# Postgraduate Seminar - 2022-2023

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 .

If you have any question, do not hesitate to get in contact with us! This seminar is organised by Alvaro Gonzalez HernandezLink opens in a new window and Katerina SanticolaLink opens in a new window.

## 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 PoincarĂ© 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.