# Mathematics Institute

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 2 - Year 2022 - 2023

##### 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.

##### 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.

##### Arshay ShethLink opens in a new window - Introduction to special values of zeta functionsLink opens in a new window
The study of special values of zeta functions is an ancient theme in number theory; nevertheless, it is still a very active and lively area of contemporary research encompassing famous unsolved problems such as the Birch and Swinnerton-Dyer conjecture. In this talk, without assuming any previous background in number theory, we will give an introduction to this fascinating branch of the subject.
We will begin by studying the Brahmagupta-Pell equation, an equation with an extremely rich mathematical history stretching to more than a millennium, and end by exploring how the fundamental properties of this equation are beautifully captured by the first major result in the area of special values of zeta functions: the analytic class number formula.
##### 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.

## Term 1 - Year 2022 - 2023

##### 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.

##### 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.

##### 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.

##### 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!

##### 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).

##### 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.

##### 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.

##### 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.

##### 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.

##### Elvira LupoianLink opens in a new window - Jacobians of curves: A brief introductionLink opens in a new window
To any algebraic curve $C$ of genus $g$, we can associate its Jacobian, a $g$-dimensional abelian variety which is functorially associated to the curve. In this talk, I will define Jacobians, assuming no previous knowledge in the subject and explore some of their properties. If time permits, I will touch on one of the ways in which rational points on a Jacobian can be used to find the set of rational points on the corresponding curve.