: Subgroup separability in 3-manifold and free-by-cyclic groups

    Abstract: A group G is said to be subgroup separable if every finitely generated subgroup of G is the intersection of finite index subgroups. It is known that a fundamental group of a compact, irreducible, closed 3-manifold M is subgroup separable if and only if M is geometric. We will discuss the problem of subgroup separability in free-by-cyclic groups by drawing a parallel between free-by-cyclic and 3-manifold groups. Time permitting, we will discuss how to extend these ideas to find non-separable subgroups in random groups

  • Probability Theory on 08 November 2023 at 16:00

    Speaker: Balint Toth (University of Bristol)

    Title: (Towards an) Invariance Principle for the Random Lorentz Gas under Weak Coupling Limit Beyond the Kinetic Time Scale

    Abstract: Kesten-Papanicolaou (1980) proved that in the weak coupling limit the random Lorentz-gas process with soft scatterers converges to the Spherical Langevin Process. Under a second, diffusive limit the spatial component of the Spherical Langevin Process converges to Brownian motion. Komorowski-Ryzhik (2006) proved that combining the weak coupling and diffusive limits, the Brownian motion is obtained, at least for a time horizon slightly beyond the kinetic time-scale. We attempt to extend this last result robustly for time scales way beyond the kinetic one. (Work in progress.)

  • Algebraic Geometry on 08 November 2023 at 15:00

    Speaker: Shengyuan Huang (Birmingham)

    Title: The orbifold Hochschild product for Fermat hypersurface

    Abstract: For a smooth scheme X, the Hochschild cohomology of X is isomorphic to the cohomology of polyvector fields as algebras. This result is claimed by Kontsevich and then proved by Calaque and Van den Bergh. In this talk, I will present my recent progress with Andrei Caldararu and Kai Xu in generalising the result above to orbifolds.

    In the projective spaces, one can consider the Fermat hypersurfaces with natural group actions. These are the main examples that we focus on in this talk. If we further assume that the hypersurface is Calabi-Yau, we prove the algebra isomorphism between its Hochschild cohomology and polyvector fields.

  • Algebraic Topology on 07 November 2023 at 16:00

    Speaker: Scott Balchin (Queen's University Belfast)

    Title: A jaunt through the tensor-triangular geometry of rational G spectra for G profinite or compact Lie

    Abstract: TBA

  • Algebraic Geometry on 07 November 2023 at 15:00

    Speaker: Alicia Dickenstein (University of Buenos Aires)

    Title: Iterated and mixed discriminants

    Abstract: Classical work by Salmon and Bromwich classified singular intersections of two quadric surfaces. The basic idea of these results was already pursued by Cayley in connection with tangent intersections of conics in the plane and used by Schafli for the study of hyperdeterminants. More recently, the problem has been revisited with similar tools in the context of geometric modeling and a generalization to the case of two higher dimensional quadric hypersurfaces was given by Ottaviani. In joint work with Sandra di Rocco and Ralph Morrison, we propose and study a generalization of this question for systems of Laurent polynomials with support on a fixed point configuration. In the non-defective case, the closure of the locus of coefficients giving a non-degenerate multiple root of the system is defined by a polynomial called the mixed discriminant. We define a related polynomial called the multivariate iterated discriminant. This iterated discriminant is easier to compute and we prove that it is always divisible by the mixed discriminant. We show that tangent intersections can be computed via iteration if and only if the singular locus of a corresponding dual variety has sufficiently high codimension. We also study when point configurations corresponding to Segre-Veronese varieties and to the lattice points of planar smooth polygons, have their iterated discriminant equal to their mixed discriminant.

  • Partial Differential Equations and their Applications on 07 November 2023 at 12:00

    Speaker: Elaine Crooks (Swansea University)

    Title: TBA

    Abstract: TBA

  • Algebra on 06 November 2023 at 17:00

    Speaker: Martin van Beek (University of Manchester)

    Title: Exotic Fusion Systems Related to Sporadic Simple Groups

    Abstract: Fusion systems offer a way to examine and express properties of the p-conjugacy of elements in finite groups. However, not every fusion system may be constructed from a finite group in an appropriate way. This gives rise to exotic fusion systems. An important research direction involves the study of the behaviour of exotic fusion systems (in particular at odd primes).

    In this talk, we describe several exotic fusion systems related to the sporadic simple groups at odd primes. More generally, we classify saturated fusion systems supported on Sylow 3-subgroups of the Conway group Co1 and the Thompson group F3, and a Sylow 5-subgroup of the Monster M, as well as a particular maximal subgroup of the latter two p-groups. This work is supported by computations in MAGMA.

  • Junior Number Theory on 06 November 2023 at 11:00

    Speaker: Alexandros Konstantinou (University College London)

    Title: Unveiling the power of isogenies: From Galois theory to the Birch and Swinnerton-Dyer conjecture

    Abstract: In this talk, we have a two-fold aim. Firstly, we illustrate a method for constructing isogenies using basic Galois theory and representation theory of finite groups. By exploiting the isogenies thus constructed, we shift our focus to the second aspect of our talk: the investigation of ranks of Jacobians with emphasis on predictions made by the Birch and Swinnerton-Dyer conjecture. Finally, we showcase the utility of our approach for studying ranks through various applications. These include a unified framework for studying classical isogenies and ranks, as well as a new proof for the parity conjecture for elliptic curves defined over number fields. This is joint work with V. Dokchitser, H. Green and A. Morgan.

  • Colloquium on 03 November 2023 at 16:00

    Speaker: Henna Koivusalo (Bristol)

    Title: The tales of aperiodic order

    Abstract: Aperiodic order is at most loosely term to describe discrete point sets (or tilings), which have no translational period but feature some signs of long-range organisation. The tale of the study of aperiodic order is fundamentally intertwined with physics, but as a field of mathematics also lies in the deep shadow of logic. My take on this story will cover the past 60-odd years in approximate chronological order, beginning with first examples of aperiodic tilesets, the Nobel prize-winning discovery of quasicrystal materials, and the quest to find wild quasicrystals, and ending with the unbelievable story, from just earlier this year, of finding the first aperiodic monotile.

    Time permitting, I will explain in further detail some results on my favourite method for producing aperiodic order, the cut and project sets, which are obtained by taking an irrational slice through a lattice and projecting it to a lower dimensional subspace. The definition of cut and project sets allows for many interpretations and generalisations, and they can naturally be studied in the context of dynamical systems, discrete geometry, harmonic analysis, or Diophantine approximation, for example, depending on one's own tastes and interests.

  • Statistical Mechanics on 02 November 2023 at 16:00

    Speaker: Igor Wigman (King's College London)

    Title: Almost sure GOE fluctuations of energy levels for hyperbolic surfaces of high genus

    Abstract: Title:
    This talk is based on a joint work with Zeev Rudnick.
    We study the variance of a linear statistic of the Laplace eigenvalues on a hyperbolic surface, when the surface varies over the moduli space of all surfaces of fixed genus, sampled at random according to the Weil-Petersson measure. The ensemble variance of the linear statistic was recently shown to coincide with that of the corresponding statistic in the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, in the double limit of first taking large genus and then shrinking size of the energy window. We show that in this same limit, the energy variance for a typical surface is close to the GOE result, a feature called "ergodicity" in the random matrix theory literature.

  • Analysis on 02 November 2023 at 16:00

    Speaker: Dario Prandi (Paris-Saclay)

    Title: Weyl's law for singular Riemannian manifolds

    Abstract: We will discuss some new results relating to the asymptotic growth of the eigenvalues of the Laplace-Beltrami operator on Riemannian manifolds. In particular, we will focus on a singular setting, where all geometrical invariants appearing in classical spectral asymptotics are unbounded, and the total volume can be infinite. In this setting, under suitable assumptions on the curvature blow-up, we will show how the singularity influences the Weyl's asymptotics and how to construct singular Riemannian metrics with prescribed non-classical Weyl's law. A key tool in our arguments is a new quantitative estimate for the remainder of the heat trace and the Weyl's function on Riemannian manifolds, which is of independent interest.
    This is a joint work with Yacine Chitour (Univ. Paris-Saclay, France) and Luca Rizzi (SISSA, Trieste, Italy).

  • Mathematics Teaching and Learning on 02 November 2023 at 16:00

    Speaker: Sue Johnston-Wilder (Warwick Education Studies)

    Title: 90% of jobs need maths 30% of the population has maths anxiety: what can we do?

    Abstract: I will seek to raise awareness of maths anxiety, how it affects an average 30% of the people around you at the University of Warwick and in the wider community and how you can begin to become part of the solution.
    I will introduce notions of prior harm, psychological safety and resilience applied to learning mathematics and share the mathematical resilience toolkit, showing how it is being adopted in several countries around the world.

  • Geometry and Topology on 02 November 2023 at 14:00

    Speaker: Adele Jackson (University of Oxford)

    Title: Algorithms for Seifert fibered spaces

    Abstract: Given two mathematical objects, the most basic question is whether they are the same. We will discuss this question for triangulations of three-manifolds. In practice there is fast software to answer this question and theoretically the problem is known to be decidable. However, our understanding is limited and known theoretical algorithms could have extremely long run-times. I will describe a programme to show that the 3-manifold homeomorphism problem is in the complexity class NP, and discuss the important sub-case of Seifert fibered spaces.

  • Probability Theory on 01 November 2023 at 16:00

    Speaker: Erlend Grong (University of Bergen)

    Title: Sub-Riemannian geometry, most probable paths and transformations.

    Abstract: Hello Everyone,

    This week's Probability Seminar speaker will be Erlend Grong from the University of Bergen. The talk will take place in B3.03 on Wednesday, November 1, 16-17. The title and abstract, as well as the MS Teams link for the talk are given below.

    Best regards,

    Vedran and Giuseppe
    Doing statistics on a Riemannian manifold becomes very complicated for the reason that we lack tools to define such things as mean and variance. Using the Riemannian distance, we can define a mean know as the Fréchet mean, but this gives no concept of asymmetry, also known as anisotropy. We introduce an alternative definition of mean called the diffusion mean, which is able to both give a mean and the analogue of a covariance matrix for a dataset on a Riemannian manifolds.

    Surprisingly, computing this mean and covariance is related to sub-Riemannian geometry. We describe how sub-Riemannian geometry can be applied in this setting, and mention some finite dimensional and infinite-dimensional applications.

    The results are part of joint work with Stefan Sommer (Copenhagen, Denmark).

  • Algebraic Geometry on 01 November 2023 at 15:00

    Speaker: Marvin Anas Hahn (Trinity College Dublin)

    Title: Mustafin degenerations of syzygy bundles

    Abstract: Mustafin varieties are degenerations of projective spaces, which are induced by point configurations in a Bruhat Tits building. In this talk, we use these degenerations to construct certain models of plane curves. Motivated by recent advances towards a p-adic Narasimhan—Sehsadri theorem, we then use these models to construct families of syzygy bundles which admit strongly semistable reduction. This talk is based on a joint work with Annette Werner.

  • Algebraic Topology on 31 October 2023 at 16:00

    Speaker: Bastiaan Cnossen (University of Regensburg)

    Title: Genuine sheaves on differentiable stacks

    Abstract: Cohomology theories for equivariant spaces typically only depend on the associated quotient stacks X//G. It would thus be desirable to have a flexible framework for cohomology theories for stacks. In this talk, I will present such a framework, following ideas from motivic homotopy theory. The main result is a version of relative Poincaré duality for differentiable stacks, which generalizes Poincaré duality for smooth manifolds, Atyah duality for equivariant manifolds, and the Wirthmüller isomorphism in equivariant stable homotopy theory.

  • Analysis on 31 October 2023 at 15:15

    Speaker: Or Hershkovits (Hebrew University of Jerusalem)

    Title: Hopf Lemma for Brakke Flows

    Abstract: In this talk, I will describe a variant of the classical Hopf lemma, that allows to show regularity (and non-vanishing angle) at (boundary) intersection points of two Brakke flows which are disjoint in a half of a parabolic ball.
    This Hopf Lemma can be used in the moving plane method, allowing to prove symmetry and regularity in tandem.
    This is based on a joint work with Kyeongsu Choi, Robert Haslhofer and Brian White.

  • Ergodic Theory and Dynamical Systems on 31 October 2023 at 14:00

    Speaker: Francois Ledrappier (Jussieu)

    Title: Dimension of limit sets for Anosov representations

    Abstract: We consider the action of discrete finitely generated subgroup of matrices on the space of flags. Under hyperbolicity and non-degeneracy conditions, we can estimate the dimension of minimal invariant sets. The proofs use properties of random walks on the group. This is joint work with Pablo Lessa (Montevideo).

  • Partial Differential Equations and their Applications on 31 October 2023 at 12:00

    Speaker: Markus Schmidtschen (TU Dresden)

    Title: TBA

    Abstract: TBA

  • Algebra on 30 October 2023 at 17:00

    Speaker: Lucia Morotti (University of York)

    Title: Self-extensions for irreducible representations of symmetric groups

    Abstract: It has been conjectured that irreducible representations of symmetric groups have no non-trivial self-extensions in characteristic different from 2, that is that the only modules V with 2 composition factors isomorphic to D for some irreducible module D and no other composition factor are those of the form D + D. This conjecture has been proved for some classes of modules by Kleshchev-Sheth and Kleshchev-Nakano. I will present joint results with Harry Geranios and Sasha Kleshchev and current work with Harry Geranios considering reduction results and generalisations of the above mentioned papers.

  • Junior Analysis and Probability Seminar on 30 October 2023 at 15:00

    Speaker: Phoebe Valentine (Warwick)

    Title: Characterising 1-rectifiability via connected tangents

    Abstract: A central concept in geometric measure theory is that of rectifiability. A set is called n-rectifiable if it can be covered almost everywhere by images of Lipschitz maps and is purely n-unrectifiable if its intersection with any rectifiable set has 0 measure. In this talk, we will start by motivating why tangents are a natural lens through which to view rectifiability. Indeed, the theory of Euclidean tangents has been well developed for some time, and in the case of 1-rectifiability we will discuss a geometric proof of a well known Euclidean linear approximability result. We will depend heavily on the inherent "gappiness" of purely 1-unrectifiable sets, as quantified by Besicovitch in 1938. We will then consider the problems in generalising this argument to hold in arbitrary metric spaces and have a gentle introduction to the theory of metric tangents. Finally, we will see how the construction of Besicovitch may be strengthened to show that the existence of connected metric tangents implies 1-rectifiability.

  • Junior Number Theory on 30 October 2023 at 11:00

    Speaker: Maryam Nowroozi (University of Warwick)

    Title: Perfect Powers in Elliptic Divisibility Sequences

    Abstract: The problem of determining all perfect powers in a sequence has always been interesting to mathematicians. The problem we are interested in is to prove that there are finitely many perfect powers in elliptic divisibility sequences. Abdulmuhsin Alfaraj proved that there are finitely many perfect powers in elliptic divisibility sequences generated by a non-integral point on elliptic curves of the from y^2=x(x^2+b), where $b$ is any positive integer. The main goal of our project is to generalize this result for elliptic divisibility sequences generated by any non-integral point on all elliptic curves y^2=x^3+ax^2+bx+c. This is a joint work with Samir Siksek.

  • Colloquium on 27 October 2023 at 16:00

    Speaker: Juergen Branke (Warwick Business School)

    Title: Bayesian Optimisation and Common Random Numbers

    Abstract: Bayesian optimisation algorithms are global optimisation algorithms for expensive-to-evaluate black-box problems, as they often occur when a solution candidate needs to be evaluated using simulation or physical experiments. They build a surrogate model, usually a Gaussian Process, based on the data collected to far, and then use this surrogate model to decide which new solution candidate to evaluate in the next iteration to maximise the value of information gained.
    This makes the algorithm very sample efficient, and in recent years, Bayesian optimisation has become very popular in particular for machine learning hyperparameter tuning and engineering design.

    This talk will start with a general introduction to Bayesian optimisation, discussing some of the key open challenges. The second part will then focus on how to effectively exploit common random numbers. Many objective functions (e.g., stochastic simulators) require a random number seed as input. By explicitly reusing a seed, the algorithm can compare two or more solutions under the same randomly generated scenario, such as a common customer stream in a job shop problem, or the same random partition of training data into training and validation set for a machine learning algorithm. Our proposed Knowledge Gradient for Common Random Numbers exploits this and iteratively determines a combination of solution candidate and random seed to evaluate next.

  • Combinatorics on 27 October 2023 at 14:00

    Speaker: Yani Pehova (LSE)

    Title: The Erdős-Rothschild problem for dichromatic triangles

    Abstract: TBA

  • Mathematics Teaching and Learning on 26 October 2023 at 16:00

    Speaker: Helena Verrill (Warwick)

    Title: Mathematics games developed by students taking the IATL course on serious table-top games

    Abstract: I will discuss the use of games in teaching mathematics. This is particularly focused on the games developed by students taking the serious table-top games IATL module (IL031/131). I will bring three of these games to the talk, and talk about how these games are played, and consider how students' learning can be impacted by use of games. I will mention how I have occasionally used games or puzzles in my own teaching.

  • Statistical Mechanics on 26 October 2023 at 16:00

    Speaker: Sabine Bögli (University of Durham)

    Title: On the discrete eigenvalues of Schrödinger operators with complex potentials

    Abstract: In this talk I shall present constructions of Schrödinger operators with complex-valued potentials whose spectra exhibit interesting properties. One example shows that for sufficiently large p, the discrete eigenvalues need not be bounded in modulus by the $L^p$ norm of the potential. This is a counterexample to the Laptev-Safronov conjecture (Comm. Math. Phys. 2009). Another construction proves optimality (in some sense) of generalisations of Lieb-Thirring inequalities to the non-selfadjoint case - thus giving us information about the accumulation rate of thediscrete eigenvalues to the essential spectrum. This talk is based on joint works with Jean-Claude Cuenin (Loughborough) and Frantisek Stampach (Prague).

  • Analysis on 26 October 2023 at 16:00

    Speaker: Denis Marti (Freibourg)

    Title: Geometric and Analytic Structures on Metric Spaces Homeomorphic to a Manifold

    Abstract: We explore geometric and analytic aspects of metric spaces homeomorphic to a closed, oriented manifold. We show that such spaces (which are sometimes called metric manifolds) admit a non-trivial integral current without boundary, provided they satisfy some weak assumptions. The existence of such an object should be thought of as an analytic analogue of the fundamental class of the space and can also be interpreted as giving a way to make sense of Stokes' theorem in this setting. We use this to establish (relative) isoperimetric inequalities in metric n-manifolds that are Ahlfors n-regular and linearly locally contractible. As an application, we obtain a short and conceptually simple proof of a deep theorem of Semmes about the validity of Poincaré inequalities in these spaces. We furthermore present applications to the problem of Lipschitz-volume rigidity in the case of metric manifolds. Based on joint work with G. Basso and S. Wenger.

  • Probability Theory on 25 October 2023 at 16:00

    Speaker: Julien Sabin (University of Rennes)

    Title: Nonlinear Hartree dynamics for density matrices

    Abstract: In this talk I will review results concerning the mean-field dynamics of fermionic quantum particles governed by the nonlinear Hartree equation. The particularity of this equation is that its unknown is a bounded operator on a Hilbert space, rather than a (wave)function as is the case for most PDEs. I will explain how to deal with setting, with a focus on the large time behaviour of solutions.

  • Algebraic Geometry on 25 October 2023 at 15:00

    Speaker: Farhad Babaee (Bristol)

    Title: Complex tropical currents

    Abstract: In this talk, I will recall basic ideas in tropical geometry and the theory of positive currents, and I will discuss why exploring the interactions of these two domains is natural and useful.

  • Analysis on 24 October 2023 at 16:00

    Speaker: Or Hershkovits (Hebrew University of Jerusalem)

    Title: Mean curvature flow in spaces with positive cosmological constant

    Abstract: In this talk, I will describe an approach of using Lorentzian mean curvature flow (MCF) to probe cosmologies satisfying the Einstein equation with positive cosmological constant with matter obeying the strong energy condition.
    Assuming surface symmetry, I will explain how such flow converges, in some sense, to the standard constant mean curvature (CMC) slicing of de Sitter space, implying in particular, that such cosmologies are themselves asymptotic to de Sitter space.
    I will then illustrate a condition, natural in the above context, such that any local graphical mean curvature flow (without symmetry) in de Sitter space satisfying that condition converges to the standard CMC slicing of the entire de Sitter space.
    Effort will be made to make the talk accessible to the wide mathematical audience. This is based on a joint work with Creminelli, Senatore and Vasy, and on a joint work with Senatore.

  • Algebraic Topology on 24 October 2023 at 16:00

    Speaker: Itamar Mor (Queen Mary University of London)

    Title: Profinite Galois descent in K(n)-local homotopy theory

    Abstract: Using condensed mathematics, I give a construction of the K(n)-local E_n-Adams spectral sequence as a HFPSS for the continuous action of the Morava stabiliser group. A modified version gives a spectral sequence computing the Picard and Brauer groups of K(n)-local spectra.

  • Ergodic Theory and Dynamical Systems on 24 October 2023 at 14:00

    Speaker: Irving Calderon (Durham University)

    Title: Explicit spectral gap for Schottky subgroups of $\mathrm{SL} (2, \mathbb{Z})$

    Abstract: TBA

  • Partial Differential Equations and their Applications on 24 October 2023 at 12:00

    Speaker: Alexandra Holzinger (University of Oxford)

    Title: TBA

    Abstract: TBA

  • Algebra on 23 October 2023 at 17:00

    Speaker: Veronica Kelsey (University of Manchester)

    Title: Nice and Nasty Numerical Invariants

    Abstract: For a permutation group G we can define the maximal irredundant base size and the relational complexity, denoted I(G) and RC(G) respectively. Roughly speaking the maximal irredundant base size is the size of the “worst” base for G, and relational complexity is a measure of when a local property extends to a global one.

    We begin by defining these numerical invariants and then cover some examples which illustrate the “nice” behaviour of I(G) and the “nasty” behaviour of RC(G). We’ll then skim through the proof of the relational complexity of a family of groups.

  • Junior Analysis and Probability Seminar on 23 October 2023 at 15:00

    Speaker: Federico Bertacco (Imperial)

    Title: Scaling limits of planar maps under the Smith embedding

    Abstract: Over the past few decades, there has been significant progress in the study of scaling limits of random planar maps. In this talk, I will provide motivation for this problem and then focus on the scaling limits of (random) planar maps under the Smith embedding. This embedding is described by a tiling of a finite cylinder by rectangles, where each edge of the map corresponds to a rectangle, and each vertex corresponds to a horizontal segment. I will argue that when considering a sequence of finite planar maps embedded in an infinite cylinder and satisfying a suitable invariance principle assumption, the a priori embedding is close to an affine transformation of the Smith embedding at larger scales. By applying this result, I will prove that the Smith embeddings of mated-CRT maps with the sphere topology converge to LQG. This is based on joint work with Ewain Gwynne and Scott Sheffield.

  • Junior Number Theory on 23 October 2023 at 11:00

    Speaker: Arshay Sheth (University of Warwick)

    Title: The Hilbert-Polya dream: finding determinant expressions of zeta functions

    Abstract: The Hilbert-Polya dream, which seeks to express the Riemann zeta function as a characteristic polynomial of an operator on a Hilbert space, is one possible approach to prove the Riemann Hypothesis. While this approach has never been successfully carried out, its core principle- finding determinant expression of zeta functions- has manifested itself in several different areas of number theory in the last century. In this talk, we will attempt to give a panoramic survey of the Hilbert-Polya dream.

  • Colloquium on 20 October 2023 at 16:00

    Speaker: Colva Roney-Dougal (St Andrews)

    Title: Counting permutation groups

    Abstract: What does a random permutation group look like? This talk will start with a brief survey of how we might go about counting subgroups of the symmetric group S_n, and talk about what is known about "most" subgroups.

    To tackle the general problem, it would clearly be helpful to know how many subgroups there are. An elementary argument gives that there are at least 2^{n^2/16} subgroups, and it was conjectured by Pyber in 1993 that up to lower order error terms this is also an upper bound. This talk will present an answer to Pyber's conjecture.

    This is joint work with Warwick’s own Gareth Tracey.

  • Combinatorics on 20 October 2023 at 14:00

    Speaker: António Girão (Oxford)

    Title: On induced C4-free graphs with high average degree

    Abstract: TBA

  • Analysis on 19 October 2023 at 16:00

    Speaker: Andrea Mondino (Oxford)

    Title: Lorentzian Ricci bounds and Einstein’s theory of gravity in a non smooth setting: an optimal transport approach

    Abstract: Optimal transport tools have been extremely powerful to study Ricci curvature, in particular Ricci lower bounds in the non-smooth setting of metric measure spaces (which can be been thought as "non-smooth Riemannian manifolds”).
    Since the geometric framework of general relativity is the one of Lorentzian manifolds (or space-times), and the Ricci curvature plays a prominent role in Einstein’s theory of gravity, it is natural to expect that optimal transport tools can be useful also in this setting.
    The goal of the talk is to introduce the topic and to report on recent progress.
    More precisely: After recalling the general setting of Lorentzian pre-length spaces (introduced by Kunzinger-Sämann, after Kronheimer-Penrose), I will discuss some basics of optimal transport theory thereof in order to define "timelike Ricci curvature and dimension bounds” for a possibly non-smooth Lorentzian space. Some cases of such bounds have remarkable physical interpretations (like the attractive nature of gravity) and can be used to give a characterisation of the Einstein’s equations for a non-smooth space and to establish new isoperimetric-type inequalities in Lorentzian signature. Based partly on joint work with S. Suhr and partly on joint work with F. Cavalletti.

  • Geometry and Topology on 19 October 2023 at 14:00

    Speaker: Clément Legrand (LaBRI)

    Title: Reconfiguration of square-tiled surfaces

    Abstract: A square-tiled surface is a special case of a quadrangulation of a surface, that can be encoded as a pair of permutations in \(S_n \times S_n\) that generates a transitive subgroup of \(S_n\). Square-tiled surfaces can be classified into different strata according to the total angles around their conical singularities. Among other parameters, strata fix the genus and the size of the quadrangulation. Generating a random square-tiled surface in a fixed stratum is a widely open question. We propose a Markov chain approach using "shearing moves": a natural reconfiguration operation preserving the stratum of a square-tiled surface. In a subset of strata, we prove that this Markov chain is irreducible and has diameter \(O(n^2)\), where \(n\) is the number of squares in the quadrangulation.

  • Probability Theory on 18 October 2023 at 16:00

    Speaker: Anna Maltsev (Queen Mary University London)

    Title: Bulk Universality for Complex Non-Hermitian Gauss-divisible Matrices

    Abstract: In this talk I will discuss universality of the k-point correlation function for Gauss divisible non-Hermitian matrices. We consider NxN matrices with centred, independent and identically distributed complex entries that have a small Gaussian component. We prove that the bulk correlation functions are universal in the large N limit using Householder transformations, supersymmetry, and Laplace method. Assuming the entries have finite moments and are supported on at least three points, the Gaussian component is removed by the four moment theorem. This is based on joint work with Mohammed Osman.

  • Algebraic Geometry on 18 October 2023 at 15:00

    Speaker: Vaidehee Thatte (KCL)

    Title: Understanding the Defect via Ramification Theory

    Abstract: Classically, the degree of a finite Galois extension of complete discrete valuation fields equals the product of two invariants measuring the change in the valuation (ramification index) and the change in the residue field (inertia degree). More generally, there is a third factor - the ‘defect’. For example, we can have a degree p extension with trivial extensions of the value group and the residue field. The defect is not yet well understood and remains the main obstruction to several long-standing open problems in positive residue characteristic (e.g., resolution of singularities). The primary reason is, roughly speaking, that the classical strategy of "objects become nicer after finitely many blow-ups" fails when the defect is non-trivial. We are thrown into an infinite loop.

    In this talk, I will discuss how techniques in arithmetic algebraic geometry, number theory, and valuation theory together can help us understand the defect and deal with the obstacles it creates. In particular, I will present a generalization of the classical invariants of ramification theory, a generalization of Gabber-Ramero's work on 'filtered union', and some recent work (joint with K. Kato) on upper ramification filtration in the general case. Any necessary background in these areas will be covered via examples and rough (practical) definitions.

  • Algebraic Topology on 17 October 2023 at 16:00

    Speaker: Daniel Kasprowski (University of Southampton)

    Title: Stable equivalence relations of 4-manifolds

    Abstract: Kreck’s modified surgery gives an approach to classify 2n-manifolds up to stable diffeomorphism, i.e., up to a connected sum with copies of S^n x S^n. In dimension 4, we use a combination of modified and classical surgery to compare the stable diffeomorphism classification with other stable equivalence relations. Most importantly, we consider homotopy equivalence up to connected sum with copies of S^n x S^n. This is joint work with John Nicholson and Simona Veselá.

  • Ergodic Theory and Dynamical Systems on 17 October 2023 at 14:00

    Speaker: Roberto Castorrini (University of Pisa)

    Title: Transfer operators, spectral gap and thermodynamic formalism: from smooth to discontinuous dynamical systems

    Abstract: I will provide a brief overview of the functional approach used to analyze the statistical properties of a dynamical system, focusing on its main objective: determining a suitable Banach space that minimizes the 'non-compact' (essential) part of the spectrum of the associated transfer operator. Optimal outcomes regarding the essential spectrum for smooth hyperbolic dynamical systems are attained by employing thermodynamic formalism techniques, which utilize a variational expression for subadditive topological pressure. Drawing from a recent joint work with V. Baladi, I will illustrate similar results for systems with discontinuities, particularly piecewise expanding maps in finite dimensions.

  • Algebra on 16 October 2023 at 17:00

    Speaker: Stacey Law (University of Birmingham)

    Title: Sylow branching coefficients for symmetric groups

    Abstract: One of the key questions in the representation theory of finite groups is to understand the relationship between the characters of a finite group G and its local subgroups. Sylow branching coefficients describe the restriction of irreducible characters of G to a Sylow subgroup P of G, and have been recently shown to characterise structural properties such as the normality of P in G. In this talk, we will discuss and present some new results on Sylow branching coefficients for symmetric groups.

  • Junior Analysis and Probability Seminar on 16 October 2023 at 15:00

    Speaker: Julian Weigt (Warwick)

    Title: Endpoint regularity bounds for maximal operators in higher dimensions

    Abstract: The classical Hardy-Littlewood maximal function theorem states that maximal operators are bounded on L^p(ℝ^n) if and only if p>1. In 1997 Juha Kinnunen proved that for p>1 also the gradient of a maximal function is bounded on L^p(ℝ^n). It is an open question in the endpoint p=1. In one dimension this endpoint gradient bound is known to hold for most maximal operators due to Tanaka, Kurka and many others.
    We prove the endpoint gradient bound in all dimensions for the maximal operator that averages over uncentered cubes with any orientation. For the uncentered Hardy-Littlewood maximal operator we can prove the endpoint Sobolev bound only in the case of characteristic functions since some of our arguments only work for cubes and not for balls. Moreover, we prove the corresponding endpoint Sobolev bound for the fractional centered and uncentered Hardy-Littlewood maximal functions.
    The key arguments are of geometric nature and rely on the coarea formula, the relative isoperimetric inequality and covering lemmas.

  • Junior Number Theory on 16 October 2023 at 11:00

    Speaker: Isabel (Izzy) Rendell (King's College London)

    Title: Rational points on modular curves

    Abstract: The problem of finding rational points on modular curves is of great interest in number theory and arithmetic geometry, with many different methods in use in the subject. This will be an introductory talk where will see some key related theorems due to Faltings, Coleman and Mazur. I will discuss some methods for finding rational points, and how they can relate to other areas such as points on elliptic curves and the congruent number problem. Throughout the talk I will try and assume as few prerequisites as possible and demonstrate methods by examples.

  • Colloquium on 13 October 2023 at 16:00

    Speaker: Rob Silversmith (Warwick)

    Title: Counting problems in algebraic geometry

    Abstract: Choose five conic plane curves randomly. There are exactly 3264 ways to draw a sixth conic that is tangent to all five. (You may need complex numbers to see all of them.) Counting problems like this one have been studied for hundreds of years, and are part of a rich interplay between geometry and combinatorics. I will discuss a very down-to-earth class of counting problems with connections to many fields, including: string theory, rigid frameworks, polyhedral geometry, matroid theory, and cluster algebras. I will also mention some other recent developments and directions in the field.

  • Combinatorics on 13 October 2023 at 14:00

    Speaker: Abhishek Methuku (ETH)

    Title: The extremal number of cycles with all diagonals

    Abstract: TBA

  • Analysis on 12 October 2023 at 16:00

    Speaker: Pak-Yeung Chan (Warwick)

    Title: Gap Theorem for Nonnegatively Curved Manifolds

    Abstract: In this seminar, we shall discuss some recent results on the gap theorem of nonnegatively curved manifolds with small curvature in an average integral sense, which can be viewed as a Riemannian analog of the optimal gap result by Ni on Kahler manifolds. In dimension 3, we also establish a gap theorem for Ricci nonnegative manifolds with pointwise quadratic curvature decay and fast average integral curvature decay. This talk is based on a joint work with Man-Chun Lee.

  • Geometry and Topology on 12 October 2023 at 14:00

    Speaker: Mark Pengitore (University of Virginia)

    Title: Residual finiteness growth functions of surface groups with respect to characteristic quotients

    Abstract: Residual finiteness growth functions of groups have attracted much interest in recent years. These are functions that roughly measure the complexity of the finite quotients needed to separate particular group elements from the identity in terms of word length. In this talk, we study the growth rate of these functions adapted to finite characteristic quotients. One potential application of this result is towards linearity of the mapping class group

  • Probability Theory on 11 October 2023 at 16:00

    Speaker: Ellen Powell (University of Durham)

    Title: Characterising the Gaussian free field

    Abstract: I will discuss recent approaches to characterising the Gaussian free field in the plane, and in higher dimensions. This is based on joint work with Juhan Aru, Nathanael Berestycki and Gourab Ray.

  • Algebraic Geometry on 11 October 2023 at 15:00

    Speaker: Calla Tschanz (Bath)

    Title: Expanded degenerations for Hilbert schemes of points

    Abstract: Let X –> C be a projective family of surfaces over a curve with smooth general fibres and simple normal crossing singularity in the special fibre X_0. We construct a good compactification of the moduli space of relative length n zero-dimensional subschemes on X\X_0 over C\{0}. In order to produce this compactification we study expansions of the special fibre X_0 together with various GIT stability conditions, generalising the work of Gulbrandsen-Halle-Hulek who use GIT to offer an alternative approach to the work of Li-Wu for Hilbert schemes of points on simple degenerations. We construct stacks which we prove to be equivalent to the underlying stack of some choices of logarithmic Hilbert schemes produced by Maulik-Ranganathan.

  • Algebraic Topology on 10 October 2023 at 16:00

    Speaker: Özgür Bayındır (City University of London)

    Title: Algebraic K-theory of the two-periodic first Morava K-theory

    Abstract: Using a root adjunction formalism developed in an earlier work and logarithmic THH, we obtain a simplified computation of the algebraic K-theory of the complex K-theory spectrum. Furthermore, our computational methods also provide the algebraic K-theory of the two-periodic Morava K-theory spectrum of height 1.

  • Ergodic Theory and Dynamical Systems on 10 October 2023 at 14:00

    Speaker: Konstantinos Tsinas (University of Crete)

    Title: Ergodic averages along primes

    Abstract: We study the limiting behavior of multiple ergodic averages along sequences evaluated at primes. Building on the result of Frantzikinakis, Host, and Kra, who established (in the most general setting known) the corresponding convergence theorem in the case that the sequences are integer polynomials, we generalize their result to other sequences of polynomial growth. The most prominent examples in our work are the fractional powers $\lfloor{n^c}\rfloor$, where $c$ is a positive non-integer. We prove that sets of positive density contain arbitrarily long arithmetic progressions with common difference $\lfloor{ p^c} \rfloor$, where $p$ denotes a prime, along with a few more mean convergence theorems and equidistribution results in nilmanifolds. Our methods rely on a recent deep theorem of Matom\"{a}ki, Shao, Tao, and Ter\"{a}v\"{a}inen on the Gowers uniformity of the von Mangoldt function in short intervals, an approximation of our functions with polynomials with good equidistribution properties and a lifting trick that allows someone to pass from ${\mathbb Z}$-actions on a probability space to ${\mathbb R}$-actions.

  • Partial Differential Equations and their Applications on 10 October 2023 at 12:00

    Speaker: Ivan Moyano (Unv. Nice Sophia Antipolis)

    Title: Spectral Uncertainty principle for Laplace-Beltrami and Schroedinger operators

    Abstract: TBAIn this talk we review some classical and recent results
    relating the uncertainty principles for the Laplacian with the
    controllability and stabilisation of some linear PDEs. The uncertainty
    principles for the Fourier transforms state that a square integrable
    function cannot be both localised in frequency and space without being
    zero, and this can be further quantified resulting in unique
    continuation inequalities in the phase spaces. Applying these ideas to
    the spectrum of the Laplacian on a compact Riemannian manifold, Lebeau
    and Robbiano obtained their celebrated result on the exact
    controllability of the heat equation in arbitrarily small time. The
    relevant quantitative uncertainty principles known as spectral
    inequalities in the literature can be adapted to a number of different
    operators, including the Laplace-Beltami operator associated to C^1
    metrics or some Schödinger operators with long-range potentials, as we
    have shown in recent results in collaboration with Gilles Lebeau (Nice)
    and Nicolas Burq (Orsay), with a significant relaxation on the
    localisation in space. As a consequence, we obtain a number of
    corollaries on the decay rate of damped waves with rough dampings, the
    simultaneous controllability of heat equations with different boundary
    conditions and the controllability of the heat equation with rough

  • Junior Analysis and Probability Seminar on 09 October 2023 at 15:00

    Speaker: Lucas Lavoyer (Warwick)

    Title: Ricci flow from spaces with edge type conical singularities

    Abstract: In this talk, we will construct a solution to Ricci flow coming out of spaces with edge type conical singularities along a closed, embedded curve, under the additional assumption that for each point of the curve, our space is locally modelled on the product of a fixed positively curved cone and a line. We also prove curvature estimates for the solution and, for edge points, we show that the tangent flow at these points is a positively curved expanding gradient Ricci soliton solution crossed with a line.

  • Number Theory on 09 October 2023 at 11:00

    Speaker: Abdul Alfaraj (University of Bath)

    Title: On the Finiteness of Perfect Powers in Elliptic Divisibility Sequences

    Abstract: TBA

  • Colloquium on 06 October 2023 at 16:00

    Speaker: Anne-Sophie Kaloghiros (Brunel)

    Title: The Calabi problem for Fano 3-folds and applications

    Abstract: Algebraic varieties are geometric shapes given by polynomial equations. They appear naturally in pure and applied mathematics: from conic sections in geometry, to cubic curves in cryptography, or non-uniform rational basis splines in computer-aided graphic design.

    To measure distances between points on an algebraic variety, we equip it with a metric - a sophisticated dot product. This then leads to the notion of curvature, and allows us to split algebraic varieties into three basic (universal) types: negatively curved, flat and positively curved varieties. Positively curved varieties are higher dimensional generalisations of a sphere; they are called Fano varieties. Fano varieties appear frequently in applications, because they are often parametrised by rational functions.

    For an algebraic variety, the choice of a metric is never unique. One can try to find a special metric with good properties: a “canonical metric". Geometers looked for a suitable condition defining a canonical metric for the first half of the 20th century. In 1957, Calabi proposed that this canonical metric should satisfy both a certain algebraic property (being Kähler) and the Einstein (partial differential) equation. Finding which compact complex manifolds admit such a metric is the object of the Calabi problem, an area of research at the crossroads of algebraic and differential geometry that has been very active for the last decades.

    A necessary condition for the existence of such a metric is that the manifold belongs to one of the three basic universal types. Yau and Aubin/Yau confirmed Calabi's prediction and showed that manifolds with negative or flat curvature always admit a Kähler-Einstein metric in the 1970s. By contrast, the Calabi problem is much more subtle for manifolds
    with positive curvature: Fano manifolds may or may not admit a Kähler-Einstein metric.

    Research on the Calabi problem for Fano manifolds culminated in the formulation and proof of the Yau-Tian-Donaldson conjecture. This conjecture, now a theorem, states that a Fano manifold admits a Kähler-Einstein metric precisely when it satisfies a sophisticated algebro-geometric condition called K-polystability. Surprisingly, the notion of K-polystability also sheds some light on their moduli theory, that is how they behave in families ( another poorly understood aspect of their geometry).

    In this talk, I will present an overview of the Calabi problem, and present its solution in small dimension ( in which we have a classification of deformation families of smooth Fano varieties). I will discuss applications to other areas such as moduli theory.

  • Combinatorics on 06 October 2023 at 14:00

    Speaker: Michael Savery (University of Oxford)

    Title: Chromatic number is not tournament-local

    Abstract: TBA

  • Analysis on 05 October 2023 at 16:00

    Speaker: Katie Gittins (Durham University)

    Title: Heat Content of Polygonal Domains

    Abstract: Let D \subset \mathbb{R}^2 be a bounded set with polygonal boundary \partial D. We impose an initial temperature condition on \mathbb{R}^2 \setminus \partial D and can also impose boundary conditions on the edges of \partial D, such as a Dirichlet (cooling) boundary condition.
    In such a setting, it is natural to ask: how much heat is left inside D at time t? This quantity is the heat content of D. The small-time asymptotic expansions for the heat content of D encode information about the geometry of D and \partial D. Our goal is to explore how these expansions depend upon the geometry and on various combinations of initial and boundary conditions.
    We first review some of the previously known results for the small-time asymptotic expansions for the heat content of D with certain initial and boundary conditions. We then present recent results for the case where D is contained in a larger set with polygonal boundary on which a Neumann (insulating) boundary condition is imposed. The latter is based on joint work with Sam Farrington. Time-permitting, we may also discuss some geometric applications of these asymptotic expansions.

  • Geometry and Topology on 05 October 2023 at 14:00

    Speaker: Raphael Zentner (Durham University)

    Title: Rational homology ribbon cobordism is a partial order

    Abstract: Last year, Ian Agol has proved that ribbon knot concordance is a partial order on knots, a conjecture that has been open for more than three decades. His proof is beautiful and surprisingly simple. There is an analog notion of ribbon cobordism for closed 3-manifolds. We use Agol's method to show that this notion of ribbon cobordism is also a partial order within the class of irreducible 3-manifolds. This is joint work with Stefan Friedl and Filip Misev.

  • Probability Theory on 04 October 2023 at 16:00

    Speaker: Tom Klose (University of Warwick)

    Title: Large deviations for the Φ^4_3 measure via Stochastic Quantisation

    Abstract: The Φ^4_3 measure is one of the easiest non-trivial examples of a Euclidean quantum field theory (EQFT) whose rigorous construction in the 1970's has been one of the celebrated achievements of the Constructive QFT community. In recent years, progress in the field of singular stochastic PDEs, initiated by the theory of regularity structures, has allowed for a new construction of the Φ^4_3 EQFT as the invariant measure of a previously ill-posed Langevin dynamics – a strategy originally proposed by Parisi and Wu ('81) under the name Stochastic Quantisation. In this talk, I will demonstrate that the same idea also allows to transfer the large deviation principle for the Φ^4_3 dynamics, obtained by Hairer and Weber ('15), to the corresponding EQFT. Our strategy is inspired by earlier works of Sowers ('92) and Cerrai and Röckner ('05) for non-singular dynamics and potentially also applies to other EQFT measures. This talk is based on joint work with Avi Mayorcas (University of Bath).

  • Algebraic Geometry on 04 October 2023 at 15:00

    Speaker: Charles Favre (École Polytechnique)

    Title: b-divisors and dynamical applications

    Abstract: b-divisors were introduced by Shokurov in the context of the minimal model program. We shall explain how to develop a positivity theory of these objects that have remarkable applications to algebraic dynamics.

  • Ergodic Theory and Dynamical Systems on 03 October 2023 at 14:00

    Speaker: Yves Benoist (Université Paris-Saclay)

    Title: Convolution and square on abelian groups

    Abstract: The aim of this talk will be to construct functions on a cyclic group of odd order whose ''convolution square'' is proportional to their square. For that, we will have to interpret the cyclic group as a subgroup of an abelian variety with complex multiplication, and to use the modularity properties of their theta functions.