# Mathematics Colloquium 2021-22 Abstracts

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

*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*

*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*

*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*

*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*

*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*

*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*

*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?*

*'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*

*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*

*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 2021: Kat Rock (Warwick) ***Cost-effective disease elimination? Modelling to guide tailored intervention strategies against African sleeping sickness*

*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.