# Postgraduate Seminar

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 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

**Nicolò Paviato **- Decay of the Transfer Operator for Maps and Flows

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.

#### 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)

**George Kontogeorgiou **- Yet Another Locker Problem* *

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.

Week 4: Wednesday 28th October

**Ryan Acosta Babb** - All Functions are Continuous! A Provocative Introduction to Constructive Analysis

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)

**Philippe Michaud-Rodgers** - Fermat's Last Theorem and the Modular Method

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)

**Simon Gabriel & Arjun Sobnack** - Topics in the Real World

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

**Nicholas Fleming** - Homogenisation of Deterministic Fast-Slow Systems

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

**Diogo Caetano** - Partial Differential Equations on Time-Dependent Spaces

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

**Sunny Sood** - Implicit Function Theorems for Lipschitz Functions

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

**Josh Daniels-Holgate** - A Brief Introduction to Singularities of the Mean Curvature Flow

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.