# Postgraduate Seminar

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

This year, all the talks will be held online via Microsoft Teams at 3 p.m. on Wednesday (except otherwise stated). The post-seminar lunch will be replaced with an online coffee afternoon at 4 p.m after the seminar.

Organisers: Simon Gabriel & Arjun 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)

**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 O(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 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 7: Wednesday 18th 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.

Week 9: Wednesday 2nd December

**Sunny Sood** - TBA

Week 10: Wednesday 9th December

n/a