# W.O.M.P.S. (2020-21)

Welcome to the webpage for the Warwick Online Mathematics Postgraduate Seminar.

This term, all the talks will be held online via Microsoft Teams at 12 noon on Wednesday (except when stated otherwise). The seminar lunch will be replaced with an online coffee afternoon at 1 p.m after the seminar.

Organisers: Simon Gabriel & Arjun Sobnack

#### Term 3 - The seminars are held online at 12 noon on Wednesdays on Microsoft Teams

Week 1: Wednesday 28th April

We identify the rate of growth of the length of the concave majorant of a one-dimensional Lévy processes $X$ on $[0,T]$ and prove CLT-type results for the errors as $T \to \infty$. In the case where $X$ has zero mean $\mathbb{E}[X_1]=0$ and finite second moment, $\mathbb{E}[X_1^{2}]<\infty$, we establish a central limit theorem where the centering, which has a stochastic component, has fluctuations of size $\sqrt{\log T}$. When normalizing by $\sqrt{\log T}$ the centered length converges weakly to a normal distribution with zero mean, and variance $\mathbb{E}[X_1^2]^2/2$. The stochastic component has its own stochastically independent central limit theorem with fluctuations of size $\sqrt{\log T}$ and a deterministic centering, converging weakly to the standard normal distribution. Distributional limit theorems are also constructed for the length of the concave majorant when $X_t$ is attracted to an $\alpha-$stable process with $\alpha \in (0,2)\setminus\{1\}$. The proofs rely on a characterisation of the law of the faces of the concave majorant for all Lévy processes in term of the Lévy process itself and an independent uniform stick-breaking process.

Week 2: Wednesday 5th May

We overview the recently developed level set approach to the existence theory of minimal submanifolds and present some joint work with A. Pigati and D. Stern.
The underlying idea is to construct minimal hypersurfaces as limits of nodal sets of critical points of functionals. In the first part of the talk we will give a general overview of the codimension one theory. We will then move to the higher codimension setting, and introduce the self-dual Yang–Mills–Higgs functionals. These are a natural family of energies associated to sections and metric connections of Hermitian line bundles, whose critical points have long been studied as a basic model problem in gauge theory. We will explain to what extend the variational theory of these energies is related to the one of the $$(n - 2)$$–area functional and how one can interpret the former as a relaxation/regularization of the latter. Time permitting we will mention some elements of the proof, with special emphasis on the gradient flow of the Yang–Mills–Higgs energies.

Invited by A. Sobnack

Week 3: Wednesday 12th May

Abelian varieties, which have the structure of both a projective variety and an abelian group, play a very central role in algebraic number theory. After recalling some classical theory of abelian varieties in characteristic $$0$$, we see that the theory of abelian varieties in characteristic $$p$$ is much richer and more involved. We then focus our attention to particularly pleasant examples coming from so-called Artin–Schreier curves. Finally, we show that established results for $$p=2$$ do not extend to odd $$p$$.

Week 4: Wednesday 19th May

In this talk, we will introduce sublinear expanders with the goal of answering the following question, posed by Erdös in 1975: what is the smallest function $f(n)$ for which all graphs with $n$ vertices and $f(n)$ edges contain two edge-disjoint cycles $C_1$ and $C_2$, such that the vertex set of $C_2$ is a subset of the vertex set of $C_1$ and their cyclic orderings of the vertices respect each other? We prove the optimal linear bound $f(n)=O(n)$ using sublinear expanders. This is joint work with Jaehoon Kim, Younjin Kim and Hong Liu. No background knowledge will be assumed.

Week 5: Wednesday 26th May

Speaking with Style

12:05 p.m.–12:40 p.m.: Marc Homs Dones - Nonlinear consensus dynamics on networks

What does a school of fish, people’s opinions and the detection of a robbery have in common? All of them can be modelled by consensus dynamics. In this framework, a time-varying state is given to each node of a network which evolves according to its neighbours. Linear consensus is the most commonly used model due to its simplicity, but it produces very restrictive dynamics. In this talk, we will introduce a nonlinear generalization that allows for more flexibility while preserving some key features. We will then study the fixed points of this system as well as their stability through the concept of effective resistance. Finally, we will show how our methods can be used to completely understand the dynamics in “trees of motifs” type of networks.

This is joint work with K. Devriendt and R. Lambiotte; arXiv:2008.12022.

12:45 p.m.–1:00 p.m.: Alberto Cassar - An overview of pinning and wetting phenomena within a probabilistic framework

I will be presenting my work on the so-called pinning and wetting models which appear naturally in the Physics literature in the contexts of semiflexible polymers and deforming rods in space. The discussion will be devoted to the construction of the models and a brief description of the phase transitions and the concentration of measure problem in various settings such as gradient interaction as opposed to Laplacian interaction and free boundary conditions as opposed to Dirichlet boundary conditions.

Week 6: Wednesday 2nd June

No speaker this week.

Week 7: Wednesday 9th June

The aim of this talk is to introduce some of the aspects needed for the implementation of the virtual element method (VEM) in two dimensions within the DUNE framework. The virtual element method is a generalisation of both finite element and mimetic finite difference methods, first introduced in 2013 to solve second-order elliptic problems. DUNE, the Distributed and Unified Numerics Environment, is free and open source software for the numerical solution of PDEs, implemented in C++. In the first part of the talk, we introduce the basic concepts of VEM applied to a simple Poisson problem and discuss properties of the discrete spaces. In the second part we present some of the details of the implementation and discuss how to use Python alongside the domain specific Unified Form Language (UFL) to carry out simulations.

Week 8: Wednesday 16th June

Curve shortening flow (or more generally mean curvature flow) is an important geometric flow with applications in geometry, topology, general relativity, material science and image processing. One can view curve shortening flow as a geometric or non-linear version of the heat equation. In 1935, Tychonoff showed a non-uniqueness result for the heat equation if you allow your solution to have sufficiently large growth at infinity. I will present an analogous result for curve shortening flow, assuming that the curvature of our surface has sufficiently large growth at infinity.

Week 9: Wednesday 23rd June

Hollis Williams - Modular Tensor Categories and Topological QFTs

We summarise Edward Witten's paper “Quantum Field Theory and the Jones Polynomial” and compare Witten's approach to topological quantum field theories via Feynman integrals and Wilson lines with the mathematical interpretation via modular tensor categories.

Week 10: Wednesday 30th June

We give an overview of Gibbs measures for the one dimensional nonlinear Schrödinger equation (NLS). In the first part of the talk, we discuss some of the properties of the NLS and its deterministic well-posed theory. In the second part, we summarise the construction of Gibbs measures given in Jean Bourgain's paper "Periodic Nonlinear Schrödinger Equation and Invariant Measures."

#### Term 2 - The seminars are held online at 12 noon on Wednesdays on Microsoft Teams

Week 2: Wednesday 20th January from 1 p.m. onwards

1 p.m. - Coffee afternoon

2 p.m. - 'THE DARING LION' - VIRTUAL ESCAPE ROOM

This is the first in a series of events created especially for Postgraduate students in Maths and Stats - more to be announced soon!

The brilliant 'Study Happy' Team at Warwick have created an atmospheric and mind-bending online escape room experience like no other – especially for Maths and Stats postgraduate students! ‘The Daring Lion’ will see you battle through a minefield of conundrums to save Joe, your fellow CIA agent.

Key information

• Players attend individually, but work as one large team
• Collaboration happens in the chat function of Teams. There is no obligation to put your camera or microphone on, although you can speak aloud if you so wish!

Please email the Postgraduate Coordinator (Reine Walker) at postgraduatemaths@warwick.ac.uk to take part and you will be added to the Team before the event.

Week 3: Wednesday 27th January

Great interest has been shown in understanding limit laws, such as the Central Limit Theorem and Donsker's Invariance Principle, for dynamical systems. A standard technique for obtaining these results relies on use of the 'transfer operator', introduced by Ruelle in 1968. The spectral properties of this operator often give exponential contraction for mean zero observables, which in turn implies exponential decay of correlations for the system. In this talk we will describe basic properties of the transfer operator, and present a new result about exponential contraction for regular observables.

Week 4: Wednesday 3rd February (Seminar starts at 2 p.m. this week)

In many cases, one can use parabolic partial differential equations to improve a given geometric object. The Ricci Flow, introduced in the '80s by Richard Hamilton, is one of these cases, where one evolves the metric on a Riemannian manifold by its Ricci curvature. Being a heat-type equation, the hope is that the Ricci Flow will improve a possibly strange initial metric, and so provide geometric and topological information about the manifold. In this talk, we will introduce the Ricci Flow and some of its properties, with the aim of giving an overview of Hamilton's first result: Every three-dimensional closed manifold that admits a metric with strictly positive Ricci curvature also admits a metric with constant positive sectional curvature. Time permitting, we will briefly comment on a more recent research topic for the Ricci Flow, where one tries to use it to smooth out singular spaces.

Week 5: Wednesday 10th February

Matthew Staniforth (University of Southampton) - An Introduction to Toric Topology

Toric Topology is a rapidly evolving branch of topology. It is highly interdisciplinary in nature, with links to algebraic topology, combinatorics, algebra, and geometry. Toric Topology, by harnessing these connections, has the potential to bring new insights, by enhancing the methods of each respective area with those of the other areas. In this talk I shall shamelessly prostitute the study of Toric Topology; we hope not only to motivate, but also to give an idea of the achievements of the study of this area, and to provide an overview of the direction of research henceforth.

Invited by A. Hodson

Week 6: Wednesday 17th February

Measures of maximal entropy are important invariant measures in ergodic theory and have been since their introduction in the ‘60s. We introduce the relevant ideas of thermodynamic formalism and hyperbolic dynamical systems through classic examples such as hyperbolic toral automorphisms or geodesic flows. Intuitively, a system is hyperbolic if the tangent space at each point can be split into expanding (unstable) and contracting (stable) directions. In line with this intuition, the unstable manifold at a point $x$ is defined to be the set of points whose backwards orbit is close to that of $x$ (and so the forward orbits essentially expand.) It is well known that looking at the unstable direction produces more interesting dynamics and in this talk we will present a construction of measures of maximal entropy as the limit of push forward measures supported on pieces of unstable manifold.

Week 7: Wednesday 24th February

In the study of minimal surfaces, if the Plateau Problem—that of finding a surface area–minimising "soap film" whose boundary agrees with a given curve in 3-D space—is the most famous, then the Bernstein Problem, suggested as conjecture in 1927 by Sergei N. Bernstein, is surely the second most.

The focus of this talk is to give an informal exposition of the second-mentioned Bernstein Problem. The works leading up to its final resolution in 1969, the Bernstein Theorem, provide both a rich history and a great collection of exemplar techniques used more generally in the study of geometric partial differential equation. Thus the true aim of this talk is to give some insight into the place these techniques hold in a wider framework; indeed, the talk could have been entitled "An Introduction to Geometric Partial Differential Equations via the Bernstein Problem" if not for the risk of sounding too serious.

The informality of this talk takes form in opting for hand-wavy picture-based, but intuitive, arguments over truly airtight but technically challenging justifications of its claims. As such, the main pre-requisite of this talk is a fair willingness to suspend disbelief. Some knowledge of calculus of variations and Riemannian geometry will be helpful but is not necessary.

Arjun has made his slides available here.

Week 8: Wednesday 3rd March

Oliver McGrath (University of Oxford) - Introduction to Sieve Theory and a Variation on the Prime k-Tuples Conjecture

Sieve methods are analytic tools that we can use to tackle problems in additive number theory. This talk will serve as a gentle introduction to the area. At the end we will discuss recent progress on a variation on the Prime k-Tuples Conjecture which involves sums of two squares. No knowledge of sieves is required!

Invited by D. Mastrostefano

Week 9: Wednesday 10th March

12 noon - Coffee Afternoon

1:20 p.m. - WOMEN IN MATHS EVENT

You are all warmly invited to our online event which aims to celebrate International Women's Day and the achievements of female and non-binary mathematicians.

When: Wednesday 10th March 2021, afternoon
Webpage: https://alicehodson.gitlab.io/women-in-maths/

Full schedule including abstracts for talks can be found on the website -

1.20 - 1.30 : Welcome and introduction!
1.30 - 1.45 : Adela Gherga - Computing elliptic curves over $\mathbb{Q}$
1.50 - 2.05 : Ellie Archer - Random fractal trees
2.10 - 2.25 : Sophie Meakin - Mathematical modelling and forecasting during the COVID-19 pandemic
2.30 - 2.45 : Coffee break
2.45 - 3.10 : Josephine Evans - Entropy and collective motion
3.15 - 3.40 : Susana Gomes - From linear control theory to nonlinear dynamics: controlling thin film flows
3.45 - 4.00 : Coffee break
4.00 - 4.45 : Carolina Araujo - Algebraic geometry - research and trajectory
4.45 - 5.30 : Panel discussion and Q&A with current PhD students

The event is open to anyone regardless of gender or background. Feel free to drop in to as many talks as you like. We look forward to seeing you there!

Alice, Diogo and Linda

Week 10: Wednesday 17th March

Alistair Miller (Queen Mary University of London) - Topological Groupoids and their $C^*$-Algebras

The construction of $$C^*$$-algebras from topological groupoids provides $$C^*$$-algebraists with a wealth of interesting $$C^*$$-algebras and new ways of viewing many $$C^*$$-algebras of interest. Topological dynamical systems fit into the framework of topological groupoids, so we can also use $$C^*$$-algebraic tools to study topological dynamical systems.

In this talk I will introduce topological groupoids and $$C^*$$-algebras with examples, and will try to give an idea of why people might want to study their interplay.

Invited by A. Sobnack

#### Term 1 - The seminars are held online at 3 p.m. on Wednesdays on Microsoft Teams

Week 1: Wednesday 7th October (Seminar starts at 11 a.m. this week)

A locker problem in the tradition of Peter Bro Miltersen! Numbered cards are contained in equinumerous lockers. Bob Seeker and Alice Heplful seek a certain card. Alice looks inside the lockers and transposes two cards before the sought card is announced. Bob opens two lockers after it is announced. If Bob finds the sought card, they win. Can they achieve a chance of victory asymptotically better than $\mathcal{O}(\frac{1}{n})$? Tune in to find out! Joint work with Artur Czumaj and Mike Paterson.

Week 2: Wednesday 14th October (Seminar starts at 1 p.m. this week)

Anna Skorobogatova (Princeton University) - How Small Can Kakeya Sets Be? An Approach Via Harmonic Analysis

Some 100 years ago, Besicovitch and Kakeya independently studied the following twin problems:

• Given a Riemann integrable function on a two-dimensional plane, does one always have a Fubini-type disintegration theorem that decomposes the integral into two orthogonal directions?
• Can one continuously rotate a unit line segment in the plane in a way such that the resulting area is arbitrarily small, or even zero?

Both problems are closely related to investigating the existence of a set the plane that contains a unit line segment in every direction, but has zero area. One can extend this to arbitrary dimensions. The natural follow-up question is: How small can we make such a set in $\mathbb{R}^n$? Can it have dimension smaller than $n$? This is a long-standing open problem, known as the Kakeya Conjecture.

Motivated by the ground-breaking work of Fefferman in the 1970s on the ball multiplier problem in dimension $2$ or larger, one can see the interplay between the geometry involved in the Kakeya Conjecture and results in harmonic analysis. We will see the links between these two seemingly different areas of mathematics.

Week 3: Wednesday 21st October

Julian Sieber (Imperial College London) - The Unreasonable Effectiveness of the Martingale Problem

Under mild regularity assumptions, functions of a Markov process can be compensated to define a martingale. Conversely, if we know that the compensated expression is a martingale for a sufficiently rich class of functions, then this uniquely characterizes the underlying Markov process. This intimate relation was first pointed out by D.W. Stroock and S.R.S. Varadhan in a series of seminal papers in the late 60s. We shall give a non-technical overview of the most important applications of this so-called martingale problem. Among them are averaging principles for stochastic fast-slow systems, which we're going to explain in the final part of the talk. There, we'll also present an averaging result of T.G. Kurtz based on the convergence of occupation measures.

Invited by S. Gabriel

Week 4: Wednesday 28th October

Hilbert once quipped that "Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer". But Le Verrier discovered Neptune without even looking out the window! The aim of this talk is to showcase constructive mathematics to see how far we can go without excluded middle, and hopefully discover some beautiful (or traumatising) new landscapes along the way. We begin by ironing out some misconceptions about constructivism and discussing some motivations behind it. We then present some basic analysis with examples of constructive proofs and definitions, as well as negative pathologies, such as the failure of the Intermediate Value Theorem. Finally, we venture into the land of choice sequences and provide a (surprisingly elementary) proof of Brouwer's infamous Continuity Theorem: all real-valued functions on the interval $[0,1]$ are continuous.

Week 5: Wednesday 4th November (Seminar starts at 12 noon this week)

Fermat's Last Theorem states that the equation $x^n+y^n=z^n$, with $n$ at least $3$, has no solution for positive integers $x$, $y$ and $z$. In this talk I will give an overview of the proof of this result. Using three 'black boxes' of Wiles, Ribet, and Mazur, I will show how the interplay between modular forms and elliptic curves led to the resolution of this 400-year-old problem. I will also discuss how the same strategy (the modular method) can be used to solve other classes of Diophantine equations. The aim of this talk is to provide an introduction to some fundamental concepts in number theory, and I will assume no background knowledge.

The pillar of generosity that is Philippe has kindly made public his slides here.

Week 6: Wednesday 11th November (Seminar starts at 4 p.m. this week)

The real world has been of concern to mathematicians since their conception. As well as being an excellent source of interesting mathematical problems, such as "How many beans make five?", the real world also provides us with deep questions in biology, philosophy and the medical sciences: Was is the chicken or the egg that came first? How string is a piece of long? What exactly makes homeopathy so effective?

We will discuss a large range of topics arising in the real world including, but not limited to, coronavirus and its consequences on university life, modern affairs of dating and relationships, and the state of the economy, finally answer the age-old question of exactly what that has to do with the price of fish. The talk will be informal and we hope to engage the audience plentifully. We will happily pursue avenues of conversation suggested by attendees.

Week 7: Wednesday 18th November

Homogenisation of deterministic fast-slow systems is an area of some interest to applied mathematicians. For $\varepsilon>0$, consider a system of ODEs on $\mathbb{R}^d\times M$ of the form $$\frac{\mathrm{d}x^{(\varepsilon)}}{\mathrm{d}t}=a(x^{(\varepsilon)},y^{(\varepsilon)})+\frac{1}{\varepsilon} b(x^{(\varepsilon)},y^{(\varepsilon)}) \quad \text{(slow)} \qquad \text{ and } \qquad \frac{\mathrm{d}y^{(\varepsilon)}}{\mathrm{d}t}=\frac{1}{\varepsilon^2} g(y^{(\varepsilon)}) \quad \text{(fast)},$$ where $y^{(1)}$ is a chaotic' flow. The initial condition $y^{(\varepsilon)}(0)$ is picked randomly, with the rest of the system being deterministic. As $\varepsilon\rightarrow 0$, the slow dynamics $x^{(\varepsilon)}$ converges in distribution to the solution of a stochastic differential equation. In the first part of our talk we motivate this problem and discuss how it relates to showing a statistical limit law for $y^{(1)}$.

We then look at a discrete-time analogue of this problem. Time permitting, we prove that the limiting stochastic differential equation for the slow dynamics can be very general, even if we only consider very simple fast dynamics.

No knowledge of stochastic calculus or dynamical systems will be assumed.

Week 8: Wednesday 25th November

The aim of this talk is to describe he mathematical analysis behind the treatment of partial differential equations whose solutions lie in time-dependent function spaces.

In the first half of the presentation, we take a general view on the problem and describe an abstract framework suitable to problems of this kind, such as PDEs on moving domains or evolving surfaces. In the absence of an inner product structure, the variational formulation of parabolic problems on time-dependent domains is non-trivial, and our methods provide the theoretical background to do so in a general Banach space setting (without assuming separability or reflexivity of the solution spaces).

The second part is devoted to a specific nonlinear PDE. We derive a Cahn-Hilliard equation on an evolving surface with a logarithmic potential, and prove existence, uniqueness, and stability of (weak) solutions. It turns out that well-posedness of the problem relies on an interplay between the moving nature of the domains and properties of the solution, and necessary conditions arise. We explore these conditions, and propose an alternative derivation of the model which is more compatible with the evolution of the surfaces and for which a general well-posedness result can be established.

Week 9: Wednesday 2nd December

Lipschitz functions are ubiquitous throughout the Sciences. Implicit Function Theorems have found significant applications within subject areas ranging from Differential Geometry to Mathematical Economics. Therefore, studying the Implicit Function Theorems of Lipschitz functions would seem like an interesting and fruitful avenue of research.

Surprisingly however, the mathematics that goes into this does not appear to be well known within the mathematical community.

The aim of the talk is to introduce the audience to an Inverse Function Theorem and two Implicit Function theorems of Lipschitz functions. Along the way, we will study the relationship between these two Implicit Function Theorems and formulate the so called generalised derivative’ of a Lipschitz function.

If time permits, we will also discuss two open problems relating to the above and partial solutions found by the speaker.

This work was done by the speaker for his final year MMath project at Warwick, supervised by Professor David Mond.

Week 10: Wednesday 9th December

The Mean Curvature Flow is a parabolic, quasi-linear system of PDEs describing the evolution of a submanifold by its mean curvature. For curves in the plane it is also known as the Curve Shortening Flow.

Short time existence and uniqueness of smooth solutions from a given hypersurface is known. Moreover, the smooth flow can be continued for as long as the curvature remains bounded. Tools such as the Avoidance Principle tell us that singularities must form.

Understanding the flow at and through these singularities is an area of on going research. I will detail how we can approach understanding these singularities, in particular, I will explain the Level Set Flow, a weak solution to the Mean Curvature Flow.