# Mathematics Colloquium 2019-20 Abstracts

4 October 2019: Edriss Titi (Cambridge, Texas A&M, Weizmann) Is dispersion a stabilizing or destabilizing mechanism? Landau-damping induced by fast background flows
Abstract:
In this talk I will present a unified approach for the effect of fast rotation and dispersion as an averaging mechanism for, on the one hand, regularizing and stabilizing certain evolution equations, such as the Navier-Stokes and Burgers equations. On the other hand, I will also present some results in which large dispersion acts as a destabilizing mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation. In addition, I will present some new results concerning two- and three-dimensional turbulent flows with high Reynolds numbers in periodic domains, which exhibit "Landau-damping" mechanism due to large spatial average in the initial data.

11 October 2019: Oscar Randal-Williams (Cambridge) Spaces of manifolds
Abstract:
Once one understands enough about individual mathematical objects, one begins to wonder how they can vary, i.e. what families of such objects can look like. This is especially natural in Topology, where families of sets (covering spaces) and families of vector spaces (vector bundles) are basic and classical objects of study. One wishes to distinguish such families by measuring how "twisted" they are (compare the Möbius strip with the cylinder). In this talk I will explain the situation when the mathematical objects involved are smooth manifolds, so that families are smooth fibre bundles. I will explain the basic invariants which measure how "twisted" such bundles are, and some recent progress in understanding them.

18 October 2019: Florian Theil (Warwick) Optimal and likely patterns
Abstract:
Energy driven pattern formation is a phenomenon where simple mathematical functions and functionals have interesting minimisers. The most classical patterns include periodic arrangements, they are relevant for sphere packing, molecular dynamics and statistical physics. I will explain the links between those view points and report recent and surprising mathematical developments.

25 October 2019: Lasse Rempe - Gillen (Liverpool) Building surfaces from equilateral triangles
Abstract:
In this talk, we consider the following natural question. Suppose that we glue a (finite or infinite) collection of equilateral triangles together in such a way that each edge is identified with precisely one other edge, each vertex is identified with only finitely many other vertices. If the resulting surface is connected, it naturally has the structure of a Riemann surface, i.e., a one-dimensional complex manifold. We ask which surfaces can arise in this fashion.

The answer in the compact case is given by a famous classical theorem of Belyi, which states that a compact surface can arise from this construction if and only if it is defined over a number field. These Belyi surfaces and their associated “Belyi functions” have found applications across many fields of mathematics.

In joint work with Chris Bishop, we give a complete answer of the same question for the case of infinitely many triangles (i.e., for non-compact Riemann surfaces).

01 November 2019: Ruth Baker (Oxford) Mathematical and computational challenges in interdisciplinary bioscience: efficient approaches for exploring models and interfacing with quantitative data.
Abstract:
Simple mathematical models have had remarkable successes in biology, framing how we understand a host of mechanisms and processes. However, with the advent of a host of new experimental technologies, the last ten years has seen an explosion in the amount and types of quantitative data now being generated. Increasingly larger and more complicated processes are now being explored, including large signalling or gene regulatory networks, and the development, dynamics and disease of entire cells and tissues. As such, the mechanistic, mathematical models developed to interrogate these processes are also necessarily growing in size and complexity. These detailed models have the potential to provide vital insights where data alone cannot, but to achieve this goal requires meeting significant mathematical challenges. In this talk, I will outline some of these challenges, and recent steps we have taken in addressing them.

08 November 2019: Jack Thorne (Cambridge) The arithmetic of simple singularities
Abstract:
Simple singularities form one of the many families of objects in mathematics that admit a classification by ADE Dynkin diagrams. I will explain how this is related to the classification of simple Lie algebras, and how in turn this relation can be used to study the arithmetic of certain families of algebraic curves over the rational numbers.

15 November 2019: Dave Benson (Aberdeen) Symmetry: A unifying thread in mathematics
Abstract:
I shall talk about groups, their representations and their actions and how this topic interacts with other parts of mathematics.

22 November 2019: Elena Celledoni (NTNU, Trondheim) Shape analysis on Lie groups with applications in computer animation
Abstract:
Shape analysis methods have in the past few years become very popular, both for theoretical exploration as well as from an application point of view. Originally developed for planar curves, these methods have been expanded to higher dimensional curves, surfaces, activities, character motions and many other objects.
In this talk, I will review recent work on shape analysis of curves on Lie groups for problems of computer animations.

29 November 2019: Eddie Wilson (Engineering Maths, Bristol) Mathematics to model the future driverless vehicle system
Abstract:
Driverless cars are presently a fashionable research topic. Significant research effort in computer vision and artificial intelligence has been devoted to the problem of safe driving for an individual vehicle. However, to date, little thought has been given to how driverless cars will operate as a *system* - perhaps competing or cooperating with each other and with "legacy vehicles" (driven by humans). These "concept-of-operation" questions will be the focus of my talk - and therein is a rich vein of quite simple mathematical models that might inform policy. The meat of the talk will focus on two specific examples: (a) the potential gains that coordinated centralised routing might offer in cities; and (b) the design of lane-sharing policies for motorways. In each of these cases, we couple Nash games with a "supply model". The mathematics is relatively straightforward, yet intriguing mathematical structure will emerge.

06 December 2019: Alessio Corti (Imperial College) Fano search
Abstract:

Fano varieties and their classification, the Fano/Landau--Ginzburg mirror correspondence, and a program to classify Fano varieties by classifying their Landau--Ginzburg mirrors first.

10 January 2020: Alexander Veselov (Loughborough) Geometrization, integrability and knots
Abstract:
After Arnold the classical integrability is usually understood in the Liouville sense as the existence of sufficiently many Poisson commuting integrals. About 20 years ago it was discovered that this does not exclude the chaotic behaviour of the system, which may even have positive topological entropy.

I will review the current situation with Liouville integrability in relation with Thurston’s geometrization programme, using as the main example the geodesic flows on the 3-folds with SL(2,R)-geometry.

A particular case of such manifold SL(2,R)/SL(2,Z) is known (after Quillen) to be topologically equivalent to the complement of the trefoil knot in 3-sphere. I will explain that the remarkable results of Ghys about modular and Lorenz knots can be naturally extended to the integrable region, where these knots are replaced by the cable knots of trefoil.

The talk is partly based on a joint work with Alexey Bolsinov and Yiru Ye and will not require any specific knowledge.

MONDAY 13 January 2020, 5pm, MS.04: Albert Schwarz (California, Davis) Geometric approach to quantum theory
Abstract:
I give a formulation of quantum theory where the starting point is a convex set of states. I show that the formulas for probabilities can be derived from first principles. Particles are defined as elementary excitations of ground state and quasiparticles as elementary excitations of translation invariant stationary state.

I am planning to discuss briefly collisions of (quasi)particles in geometric approach.

17 January 2020: John Toland (Bath) Finitely Additive Measures and Weak Convergence in $L_\infty(X,\cal{L},\lambda)$

24 January 2020: Peter
Ashwin (Exeter) Tipping points of non autonomous dynamical systems: from theory to application
Abstract:
Dynamical systems have been immensely successful in describing and predicting behaviours of a wide range of applications, especially in cases that can be well-modelled by a closed (unforced or autonomous) dynamical system. However, not all systems are closed; for example, elements in the climate system respond to changes in greenhouse gas forcing in quite a nontrivial manner over a range of timescales. For such open (forced or nonautonomous) systems with time-varying inputs, it can be hard to get much insight out of dynamical systems theory: usually one resorts to numerical simulations. This is especially the case when inputs vary on a similar timescale to the system itself. I will discuss recent work in understanding critical transitions or tipping points for some classes of nonautonomous dynamical systems, with applications to modelling possible climate tipping points.

31 January 2020: John Parker (Durham) Kleinian groups with two parabolic generators

Abstract:

I will survey the problem of determining when two non-commuting parabolic Moebius transformations generate a discrete group. In particular I will discuss the classification of non-free discrete groups of this kind. This classification follows ideas of Ian Agol and is joint work with Hirotaka Akiyoshi, Ken'ichi Ohshika, Makoto Sakuma and Han Yoshida.

7 February 2020: Gabriel Paternain (Cambridge) The non-abelian X-ray transform
Abstract:
I will discuss the problem of how to reconstruct a matrix-valued potential from the knowledge of its scattering data along geodesics on a compact Riemannian manifold with boundary.

The problem arises in new experiments designed to measure magnetic
fields inside materials by shooting them with neutron beams from
different directions, like in a CT scan.
14 February 2020: Adam Harper (Warwick) The Riemann zeta function and probability
Abstract:

The Riemann zeta function is one of the most fascinating objects in mathematics. Analytic number theorists have spent more than 150 years investigating its zeros, average and maximum size, and value distribution, and have found probabilistic ideas and methods to be very powerful tools for attacking some of these questions. Recently, connections have been found with quite subtle probabilistic issues such as branching random walk and multiplicative chaos. I will try to explain some of these connections, ideas from the proofs, and what they really tell us about the zeta function.

21 February 2020: Peter Kropholler (Southampton) How and why soluble groups became so important

Abstract:

From the middle ages to the present day, mathematical calculation has proved to be a compelling pastime with many applications. But group theory only emerged in the early nineteenth century in response to the desire to understand
how to solve polynomial equations in one variable. A hundred years later in the early 20th century, von Neumann showed how group theory again played a role, this time with the bizarre Banach Tarski paradox which apparently showed that a solid ball in space could be doubled in size by cutting it into four pieces and moving them around.
In this talk we'll explore these ideas, learn why soluble groups are so called ( or why they are sometimes called solvable) and what is the current state of the art in understanding these groups.

28 February 2020: Benedikt Wirth (Munster) Variational models for transportation networks: old and new formulations

Abstract:

A small number of models for transportation networks (modelling street, river, or vessel networks, for instance) has been studied intensely during the past decade, in particular the so-called branched transport and the so-called urban planning. They assign to each network the total cost for transporting material from a given initial to a prescribed final distribution and seek the cost-optimal network. Typically, the considered transportation cost per mass is smaller the more mass is transported together, which leads to highly patterned and ramified optimal networks. I will present novel formulations of these models which allow a better interpretation as an optimal design problem.

6 March 2020: Oleg Zaboronski (Warwick) Reaction diffusion particle systems: from chemistry to algebra and analysis

Abstract:

Strong fluctuation effects make the perturbative analysis of reaction-diffusion particle systems in one and two dimensions pretty useless. In the talk I will explain how the one-dimensional models can be studied if one is able to build representations of Hecke algebras, solve infinite systems of PDE's and calculate Fredholm determinants. As a byproduct, we will also find out what is the probability that a large real random matrix has no real eigenvalues. This colloquium is a summary of some of the research Roger Tribe and I have been doing over the last ten years.

13 March 2020: Samir Siksek (Warwick) Which numbers are sums of seven cubes?

Abstract:

In 1851, Carl Jacobi made the experimental observation that all integers are sums of seven non-negative cubes, with precisely 17 exceptions, the largest of which is 454. Building on previous work by Maillet, Landau, Dickson, Linnik, Watson, Bombieri, Ramare, Elkies and many others, we complete the proof of Jacobi's observation.

1 May 2020: Björn Stinner (Warwick) Free Boundary Problems with Surface Phenomena
Abstract:
Free boundary problems refer to partial differential equations in domains with unknown (free) boundaries and appear in various areas such as solidification processes, tumour growth, or multi-phase flow. Sometimes these problems involve phenomena or processes on the boundaries that can be modelled by surface PDEs.

In this talk, we focus on diffuse interface approaches based on the phase field methodology to such problems. The boundaries are represented by thin layers, across which quantities rapidly but smoothly change their values, and are

described in terms of order parameters. The equations governing these order parameters can be numerically solved using relatively straightforward methods, such as finite elements with adaptive mesh refinement, as no boundary tracking is required. We discuss how equations adjacent to or on a free boundary can be incorporated into phase field models in a generic way. This underpins a software framework where specific models can be conveniently implemented or amended in a high-level language. A few computational examples and code listings will serve to illustrate the approach.

15 May 2020: John Cremona (Warwick) Random Diophantine Equations
Abstract:
How likely is it that a two random integers are coprime? What is the probability that a random quadratic equation has a root? Or that a random quadratic form in n variables is isotropic?

Using questions like these as examples, I will show how a variety of methods from real and p-adic analysis and number theory can be used to both formulate such questions more precisely and give answers in many cases. I will include some recent results about binary forms which exhibit some unexpected and (so far) unexplained symmetries.

This is joint work with Manjul Bhargava (Princeton) and Tom Fisher (Cambridge) together with a guest appearance by Jon Keating (Bristol/Oxford).

22 May 2020: Kirsten Wickelgren (Duke) There are 160,839<1> + 160,650<-1> 3-planes in a 7-dimensional cubic hypersurface
Abstract:
Given a generic choice of polynomials with complex coefficients, one can compute the dimension of the straight lines, planes, or d-dimensional planes contained in the common zeros of the polynomials. When this dimension is 0, there is some finite number of d-planes. For example, there are 321,489 3-dimensional planes in the zero locus of a degree 3 homogeneous polynomial in 9 variables over the complex numbers. This number can be identified with the topological Euler number of a certain vector bundle. However, it only corresponds to the count of d-planes over an algebraically closed field like the complex numbers. We can get information over other fields like the real numbers, the rational numbers, or finite fields, by using an Euler number from A1-homotopy theory instead. This Euler number is no longer an integer; instead it is a bilinear form, and invariants of bilinear forms record information about the arithmetic and geometry of the planes. In this talk, we will introduce these enumerative problems and A1-Euler numbers. We establish integrality results for the A1-Euler class, and use this to compute the Euler numbers associated to arithmetic counts of d-planes on complete intersections in terms of topological Euler numbers over the real and complex numbers. The example in the title then follows from work of Finashin--Kharlamov. This is joint work with Tom Bachmann.

29 May 2020: Kannan Soundararajan (Stanford) Equidistribution from the Chinese Remainder Theorem
Abstract:
One important theme in number theory concerns the equidistribution (or not) of natural sequences (modulo 1). A classical example is Weyl's theorem that the fractional parts of n\alpha for any irrational number \alpha are equidistributed. In this talk I will discuss an equidistribution result which arises from the Chinese Remainder Theorem.

Suppose for each prime p we are given a set A_p (possibly empty) of residue classes mod p. Use these and the Chinese Remainder Theorem to form a set A_q of residue classes mod q, for any integer q. Under very mild hypotheses, we show that for a typical integer q, the residue classes in A_q will become equidistributed. The prototypical example (which this generalises) is Hooley's theorem that the roots of a polynomial congruence mod n are equidistributed on average over n. I will also discuss generalisations of such results to higher dimensions, and when restricted to integers with a given number of prime factors.
(Joint work with Emmanuel Kowalski.)

5 June 2020: Eric Vanden Eijden (Courant Institute) Trainability and accuracy of artificial neural networks
Abstract:
The methods and models of machine learning (ML) are rapidly becoming de facto tools for the analysis and interpretation of large data sets. Complex classification tasks such as speech and image recognition, automatic translation, decision making, etc. that were out of reach a decade ago are now routinely performed by computers with a high degree of reliability using (deep) neural networks. These performances suggest that it may be possible to represent high-dimensional functions with controllably small errors, potentially outperforming standard interpolation methods based e.g. on Galerkin truncation or finite elements that have been the workhorses of scientific computing. In support of this prospect, in this talk I will present results about the trainability and accuracy of neural networks, obtained by mapping the parameters of the network to a system of interacting particles relaxing on a potential determined by the loss function. This mapping can be used to prove a dynamical variant of the universal approximation theorem showing that the optimal neural network representation can be attained by (stochastic) gradient descent, with a approximation error scaling as the inverse of the network size. I will also show how these findings can be used to accelerate the training of networks and optimize their architecture, using e.g nonlocal transport involving birth/death processes in parameter space.

12 June 2020: Charles Elliott (Warwick) PDEs and geometric biomembranes: A lockdown mashup
Abstract:
I will survey some work on developing mathematical methodology for studying the well-posedness and approximation of systems of partial differential equations arising in mathematical models pertaining to biomembranes and biological cells.
The content will include discussion of surface energies (Canham-Helfrich and Willmore) and conservation laws leading to geometric PDEs (curvature dependent surface motion) and nonlinear parabolic systems in complex and evolving domains. This will be linked to the concept of evolving finite element spaces developed for numerical simulation.

This will be a non-technical talk accessible to all levels from Math Ugs onwards. There will be some calculus and functional analysis but no long calculations or proofs- there may be long equations! There will be cute figures and animations - all thanks to collaborators who will be cited in the talk. There may be digressions.

19 June 2020: Peter Sarnak (IAS, Princeton) The topologies of random real algebraic hyper-sufaces
Abstract:
The topology of a hyper-surface in P^n(R) of high degree can be very complicated. However if we choose the surface at random there is a universal law for its distribution over its components . Little is known about this law and it appears to be dramatically different for n=2 and n>2 .
There is a similar theory for zero sets of monochromatic waves which model nodal sets of eigenfunctions of quantizations of chaotic hamiltonians .
Joint work with Y.Canzani and I.Wigman