Warwick Mathematics Institute Events

Seminar List Entry | Seminars by subject

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Upcoming Seminars

Junior Number Theory on 04 December 2023 at 11:00 in B3.01

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.

Junior Analysis and Probability Seminar on 04 December 2023 at 16:00 in D1.07

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.

Algebra on 04 December 2023 at 17:00 in B3.02

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.

Ergodic Theory and Dynamical Systems on 05 December 2023 at 14:00 in B3.02

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.

Algebraic Topology on 05 December 2023 at 16:00 in B3.03

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.

Probability Theory on 06 December 2023 at 16:00 in B3.03

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.

Geometry and Topology on 07 December 2023 at 14:00 in B3.02

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.

Analysis on 07 December 2023 at 16:00 in B3.02

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.

Statistical Mechanics on 07 December 2023 at 16:00 in MS.03

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.

Combinatorics on 08 December 2023 at 14:00 in B3.02

Speaker: Ryan Martin (Iowa State)
Title: Counting cycles in planar graphs

Abstract: TBA

Colloquium on 08 December 2023 at 16:00 in B3.02

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.

Past Seminars

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
controls.

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.