# 2021-22

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.

#### Term 3 2021-22 - The seminars are held on Wednesday 12:00 - 13:00 in B3.02 - Mathematics Institute.

Week 1: Wednesday 27th April

Trees of uniform exponential growth have been known since the dawn of time. Trees of uniform polynomial growth have been known since 2000, which is when Benjamini and Schramm constructed them. Both the former and the latter include examples of unimodular trees.
In this talk I will explain why (very many) unimodular trees of uniform intermediate growth exist. This is joint work with Martin Winter, and answers a question by Itai Benjamini.

Week 2: Wednesday 4th May

Given a hyperelliptic curve $C : y^2 = f(x)$ over a number field $K$, we can study its reduction behaviour at odd primes using the machinery of cluster pictures, introduced by Dokchitser--Dokchitser--Maistret--Morgan. This allows us to compute a wide range of arithmetic invariants for both $C$ and its Jacobian, including the conductor, minimal discriminant, Tamagawa numbers, whether the curve is semistable, and much more! In this talk, we'll go through several examples of cluster pictures and will prove a couple of neat results about hyperelliptic curves along the way.

Week 3: Wednesday 11th May

According to Feller, Karamata's Tauberian theorem has a 'glorious history', even though it is often omitted from modern books on probability theory. In this talk, we outline the proof of this theorem and show how it can extend to work for generalised signed measures. This extension has a surprising application in the theory of stochastic control. On route, we highlight the nuances between the vague convergence of signed measures and the pointwise convergence of their distribution functions.

Week 4: Wednesday 18th May

In this talk we discuss nonconforming virtual element methods (VEMs) for fourth-order problems. At present, the available VEM literature on fourth-order problems only includes defining projection operators based on the underlying variational problem. This approach involves constructing only one projection which depends on the local contribution to the bilinear form. Instead, we follow the approach of defining a hierarchy of projection operators for the necessary derivatives with the starting point being a constraint least squares problem. By defining the projection operators in this way, we show that we can directly apply our method to nonlinear fourth-order problems. This approach can also be easily included into existing software frameworks.
This talk showcases the application of our generalised method to the Cahn-Hilliard equation. As a consequence of our approach, we do not require any special treatment of the nonlinearity. Our method is shown to converge with optimal order also in the higher order setting. The theoretical convergence result is verified numerically with standard benchmark tests from the literature.

Week 5: Wednesday 25th May

If a homogeneous Diophantine equation has a nontrivial solution in the integers, then it also has a nontrivial solution in the reals and in the integers modulo $p^{n}$ for ever prime $p$ and $n>0$. The Hasse Principle is said to hold for an equation if the converse also holds.
In their 2014 paper, Brüdern and Dietmann proved that for diagonal homogenous equations of degree $k$ in at least $3k+2$ variables, the Hasse Principle almost always holds, in the sense that if one chooses the coefficients uniformly in a box of size $A$, then the probability that the Hasse Principle holds tends to 1 as $A$ tends to infinity.
In this seminar, we will outline the main ideas of their proof (which uses the Circle Method), and then talk about the analogous result for solubility of equations in the primes (solutions consisting of prime numbers). It turns out that one of the main challenges of doing this is coming up with an analogue to the Hasse Principle, which I am calling a "Prime Hasse Principle".

Week 6: Wednesday 1st June

Spheres of negative dimension sound like something only mathematicians could come up with; and that's why nobody will talk to us at parties. But that's OK, because we have stable homotopy theory to keep us entertained. In this talk, we will introduce some key ideas like categorification, stabilization, and spectra; and explain how formally inverting spheres leads us to the study of all possible cohomology theories and to a vast generalization of commutative algebra. Finally, we will discuss how to incorporate group actions into this, and some in-progress effort to understand more things about the group of order 2.

Week 7: Wednesday 8th June

In this talk I will discuss two extremes of Dehn functions. On one end we have every geometric group theorist’s favourite groups: hyperbolic groups, which have linear Dehn functions. On the other end, we will look at a group $\Gamma$ with Ackermann Dehn function (a function that is not even primitive recursive). I will firstly give a brief introduction to hyperbolic groups and give the definition of a Dehn function for a group. I will then show the construction of $\Gamma$ and give a sketch proof of the lower bound of its Dehn function. $\Gamma$ is an interesting example of a group built as an HNN extension of a free-by-cyclic, one-relator, CAT(0) group $G$, relative to its free subgroup $H$. It shows that even groups that may seem ‘nice’ can have wild properties.

Week 8: Wednesday 15th June

A complex representation of a group $G$ is a group homomorphism from $G$ to $GL_n(\mathbb{C})$ and a character is the trace of the of the matrix corresponding to an element of the group. About 40 years ago, Knutson conjectured that for every irreducible character, there is a generalised character such that their tensor product is the regular character. Savitskii disproved this in 1993 and posed a new conjecture. In this talk, after a brief introduction to Character Theory, we will disprove Savitskii's Conjecture and discuss its relations to Kaplansky's Sixth Conjecture.

Week 9: Wednesday 22nd June

Let $M$ be a closed manifold, let $\phi_{t}$ be a flow on $M$ such that all of its orbits are periodic. A natural question is to ask whether or not the length of the orbits of $\phi$ is bounded (this is the periodic orbit conjecture). It turns out it isn’t necessarily true when the dimension of $M$ is greater than or equal to 4. I will explain one of the first counterexamples to this conjecture, given by Sullivan in dimension 5 in 1976.

Week 10: Wednesday 29th June

In this talk I will give an introduction to complex dynamics and discuss the monodromy problem in polynomial parameter space. This relates the topology of the parameter space of degree $d$ polynomials to the automorphisms of the space of one-sided infinite strings on $d$ symbols. I will discuss how this was proven and, time permitting, discuss the analogous problem in Hénon parameter space, computational work to understand the higher dimensional problem, and interesting phenomena arising in the course of experimentation.

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

Week 1: Wednesday 12th January

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

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

Cryptography is the study of secure communication techniques used in increasingly many everyday tasks such as instant messaging and online transactions. Modern cryptography is heavily based on number theory, with the latest research using elliptic curves to construct quantum-resistant cryptosystems. In this talk, we will review basic theory of elliptic curves and then we will see how to construct two public key cryptosystems using it, discussing their security at the same time.

Week 4: Wednesday 2nd February

The general aim of representation theory is to classify all representations of an object up to equivalence, where the type of representation considered can vary depending on the context. In this talk, I will discuss one way this can be achieved for abstract representations of $GL_{2}$ over a finite field, and smooth representations of $GL_{2}$ over a $p$-adic field. Both approaches involve natural actions of $GL_{2}$ on some space, and these motivate studying actions of $GL_{2}$ (over a $p$-adic field) on certain rigid analytic spaces in order to better understand a larger class of representations than just smooth. I will talk about the representations which arise from these constructions, and current work which attempts to better understand them.

Week 5: Wednesday 9th February

Can you find all continuous functions of the real line such that $f^{2}:=f \circ f = id$? And what about finding all periodic functions, i.e. $f^{n} = id$? Or even all solutions to $f^{n }= f^{k}$? In this session, we will try to answer these questions together.

This is joint work with Armengol Gasull, presented in doi:10.3934/dcds.2020303 (Looking at it ahead of the presentation is considered cheating!).

Week 6: Wednesday 16th February

In 1966, Mark Kac asked if one can determine the shape of a drum from the sound it makes. It turned out that this is in general not possible. In this talk, we approach a slightly twitched problem: can we determine a curve over a finite field (up to isomorphism) from its number of points? Continuing the striking similarity between both questions, the answer is again no; we call curves with the same point count isogenous. Instead we study 'doubly isogenous' curves, which are even more alike than isogenous curves. A natural question arises: are two doubly isogenous curves necessarily isomorphic? We treat this question in great detail for a family of curves with prescribed automorphism groups.

This summarises a joint paper (https://arxiv.org/abs/2102.11419Link opens in a new window) with Vishal Arul, Jeremy Booher, Everett Howe, Wanlin Li, Vlad Matei, Rachel Pries and Caleb Springer.

Week 7: Wednesday 23rd February

In this talk, we will attempt to shed light on a typical night of the many drunks of Flatland. We will show that although it can be pretty eventful, the many drunks are quite forgetful. Surprisingly, using tools and ideas from functional analysis, quantum mechanics and random polymers can explain a lot about their behaviour.

This is joint work with N. Zygouras and can be found in arXiv:2109.06115 and arXiv:2202.08145.

Week 8: Wednesday 2nd March

A complex number is said to be algebraic if it is the root of a non-zero polynomial with integer coefficients. A complex number that is not algebraic is called transcendental. Broadly speaking, transcendental number theory is the study of transcendental numbers. In this talk, without assuming any previous background in number theory, we will give an introduction to this subject from a historical point of view; we will begin by discussing some of the earliest results of the subject and end by discussing important open problems that remain shrouded in mystery to this day.

Week 9: Wednesday 9th March

In this talk we will see how a couple of krakens can be very useful to embed a pillar in a graph with large minimum degree (solving a conjecture of Thomassen, 1989). A pillar is a graph that consists of two vertex-disjoint cycles of the same length, $s$ say, along with $s$ vertex-disjoint paths of the same length which connect matching vertices in order around the cycles. Despite the simplicity of the structure of pillars and various developments of powerful embedding methods for paths and cycles in the past three decades, this innocent looking conjecture has seen no progress to date. In this talk, we will try to give an idea of the proof of such embedding, which consists of building a pillar (algorithmically) in sublinear expanders. This is joint work with Hong Liu.

Week 10: Wednesday 16th March

We construct smooth flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such singularities, with suitable barriers to flows with surgery. A limiting argument is then used to control these approximating flows. We demonstrate an application of this surgery flow by improving the entropy bound on the low-entropy Schoenflies conjecture.

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

Week 1: Wednesday 6th October

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

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

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

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

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

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

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

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

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

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.