# Postgraduate Seminar

Welcome to the webpage for the Warwick Mathematics Postgraduate Seminar.

This term, all talks will be held in B3.02 at 12 noon on Wednesday (except when stated otherwise). The seminar will take a hybrid format so that students can join us virtually if they have (or prefer) too. The link to join the seminar virtually is:

In addition, there will be `coffee afternoon' at 1 p.m immediately after the seminar, for those of us who still enjoy socialising.

Organisers: Lucas Lavoyer and Sunny Sood.

#### Term 2 2021-22 - The seminars are held on Wednesday 12:00 - 13:00 in B3.02 - Mathematics Institute.

Week 1: Wednesday 12th January

**Hollis Williams** **- **The Penrose inequality for perturbations of Schwarzschild spacetime

The Penrose inequality is a remarkable geometric inequality that relates the mass of a black hole spacetime to the total area of its black holes. Penrose suggested the inequality on physical grounds in the 1960s, but a rigorous mathematical proof in the general case is still lacking. We present some new ideas towards a proof for the special case of perturbations of Schwarzschild spacetime using an elliptic PDE called the Jang equation.

Week 2: Wednesday 19th January

**Alvaro Gonzalez Hernandez - **An introduction to the Hasse principle through examples

The Hasse principle asks a very important question in the study of Diophantine equations: does the existence of real and *p*-adic solutions imply the existence of rational solutions? In this talk I will use examples of equations to motivate why this principle is useful and how it is linked to the geometry of the varieties defined by such equations. In particular, the connections between the Hasse principle and the arithmetic structure of elliptic curves will be discussed.

If time permits, I will explain how to construct explicit counterexamples to the Hasse principle as the homogeneous spaces associated to elliptic curves with non-trivial Tate-Shaferevich groups.

Week 3: Wednesday 26th January

**Diana Mocanu**-TBA

Week 4: Wednesday 2nd February

Week 5: Wednesday 9th February

**Marc Homs** **Dones** - TBA

Week 6: Wednesday 16th February

**Steven Groen** - TBA

Week 7: Wednesday 23rd February

**Dimitris Lygkonis - **TBA

Week 8: Wednesday 2nd March

**Arshay Sheth**-TBA

Week 9: Wednesday 9th March

**Irene Gil Fernández- **TBA

Week 10: Wednesday 16th March

#### Term 1 2021-22 - The seminars are held on Wednesday 12:00 - 13:00 in B3.02 - Mathematics Institute.

Week 1: Wednesday 6th October

**Hollis Williams **- Noncommutative Geometry and CFTs

We give the abstract definition for QFTs and CFTs and then outline the applications of spectral triples (data sets which encode noncommutative geometries) in theoretical physics and field theory. We discuss an interesting connection between spectral triples and two-dimensional superconformal field theories which is relevant for string theory.

*After the seminar, the first year PhD students are invited to take part **in a 'Treasure Hunt' around the campus. This will be a great opportunity **to have a bit of mid-week fun, learn where the important parts of the **campus are and to get to know your fellow colleagues (in particular, who **you'd rather not share an office with next year!).*

*In addition to this, the first year PhD students will also have to **opportunity to take part in a `photoshoot' organised by the department. If you are interested, please make sure you are well dressed for the occasion!*

Week 2: Wednesday 13th October

**Ryan Acosta Babb** - Lᵖ Convergence of Series of Eigenfunctions for the Equilateral Triangle

The eigenfunctions of the Laplacian on a rectangle give rise to the familiar Fourier series, whose Lᵖ convergence is well known. We will use this result to obtain Lᵖ convergence of series of trigonometric eigenfunctions of the Dirichlet Laplacian on an equilateral triangle. Along the way we will discuss some of the limitations of the argument owing to symmetry considerations.

Week 3: Wednesday 20th October

**William O'Regan**- Efficiently covering the Sierpinski carpet with tubes

We call a delta/2-neighbourhood of a line in R^d a tube of width delta. For a subset K of R^d it is an interesting problem to try and efficiently cover K with tubes as to try and minimise the total width of the tubes used. If for every epsilon > 0 we are able to find a collection of tubes which cover K with their total width less than epsilon we say that K is *tube-null.* The notion of tube-nullity has its roots in harmonic analysis, however, the notion is interesting in its own right. In the talk I will give an example of a set which is tube-null, the Sierpinski carpet, along with a rough sketch of its proof. If time permits I will discuss some open problems in the area along with their progress.

Week 4: Wednesday 27th October

**No talk this week- **Non-academic careers for postgraduate mathematicians & researchers

What are the career options for Masters & PhD Mathematicians who decide not to pursue an academic career? Post graduate mathematicians have an analytical and problem-solving skill set that is in demand, and there are a variety of very interesting career opportunities. In this interactive Q&A a panel of Warwick alumni will describe their career journey since graduation and share their hints and tips to help you plan a non-academic career. The panel will feature:

**Mattia Sanna ***(Data Scientist at Methods Analytics*: Warwick PhD Computational Algebraic Number Theory 2020)

**Chris Gamble ***(Applied Engineering co-lead, DeepMind: *Warwick MORSE 2009, University of Oxford DPhil Machine Learning & Bayesian Statistics 2014)

**Huan Wu ***(Project Leader at Numerical Modelling and Optimisation Section, TWI: *Warwick PhD Mathematics & Statistics 2017 Atomistic-to-Continuum Coupling for Crystal Defects)

**Zhana Kuncheva ***(Senior Scientist - Statistical Genetics at Silence Therapeutics plc*): Warwick BSc MORSE, Imperial PhD Mathematics & Statistics, modelling populations of complex networks)

Week 5: Wednesday 3rd November

**Patience Ablett **- Constructing Gorenstein curves in codimension four

Projectively (or arithmetically) Gorenstein varieties are a frequently occurring subset of projective varieties, whose coordinate rings are Gorenstein. Whilst there exist concrete structure theorems for such varieties in codimension three and below, the picture is less clear for codimension four. Recent work of Schenck, Stillman and Yuan outlines all possible Betti tables describing the minimal free resolution of the coordinate ring for Gorenstein varieties of codimension and Castelnuovo-Mumford regularity four. We explain how to interpret these Betti tables as a recipe book for constructing Gorenstein curves in $\mathbb{P}^5$, and give an example construction utilising the Tom and Jerry matrix formats of Brown, Kerber and Reid.

Week 6: Wednesday 10th November

**Muhammad Manji** - The Bloch-Kato Conjecture and the method of Euler Systems

The Bloch-Kato conjecture is a wide reaching conjecture in number theory relating in great generality algebraic objects (Selmer groups) and analytic objects (zeros of L-functions). It generalises well known phenomena in number theory, most notably the Birch—Swinnerton-Dyer conjecture about elliptic curves; one of the Clay institute millennium problems. I hope to provide a low tech introduction to the conjecture, defining the key concepts, and discuss important cases. If time permits, I will briefly discuss a modern approach to solving the conjecture for a range of cases using Euler systems.

Week 7: Wednesday 17th November

**Solly Coles** - Knots in dynamics: Linking numbers for geodesic flows

Knot theory is the study of topological characteristics of circles embedded in 3-dimensional space (knots). Often, invariants such as the linking number can be used to tell apart different configurations of knots. In continuous-time dynamical systems, knots may arise as orbits of flows. In this talk I will discuss existing results for knots which come from dynamical systems, as well as recent work on linking numbers for geodesic flows. If time permits, I will mention the more general case of Anosov flows.

Week 8: Wednesday 24th November

**Nuno Arala Santos** - Fourier Analysis methods in Number Theory

We will explain the Fourier-analytic ideas behind the Hardy-Littlewood circle method and describe their role in the proof of Roth's Theorem on 3-term arithmetic progressions. We will also give a rough sketch of the limitations that make these classical techniques unsuitable for tackling longer arithmetic progressions, and motivate the introduction by Gowers of the eponymous norms that led to his celebrated new proof of Szemeredi's Theorem in 2001.

Week 9: Wednesday 1st December

**Jakub Takac** - The Mordor theorem for Orlicz spaces

Given a measurable space, one may consider the Lebesgue space $L^{p}$ consisting of all measurable functions $f$ for which $|f|^{p}$ is integrable. We shall define so-called Orlicz spaces, which serve as a successful attempt at replacing the function $t \mapsto t^{p}$ in the definition of $L^{p}$ spaces with a more general Young function.

We shall explore some elementary properties of these Orlicz spaces, in particular their rearrangement-invariance. This leads to an axiomatic definition of a far more general object, a so-called rearrangement-invariant (r.i. for short) function space. If time allows, we will discuss the relation of the class of all Orlicz spaces to the class of all r.i. spaces, in particular, we will present the so-called Mordor theorem, a yet unpublished result which describes this relation in great detail.

A short artistic interlude will take place midway through the talk.

*During and after the seminar, a photographer will be present to take semi-candid photographs of PhD students socialising and talking about maths. If you would like the chance to be the face of the subsequent maths propaganda, you are more than welcome to come and have your picture taken. *

Week 10: Wednesday 8th December

**Katerina Santicola** - The Markoff Unicity Conjecture: when one door closes, another opens!

The Markoff Unicity Conjecture is a 108-year old conjecture about the solution set of the Diophantine equation $x^{2}+y^{2}+z^{2}=xyz$. The solutions, called Markoff numbers, turn up in a variety of settings, from combinatorics, to number theory, to geometry and graph theory. In this talk, we will look at the translation of the conjecture to the world of hyperbolic geometry, arguing why this approach fails to bring us closer to a proof of unicity. Then, we will look at a more promising translation to analytic number theory. Time permitting, we will go through the elementary proof that the MUC holds for all prime powers.