# Postgraduate Seminar

Hello!

Welcome to the Warwick Mathematics Postgraduate Seminar, where graduate students share the outcomes of their research to their peers.

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. This is the linkLink opens in a new window to join the seminar virtually.

For those who will join us in person, we will provide at the end of the seminar a light lunch :)

Do you want to give a talk in this seminar? This is what you have to doLink opens in a new window .

This seminar is organised by Alvaro Gonzalez HernandezLink opens in a new window and Katerina SanticolaLink opens in a new window. If you have any question, do not hesitate to get in contact with us (if possible, when you send us an email please add both of us as recipients)!

## Term 3 - Year 2022 - 2023

###### Week 1: Wednesday 26th April

##### George KontogeorgiouLink opens in a new window - Separators of Cayley graphsLink opens in a new window

**Week 2 : Wednesday 3rd May**

##### Daniel MarloweLink opens in a new window - Homotopy type theory and univalent foundationsLink opens in a new window

Voevodsky's quest to make amends led to renewed interest in type theory, the discovery of the univalence axiom, and ultimately to the hugely successful program to formalise mathematics we know and love today (think Lean, AGDA, Coq). In this talk, I'll give an overview of Martin-Löf dependent type theory and its homotopical interpretation; define $\Pi-$, $\Sigma-$, identity types and contractibility; and hopefully mention univalence. If time permits, I will define the circle, $S^1$.

###### Week 3 : Wednesday 10th May

##### Zhuo WuLink opens in a new window - Disjoint isomorphic balanced clique subdivisionsLink opens in a new window

A subdivision of a graph H is obtained by replacing each edge of H by a path, and it is called balanced if every added path is of the same length. How can we find a balanced subdivision of a complete graph if we have enough edge density? In this talk, I will present the idea behind this question. This is joint work with Irene Gil Fernández, Joseph Hyde, Hong Liu, Oleg Pikhurko.

###### Week 4 : Wednesday 17th May

##### Hamdi DërvodeliLink opens in a new window - What is tropical geometry?Link opens in a new window

The aim of this talk is to introduce tropical varieties, which are the main objects of study of tropical geometry. We give examples and describe how to construct them. They can be thought of as combinatorial shadows of classical varieties in algebraic geometry that sometimes preserve important information. This makes our lives easier because we are often able to understand valuable information about our original varieties by only looking at their shadows.

If time permits, we will mention another connection of tropical varieties with matroids. This is a useful link not only because matroids generalize graphs and linear subspaces, but also because they know about tropical varieties.

###### Week 5 : Wednesday 24th May

##### Tereso del Río AlmajanoLink opens in a new window (Coventry University) - Speeding up Cylindrical Algebraic DecompositionLink opens in a new window

Cylindrical Algebraic Decomposition (CAD) is an algebraic algorithm that comes in handy, among other things, to disprove statements about polynomials (e.g. $\exists x\ \exists y\ \exists z\ $ such that $x^3yz-yz^2>0$ and $x-z^5+3=0$) and to count the connected components of polynomial varieties. However, CAD has doubly exponential complexity on the number of variables, limiting its practical application.

In this talk, we will see how machine learning and a recent algorithm in validated numerics and machine learning can be used to radically increase the practical potential of CAD.

###### Week 6 : Wednesday 31st May

##### Grega SaksidaLink opens in a new window - How to solve the heat equation using Brownian motionLink opens in a new window

The heat equation, which describes how heat moves through space, is one of the most fundamental partial differential equations. It has attracted a lot of attention since its first formulation, and many methods were developed to solve it, including Fourier analysis.

One way of solving the heat equation is to use Brownian motion, a famous random process in probability theory. We will explain intuitively how the solution can be constructed, defining Brownian motion along the way. No special prior knowledge is required, although the basics of probability theory and analysis would be useful.

###### Week 7 : Wednesday 7th June

##### Marc TruterLink opens in a new window - Is this curve a sphere or a donut?Link opens in a new window

What does it mean for a curve to be a sphere or even a donut? This question lies in the area of classification, one of the big topics of study in algebraic geometry. A lot of information can in fact be drawn about curves just by knowing how many holes their topological pictures have, that being, whether they are spheres, donuts or even multi holed donuts.

No prior knowledge is expected, with the aim being to give a very visual and introductory explanation of the area. For example, we will try to understand what it means to be a manifold by thinking about sticking paper onto balloons!

###### Week 8 : Wednesday 14th June

##### Seth HardyLink opens in a new window - Random multiplicative functionsLink opens in a new window

###### Week 9 : Wednesday 21st June

##### Eva ZaatLink opens in a new window - Modelling the behaviour of materials, specifically metal sheets, from a continuum mechanics point of viewLink opens in a new window

Metal sheets are everywhere around us, and due to their versatility, we can find them in the architecture of buildings, manufacturing of transportation and in decorative art. In this talk we will look into the mathematics behind the bending of sheets.

We will zoom in on the physical mechanics of elastic deformation and the difficulties of mathematically modelling the reshaping of the sheets by means of two practical examples.

###### Week 10 : Wednesday 28th June

##### Cameron HeatherLink opens in a new window - Finding the brightest galaxies in the early universeLink opens in a new window

By looking deeper into the universe, we are able to look back in time to when the universe was younger. This allows us to see the properties of young galaxies, which we can compare to modern galaxies to understand how they form and evolve.

In this talk, we will look at their brightness, in particular the maximum value they can take, which we can predict using extreme value statistics.

## Term 2 - Year 2022 - 2023

###### Week 1: Wednesday 11th January

##### Hollis WilliamsLink opens in a new window - Fourier analysis for rarefied gas flowsLink opens in a new window

Fourier analysis is widely used in applied mathematics, engineering and physics. In this talk, we explain how it can be used to derive some new exact solutions for non-equilibrium rarefied gas flows. These flows fall into a regime which is inaccessible both to the Boltzmann and Navier-Stokes equations, so a different set of equations must be used known as the Grad equations.

###### Week 2 : Wednesday 18th January

##### Peize LiuLink opens in a new window - Introduction to deformation quantisation and formalityLink opens in a new window

In 1997, Kontsevich solved the problem of deformation quantisation on Poisson manifolds, which contributed to his winning of the 1998 Fields Medal.

This talk is an introduction to deformation quantisation. This is an approach of going from classical mechanics to quantum mechanics through deformation of the algebra of smooth functions on the phase space. I will explore the original idea from physics and go through some historical developments. Then I will give a crash course on deformation theory based on differential graded Lie algebras and $L_\infty$-algebras, and show its connection with deformation quantisation via Kontsevich’s formality theorem.

###### Week 3 : Wednesday 25th January

##### Arshay ShethLink opens in a new window - Introduction to special values of zeta functionsLink opens in a new window

###### Week 4 : Wednesday 1st February

##### William O'ReganLink opens in a new window - Introduction to fractal geometryLink opens in a new window

While there is no agreed definition of a fractal, broadly speaking, a fractal is a geometric shape containing detailed structure at arbitrarily small scales. They will also usually have a ‘fractal dimension’ which differs from its topological dimension. The aim of this. talk is to introduce fractal geometry to those unacquainted.

Time dependent, I will cover some of the following: box dimension, Hausdorff measure and dimension, mass distribution principle, Frostman’s Lemma, energy, projection theorems, iterated function systems et cetera, all whilst using concrete examples to get a feel for the theory. No prior knowledge will be required, but knowing what a measure is would be helpful.

###### Week 5 : Wednesday 8th February

##### Pietro WaldLink opens in a new window - Heisenberg Kakeya setsLink opens in a new window

A subset of $\mathbb{R}^n$ is called a Kakeya set if for every direction in the unit sphere there is a unit segment contained in the set and parallel to such direction.

Perhaps surprisingly, questions about their dimension turn out to be linked to several open problems in harmonic analysis and PDE. Recently, J. Liu introduced a notion of Kakeya set in the context of the Heisenberg group and proved a sharp lower bound for their "dimension''.

In this talk, I will introduce: Kakeya sets (in $\mathbb{R}^n$), the Heisenberg group, and Heisenberg Kakeya sets. After reviewing a definition of dimension, I will sketch how Liu's theorem can be proven in a conceptually simpler way. This is based on a joint work with K. Fässler and A. Pinamonti.

###### Week 6 : Wednesday 15th February

Cancelled due to illness of the speaker.

###### Week 7 : Wednesday 22th February

##### Hefin LambleyLink opens in a new window - An introduction to inverse problemsLink opens in a new window

Given an observed effect, the inverse problem is to determine the cause. These problems are hard to solve because they are unstable: small errors in the observed effect lead to big errors in the reconstructed cause.

I will give some examples of inverse problems and discuss practical ways to solve them. We'll also see how this links to uncertainty quantification, which is an active research topic both for applied mathematicians and for engineers modelling real-world phenomena in the presence of noise and uncertainty.

###### Week 8 : Wednesday 1st March

##### Patricia Medina CapillaLink opens in a new window - Crowns and their uses in generation problemsLink opens in a new window

The generation properties of a group reveal a lot of information about its structure. As such, these properties have been investigated very thoroughly in the last century. This has led to some very beautiful and surprising results being proven, such as the fact that all simple groups are generated by two elements. Relatively recently, Dalla Volta and Lucchini developed the theory of crowns in order to tackle such problems.

In this talk, we will introduce the set of crowns of a group, describe how it is utilised in generation problems, and showcase its strength via some examples. Time allowing, I will explain how this approach has been applied in recent research in order to bound the number of generators of the maximal subgroups of simple groups.

###### Week 9 : Wednesday 8th March

##### Diana MocanuLink opens in a new window - Mary Somerville's Diophantine equationsLink opens in a new window

To celebrate International Women’s day, I will be talking about “the queen of sciences” of the nineteenth century, Mary Somerville. Coincidentally, one of her first contributions to the scientific world consists of solutions to a few Diophantine equations proposed in *The New Series of the Mathematical Repository*, a famous mathematical gazette of those times.

In this talk, we will carefully discuss Mary’s original solutions and then, we will examine them through the lenses of modern algebraic geometry. Throughout, I will briefly introduce basic notions of algebraic geometry and translate Mary’s solutions into this language.

###### Week 10 : Wednesday 15th March

##### Ryan L. Acosta BabbLink opens in a new window - A tourist's guide to the (first) incompleteness theoremLink opens in a new window

Trisecting an angle, squaring a circle, doubling a cube or solving a general polynomial equation using "elementary" operations all turned out to be impossible problems. But the crowning achievement of such "impossibility" results was Gödel's celebrated *i**ncompleteness theorem:* there will always remain unprovable theorems in mathematics, no matter how many tricks we devise, or axioms contrive in vain attempts to patch the holes.

The main goal of this talk will be to explain exactly what this means (and what it *does not*), and to give a not-too-technical presentation of the surprisingly ingenious ideas which lie behind the proof.

## Term 1 - Year 2022 - 2023

###### Week 1: Wednesday 5th October

##### Sunny SoodLink opens in a new window - Homological stability for $O_{n,n}$Link opens in a new window

Motivated by Hermitian K-Theory, we study the homological stability of the split orthogonal group $O_{n,n}$.

Specifically, let $R$ be a commutative local ring with infinite residue field such that $2 \in R^{*}$. We prove that the natural homomorphism $H_{k}(O_{n,n}(R) ; \mathbb{Z}) \rightarrow H_{k}(O_{n+1,n+1}(R); \mathbb{Z})$ is an isomorphism for $k \leq n-1$ and surjective for $k \leq n$.

This will be an excellent opportunity to introduce esoteric concepts such as group homology and hyperhomology spectral sequences at the postgraduate seminar.

This is all joint work with my supervisor Dr Marco Schlichting.

###### Week 2 : Wednesday 12th October

##### Paul PanteaLink opens in a new window - Keeping exotic spheres as petsLink opens in a new window

The discovery of manifolds homeomorphic but not diffeomorphic to the standard sphere in the fifties sent shockwaves through the world — and some mathematicians never fully recovered.

These exotic spheres are cute, but they are best admired in their natural habitat. We will venture out in the wild and introduce ideas like topological K-theory, Bott periodicity, the J-homomorphism, and the Adams spectral sequence. Then, we will explore how stable homotopy theory helps us understand exotic spheres and their connection with the Poincaré conjecture, the Riemann zeta function, and the meaning of life.

###### Week 3 : Wednesday 19th October

##### Ruzhen Yang - Beilinson spectral sequence and its reverse problems on $\,\mathbb{P}^2$

Derived category is widely accepted as the natural environment to study homological algebra. We will study the structure of the bounded derived category of coherent sheaves on projective space via the semi-orthogonal decomposition (based on the Beilinson's theorem) and comparison (by a theorem by A. Bondal).

As an example we will give explicit free resolutions of some sheaves on $\,\mathbb{P}^2$ using the Beilinson spectral sequence. We will also discuss the reverse problem where we give a condition to when the complex given by the spectral sequence is a resolution of the ideal sheaf of three points.

###### Week 4 : Wednesday 26th October

##### Robin VisserLink opens in a new window - Hilbert's tenth problemLink opens in a new window

Can you find four distinct positive integers $w, x, y, z$ such that $w^3 + x^3 = y^3 + z^3$ ?

If that's too easy, try finding a non-trivial integer solution to $x^4 + y^4 + z^4 = w^4$.

And good luck finding any integral solution to $x^3 + y^3 + z^3 = 114$.

This all begs the question of whether we can construct a general algorithm to determine whether any given Diophantine equation has integer solutions. David Hilbert posed this exact question at the second ICM in 1900, where a negative answer was finally proven 70 years later by Yuri Matiyasevich building on work by Martin Davis, Hilary Putnam and Julia Robinson. In this talk, we'll explore the mathematical ideas behind Hilbert's tenth problem as well as go over many surprising applications, extensions to other number fields, and how this relates to several other famous open problems!

###### Week 5 : Wednesday 2nd November

##### James RawsonLink opens in a new window - Solvable points on higher genus curvesLink opens in a new window

Much of modern number theory is focused on trying to solve equations in the rational numbers. One case of interest is when the equations define a curve, where it turns out that the structure of the solutions is determined by a geometric invariant, the genus. Falting's theorem shows that if the genus is greater than 2, there are at most finitely many solutions. There are few results when the values of the solutions are allowed to be more general. This talk will focus on the case where the solutions are expressible in terms of addition, multiplication and $n$th-roots. I will review the background content from number theory (such as Galois groups) and algebraic geometry (mostly the concept of varieties).

###### Week 6 : Wednesday 9th November

##### Layne HallLink opens in a new window - Knotted orbits of flowsLink opens in a new window

Knots and their complements play a fundamental role in the study of 3-manifolds. Elsewhere, in dynamical systems, flows are a central object of study. These notions come together with an observation: given a flow on a 3-manifold, the periodic orbits form knots. Such knots have the extra structure of the flow from which they came, and we can use this to deduce information about them.

We will use examples such as the famous Lorenz attractor to discuss how this approach has been taken for a well-studied class of flows. Such flows allow us to draw and encode their knots, which will help us understand their topological and geometric properties.

###### Week 7 : Wednesday 16th November

##### Andrew RonanLink opens in a new window - Exact couples and nilpotent spacesLink opens in a new window

We will introduce spectral sequences via exact couples and outline how to derive the Serre spectral sequence from algebraic topology. Then, we will introduce nilpotent spaces, which are a type of space in many ways dual to a CW complex, before explaining how the Serre spectral sequence can be used to derive some of their properties. For example, the homology groups of a nilpotent space are finitely generated if and only if its homotopy groups are finitely generated.

###### Week 8 : Wednesday 23rd November

##### Alexandros GroutidesLink opens in a new window - Galois representations attached to elliptic curves and the Open Image TheoremLink opens in a new window

A Galois representation is a homomorphism $\rho:Gal(\bar{K}/K)\longrightarrow Aut(V)$ where $V$ is a finite dimensional vector space or a free module of finite rank. These objects are of great importance in number theory due to their connections with elliptic curves, modular forms and $L$-functions. We will introduce the mod-$\ell$, $\ell$-adic and adelic Galois representations attached to a non-CM elliptic curve and discuss the structure of their image. The $\ell$-adic open image does not a priori imply the adelic open image but as we will see, it all boils down to the surjectivity of the more innocent sounding mod-$\ell$ representation.

###### Week 9 : Wednesday 30th November

##### Nuno Arala SantosLink opens in a new window - Counting rational points on cubic surfacesLink opens in a new window

A fundamental problem in Diophantine geometry is to understand the asymptotic behaviour of the number of solutions to a Diophantine equation when we impose a boundedness condition on the variables. We will explain some progress in this problem for equations defining cubic surfaces in 3-dimensional space, following Roger Heath-Brown.