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# Mathematics Colloquium 2021-22 Abstracts

##### 8 October 2021: Harald Helfgott (Paris, Göttingen), Expander graphs: from telecommunications to number theory

The notion of "expander graphs" arose from potential applications to the construction (on both sides of the Iron Wall) of communication networks with good connectivity properties. One can define them combinatorially (in terms of boundaries in graphs) or spectrally (in terms of the graph Laplacian or the adjacency operator), with both definitions being essentially equivalent. By now, expander graphs are applied in many different contexts in theoretical computer science and pure mathematics.

We will discuss a new application to number theory - joint work by myself and M. Radziwiłł (Caltech). Let us define a graph with integers as vertices, and edges corresponding to prime divisors common to two vertices. Proving that this is an expander graph (or rather, a strong local expander almost everywhere) turns out to be a challenging task, involving random walks, number theory and graph theory. As consequences, we obtain several results in number theory beyond the traditional parity barrier.

##### 15 October 2021: Hugo Touchette (Stellenbosch University, South Africa), The large deviation approach to statistical physics

I will give in this colloquium a basic overview of the theory of large deviations and of its applications in statistical physics. In the first part, I will discuss the basics of this theory and its historical sources, which can be traced back in mathematics to Cramer (1938), Sanov (1960) and Varadhan (1970s) and, on the physics side, to Einstein (1910) and Boltzmann (1877). In the second part, I will discuss how the theory has been applied in recent years to study various equilibrium, nonequilibrium, and complex systems such as random graphs. Some of these systems, eg interacting particle systems and turbulent flows, are studied by different groups at Warwick.

##### 22 October 2021: Antonio Lerario (SISSA), Probabilistic algebraic geometry

In this seminar I will discuss a modern point of view on algebraic geometry over non-algebraically closed fields (e.g. over the reals or the p-adics), which introduces ideas from probability for approaching classical problems. The main idea of this approach is the shift from the notion of "generic", from classical algebraic geometry, to the notion of "random". This change of perspective brings many interesting subjects into the picture: convex geometry, measure theory, representation theory, asymptotic analysis...

##### 29 October 2021: Kurt Debattista (WMG, Warwick), Applications of Visual Computing

Visual computing deals with all aspects of computational imaging, including computer graphics, virtual reality, image processing, computational HCI and vision. In this talk applications of visual computing for use in high-fidelity graphics and interactive computer graphics (such as VR or games) will be presented. Furthermore, some uses of computer vision for industrial applications will also be shown.

##### 5 November 2021: Emanuele Dotto (Warwick Maths), Hermitian forms in algebra and homotopy theory

I will give a gentle overview on the classification problem of symmetric and quadratic forms up to certain relations, and the role that these play in the topology of manifolds. We will start in algebra and gradually build up from Sylvester's law of inertia to discover the Grothendieck-Witt group of a ring. Motivated by the surgery exact sequence, we will define L-groups and be led to discuss various notions of forms on chain complexes. We will conclude with a result obtained in joint work with Calmès-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle relating Grothendieck-Witt groups and L-groups.

##### 12 November 2021:Ana Caraiani (Imperial) Reciprocity laws and torsion classes

The Langlands program is a vast network of conjectures that connect many areas of pure mathematics, such as number theory, representation theory, and harmonic analysis. At its heart lies reciprocity, the conjectural relationship between Galois representations and modular, or automorphic forms.

A famous instance of reciprocity is the modularity of elliptic curves over the rational numbers: this was the key to Wiles’s proof of Fermat’s last theorem. I will give an overview of some recent progress in the Langlands program, with a focus on new reciprocity laws over imaginary quadratic fields.

##### 19 November 2021: Simona Paoli (Aberdeen) From Homotopy Theory to Higher categories

Topological spaces can be studied by breaking them into building blocks, called n-types, using a classical construction in homotopy theory, the Postnikov decomposition. The desire to model algebraically the building blocks of spaces was one of the motivations for the development of higher groupoids, generalizing the fundamental groupoid of a space. In this talk I will first illustrate how this naturally leads to the need to encode weakly associative and weakly unital compositions of higher morphisms in a higher groupoid and I will discuss the challenges that this poses.

More generally, structures arising in mathematical physics, namely topological quantum field theories, call for the need to define a notion of higher category, in which higher morphisms are not necessarily invertible.

The precise formalization of the notions of higher groupoids and higher categories can be achieved through several combinatorial machineries. I will introduce one of the approaches arising from homotopy theory, based on the notion of multisimplicial sets. I will finally briefly discuss why this approach is promising in terms of proving a long-standing open conjecture in higher category theory.

##### 26 November 2021: Jane Hutton (Statistics, Warwick) 'I was too fat.' Is the lost data more important than the lost pounds?

The Prime Minister's statement relied on reports that obese people suffered more severely from covid-19. At the time of his admission, data for both obesity fields was missing for 71% of patients. To assess the impact of obesity, we have to consider the lost data.

The precise sets of assumptions made about why particular variables are not recorded will influence the conclusions we reach. Evaluation of the assumptions required to imput missing data using conditional distributions, or models to explore missing not at random, are essential. As the number of variables increases, keeping track of the assumptions in, for example, chained equations for imputation, is not easy. When there are several variables, subgroups defined by categorical variables such as sex might have different patterns of missingness.

Graphical models are useful in communicating our assumptions to colleagues, to allow informed discussion from clinical and other experts. To express this, graphical models which focus on events rather than random variables are required. Event trees and Chain event graphs provide a clear description of different patterns of missing data. These graphs can illustrate the difference between selection models and pattern-mixture models. Although the definitions are probabilistically equivalent, one might be more relevant to a particular application. Rubin's definition of Missing At Random (MAR) implicitly gave a causal ordering consistent with a selection model. Ordinal CEGs illustrate different patterns of conditioning. For example, we can distinguish between obesity being MAR given age for women and being MCAR with respect to age for men.

For observational studies, CEGs help to investigate selection of units or variables, and confounding, as well as missing data. There was substantial secular variation missingness of obesity data, and of obesity for patients recorded as covid-19 hospital patients in England. Evidence that obesity data for covid-19 patients is not missing at random, and possible implications for the association of obesity and severity will be given through CEGs.

##### 3 December 2021: Tom Montenegro-Johnson (Birmingham) Mathematical Mysteries of Microscale Motility

The mathematical study of how things swim at microscopic scales turned 70 years old last week. In this talk, I will attempt a broad introduction to this field, motivate its study, run through physical principles, and finally offer some personal perspectives on future avenues of research, including microscale swimming robots.

##### 10 December 2021: Demetrios Papageorgiou (Imperial) Evolution PDEs arising in multiphase-multiphysics fluid mechanics

The talk will be devoted to the description of flows involving immiscible liquids separated by moving interfaces. The interfaces are free boundaries that must be determined as part of the solution, leading to intricate nonlinear moving boundary problems. Guided by applications (e.g. coating flows, microfluidics) additional physics will be incorporated. Asymptotic analysis leads to a rich family of nonlinear evolution PDEs that can be one- or two-dimensional in space, can contain nonlocal terms, and be quasilinear or of degenerate parabolic type. In all cases the PDEs are high order (e.g. 4th order in space), and can appear as single equations or systems of coupled equations.

I will introduce such PDEs and discuss their solutions using a mix of analysis and computations. An intriguing aspect is that spatiotemporal chaos can arise even on the microscale where inertia is absent, and such features will be explored fully. Open challenging problems will be outlined throughout the talk.

##### 14 January 2022: Kat Rock (Warwick) Cost-effective disease elimination? Modelling to guide tailored intervention strategies against African sleeping sickness

In this colloquium I will provide an overview of the “HAT MEPP” project which is led by a modelling team in the Mathematics Institute and SBIDER. The project focuses on modelling of the infectious disease human African trypanosomiasis (HAT), which is more commonly known as sleeping sickness.

I will describe how the team use methods from dynamical systems, Bayesian statistics and health economics to provide decision support for five country programmes looking to eliminate sleeping sickness. I’ll focus on some of the computational challenges with fitting models to real-world data from the Democratic Republic of Congo and how our custom-built, online graphical user interface has played an essential role in the dissemination of this large set of results to a policy audience with minimal mathematical background.

##### 21 January 2022: Sara Maloni (University of Virginia) Some new results in Higher Teichmüller Theory

The Teichmüller space of a surface S is the space of marked hyperbolic structure on S, up to equivalence. By considering the holonomy representation of such structures, the Teichmüller space can also be seen as a connected component of (conjugacy classes of) representations from the fundamental group of S into PSL(2,R), consisting entirely of discrete and faithful representations. Generalizing this point of view, Higher Teichmüller Theory studies connected components of (conjugacy classes of) representations from the fundamental group of S into more general semisimple Lie groups which consist entirely of discrete and faithful representations.

We will give a survey of some aspects of Higher Teichmüller Theory, and will make links with the recent powerful notion of Anosov representation. Time permitting, we will conclude by focusing on two separate questions:

1. Do these representations correspond to deformation of geometric structures?

2. Can we generalize the important notion of pleated surfaces to higher rank Lie groups like PSL(d, C)?

The answer to question 1 is joint work with Alessandrini, Tholozan and Wienhard, while the answer to question 2 is joint work with Martone, Mazzoli and Zhang.

##### 28 January 2022: Josephine Evans (Warwick) Introduction to hypocoercivity theory for long time behaviour of kinetic equations.

Hypocoercivity is a collection of techniques developed to study the rate of convergence to equilibrium for kinetic equations. My plan is to begin with a very brief history of collisional kinetic theory to give a context. Then I'll discuss the particular difficulties that arise when studying long time behaviour for these problems, and present the key theories developed in order to overcome them. Lastly, I'll talk briefly about the open problems that remain, particularly my own research relating to open systems.

##### 4 February 2022: Claudia de Rham (Imperial) Physics under the Gravitational Rainbow

The direct detection of gravitational waves marks the beginning of a new era for physics and astronomy with an opportunity the probe gravity at its most fundamental level. I will discuss how gravitational waves can be used as a privileged channel of communication with the secret dark components of our Universe.

##### 11 Feb 2022: Carina Curto (Penn State) Graph rules for inhibitory network dynamics

Many networks in the nervous system possess an abundance of inhibition, which serves to shape and stabilize neural dynamics. The neurons in such networks exhibit intricate patterns of connectivity whose structure controls the allowed patterns of neural activity. In this work, we examine inhibitory threshold-linear networks whose dynamics are dictated by an underlying directed graph. We develop a set of parameter-independent graph rules that enable us to predict features of the dynamics from properties of the graph. These rules provide a direct link between the structure and function of inhibitory networks, yielding new insights into how connectivity may shape dynamics in real neural circuits. Graph rules also lead us to consider some natural topological structures, such as nerves and sheaves, stemming from various graph covers. We will illustrate the theory with some applications to central pattern generator circuits and other examples of neural computation.

##### 18 Feb 2022: Sam Chow (Warwick) On Littlewood's conjecture

Littlewood's conjecture is a famous problem in diophantine approximation that also has a dynamical interpretation. I'll discuss what's known about it, including the measure-theoretic aspects. Finally, I'll mention some of my work on the subject, which is based on the correspondence between Bohr sets and generalised arithmetic progressions.

##### 25 Feb 2022: Gwyneth Stallard (Open University) Wandering domains in complex dynamics

Complex dynamics concerns the iteration of analytic functions of the complex plane. For each function, the plane is split into two sets: the Fatou set (where the behaviour of the iterates is stable under local variation) and the Julia set (where the behaviour is chaotic). One of the most dramatic breakthroughs was given by Sullivan in the 1980s when he proved that, for rational functions, all components of the Fatou set are eventually periodic and there are no so-called wandering domains. For transcendental functions, however, wandering domains can exist and the rich variety of possible behaviours that can occur is only just becoming apparent. We discuss a recent classification of the dynamical behaviour of the iterates inside wandering domains and the relationship between the behaviour of orbits of points in the interior, and orbits of points on the boundary. These new results hold in a very general context including non-autonomous dynamical systems of holomorphic maps on simply connected domains.

This is joint work with Anna Miriam Benini, Vasiliki Evdoridou, Nuria Fagella and Phil Rippon.

##### 4 March 2022: Caroline Series (Warwick) Hyperbolic geometry in data science

Hyperbolic space is very big: the area of a circle or sphere expands exponentially with its radius. Thus it contains plenty of room for expanding tree-like structures. Indeed, hyperbolic space can be coarsely likened to a tree. For this reason, hyperbolic space lends itself to organising and representing data with hierarchical structure, ranging from biological classification, cell development, communication or social networks, to linguistic relationships.

Features of hyperbolic space can also be used to model other observed behaviours of real world complex networks. Recent years have seen an explosion of both techniques and applications. This talk will attempt to give an historical overview of some of the main ideas.

##### 11 Mar 2022: Gigliola Staffilani (MIT) The Schrodinger equations as inspiration of beautiful mathematics.

In the last two decades great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a collection of techniques: Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of results using as model problem mainly the periodic 2D cubic nonlinear Schrodinger equation. I will start by giving a physical derivation of the equation from a quantum many-particles system, I will introduce periodic Strichartz estimates along with some remarkable connections to analytic number theory, I will move on the concept of energy transfer and its connection to dynamical systems, and I will end with some results on the derivation of a wave kinetic equation.

##### 18 Mar 2022: Linda Westrick (Penn State) Luzin's (N) and randomness reflection

A function f:R->R has Luzin's property (N) if f(A) has Lebesgue measure 0 for every set A of Lebesgue measure 0. We give a new characterization of this old property in terms of higher algorithmic randomness. A computable f has Luzin's (N) if and only if it satisfies the following pointwise condition: for all reals x, if f(x) is Pi^1_1-random, then so is x. Here an individual real x is called Pi^1_1-random if it is not contained in any computably describable co-analytic set of measure 0. All these notions will be defined in the talk without assuming any prior knowledge of them. Joint work with Arno PAULY and YU Liang.

##### 29 April 2022: Demi Allen (Exeter) Diophantine Approximation for Systems of Linear Forms

Diophantine Approximation is a branch of Number Theory in which the central theme is understanding how well real numbers can be approximated by rationals. In the most classical setting, a $\psi$-well-approximable number is one which can be approximated by rationals to a given degree of accuracy specified by an approximating function $\psi$. Khintchine's Theorem provides a beautiful characterisation of the Lebesgue measure of the set of $\psi$-well-approximable numbers and is one of the cornerstone results of Diophantine Approximation. In this talk I will discuss the generalisation of Khintchine's Theorem to the setting of approximation for systems of linear forms. This talk is based on recent joint work with Felipe Ramírez (Wesleyan, US).

##### 6 May 2022: Maria Bruna (Cambridge) Continuum models of strongly interacting Brownian particles

I will discuss many-particle systems with strong interactions. These models are motivated by the study of many-particle systems in biology or industrial applications, where it is crucial to account for the finite size of particles. I will explain how these interactions can be included in the models and different methods to derive continuum PDE descriptions. In the second part of the talk, I will show how these methods can be used to model active matter or self-propelled particles such as bacteria or ants.

##### 13 May 2022: Ewelina Zatorska (Imperial) Analysis of hydrodynamic models of interacting agents

Today, more and more modelling is being carried out using coarse graining of multi-agent or multi-particle systems. This leads to the systems of partial differential equations (PDEs) describing the evolution of macroscopic averaged quantities, like density, velocity, or energy. These equations are often called the “hydrodynamic” models because they resemble the classical fluid systems, except that the interaction terms are usually more complex. Although the macroscopic models are extremely useful in most of the practical situations, there is still little underlying theory regarding their well-posedness, asymptotic stability of solutions, or convergence of numerical schemes.

In this talk I will discuss mathematical theory of hydrodynamic PDEs arising in the modelling of collective behaviour that I have been developing over the past 7 years. I will focus on a couple of models of crowd motion and traffic flow, and present the analytical and the numerical frameworks that we use to study them. I will give some examples of the existence and the ill-posedness results, along with a few conjectures that we are currently working on.

##### 20 May 2022: Heather Harrington (Oxford) Computational algebra and topology for spatial structures arising in biology

Biological processes are governed by interactions at multiple scales (genomic, molecular, cellular), which are now captured by multiple modalities (multi-indexed data) and/or multi-scale spatial data. Understanding these complex biological phenomena require mathematical approaches to elucidate dynamics, predict mechanisms and reveal function. With the wealth of state-of-the-art data available at unprecedented depth and scales, new approaches are required to extract meaningful and interpretable biological insights. This talk will present computational topology methods, relying on persistent homology, that provide insight and quantification to geometric structures arising at multiple scales in biology, such as protein structure and cancer ecosystem.

##### 27 May 2022: Shreyas Mandre (Warwick) Functional interpretation for transverse arches of the human foot

The fossil record indicates that the emergence of arches in human ancestral feet coincided with a transition from an arboreal to a terrestrial lifestyle. Propulsive forces exerted during walking and running load the foot under bending, which is distinct from those experienced during arboreal locomotion. I will present mathematical models with varying levels of detail to illustrate a simple function of the transverse arch. Just as we curve a dollar bill in the transverse direction to stiffen it while inserting it in a vending machine, the transverse arch of the human foot stiffens it for bending deformations. A fundamental interplay of geometry and mechanics underlies this stiffening -- curvature couples the soft out-of-plane bending mode to the stiff in-plane stretching deformation. This result overturns a century-old theory that the longitudinal arch underlies the bending rigidity of the human foot. In addition to presenting a functional interpretation of the transverse arch of the foot, this study also furnishes an interpretation of the fossil record implying that the evolution of the transverse arch may have preceded the longitudinal arch.

##### 10 June 2022: Susana Gomes (Warwick) From linear control theory to nonlinear dynamics: controlling thin film flows

The flow of a thin fluid down an inclined plane is a canonical problem in fluid mechanics, with technological applications such as manufacturing LCD screens or microchips. Mathematically, it provides a very rich framework for modelling, analysis, and control. In this talk, I will use this problem to motivate and explore the use of common control theory results beyond ODEs and linear PDEs. I will summarise a hierarchy of models, usually consisting of fourth order nonlinear partial differential equations describing the evolution of the interfaces of these flows. Depending on the application, we would like these flows' interface to have some desired shape, and I will show how a simple control methodology developed for the simplest models can be used to this end, and adapted across the hierarchy of models to efficiently control the fully nonlinear dynamics.

##### 17 June 2022: Melanie Rupflin (Oxford) Quantitative estimates for geometric variational problems

Many interesting geometric objects are characterised as minimisers or critical points of natural geometric quantities such as the length of a curve, the area of a surface or the energy of a map. For the corresponding variational problems it is often important to not only analyse the existence and properties of potential minimisers, but to obtain a more general understanding of the energy landscape.

It is for example natural to ask whether an object that has energy very close to the minimal possible energy must also essentially "look like" a minimiser, and if so whether this holds in a quantitative sense, i.e. whether one can bound the distance to a minimiser in terms of the energy defect. Similarly one would like to understand whether for points where the gradient of the energy is very small one can hope to bound the distance of this point to the set of critical points in terms of the size of the gradient.

In this talk we will discuss some aspects of such quantitative estimates for geometric variational problems and their role in understanding the dynamics of the associated gradient flows.

##### 24 June 2022: Bérengère Dubrulle (CNRS Paris-Saclay) Regularity of Turbulence: numerical and experimental explorations

Turbulence, a phenomenon observed by physicists in natural and laboratory flows, is thought to be described by Navier-Stokes equations (NSE). Mathematicians are wondering whether NSE are well posed, namely whether they can develop a singularity in finite time from regular initial conditions. Given that, it is natural to investigate regularity properties of turbulence.

This talk describes experimental and numerical investigations that are devised to explore regularity issues in turbulence. In particular, I will show how to detect and characterize areas of “lesser regularity” and how to connect them with possible loss of uniqueness.