Thu, 01 May 2025

17:00 - 18:00
L3

C*-algebras satisfying the UCT form an analytic set

Michał Szachniewicz
(University of Oxford)
Abstract

I will sketch a proof of the statement in the title and outline how it is related to Ehrenfeucht–Fraïssé games on C*-algebras. I will provide the relevant background on C*-algebras (and descriptive set theory) and explain how to construct a standard Borel category X that can play a role of their `moduli'. The theorem from the title is an application of the compactness theorem, for a suitable first-order theory whose models correspond to functors from X. If time permits, I will mention some related problems and connections with conceptual completeness for infinitary logic. This talk is based on several discussions with Ehud Hrushovski, Jennifer Pi, Mira Tartarotti, and Stuart White after a reading group on the paper "Games on AF-algebras" by Ben De Bondt, Andrea Vaccaro, Boban Velickovic and Alessandro Vignati.

Thu, 01 May 2025

11:00 - 12:00
C5

Introduction to Arakelov theory

Michał Szachniewicz
(University of Oxford)
Abstract

I will talk about preliminaries in Arakelov geometry. Also, a historical overview will be provided. This talk will be the basis of a later talk about the theory of globally valued fields.

Mon, 16 Jun 2025
16:00
C3

Counting solutions to (some) homogeneous quadratic forms in eight prime variables

Aleksandra Kowalska
(University of Oxford)
Abstract
In 2014, Lilu Zhao counted the solutions to non-degenerate, homogeneous quadratic forms in at least nine prime variables, using the circle method. However, while the suggested formula for the number of solutions is believed to hold for forms in at least five variables, his method seems to break for general forms in less than nine variables.
In 2021, Ben Green solved the problem for forms in eight prime variables (using a very different approach), satisfying a 'genericity' condition. The aim of my project was to solve some forms in eight variables not satisfying this condition.
In the talk, I will describe my findings, which allowed me to count the number of solutions to forms in eight prime variables with off-diagonal rank 3 (i.e., which have an invertible 3x3 submatrix without diagonal entries), which is a subset of non-generic forms.
Mon, 02 Jun 2025
16:00
L6

On the largest $k$-product-free subsets of the Alternating Groups

Anubhab Ghosal
(University of Oxford)
Abstract

A subset $A$ of $A_n$ is $k$-product-free if for all $a_1,a_2,\dots,a_k\in A$, $a_1a_2\dots a_k$ $\notin A$.
We determine the largest $3$-product-free and $4$-product-free subsets of $A_n$ for sufficiently large $n$. We also obtain strong stability results and results on multiple sets with forbidden cross products. The principal technical ingredient in our approach is the theory of hypercontractivity in $S_n$. Joint work with Peter Keevash.

Mon, 26 May 2025
16:00
L6

Large values of Dirichlet polynomials with characters

Vishal Gupta
(University of Oxford)
Abstract

Dirichlet polynomials are useful in the study of the Riemann zeta function & Dirichlet L functions, serving as approximations to them via the approximate functional equation. Understanding how often they can be large gives bounds on the number of zeroes of these functions in vertical strips - known as zero density estimates - which are relevant to the distribution of primes in short intervals. Based on Guth-Maynard, we study large values of Dirichlet polynomials with characters, relevant to Dirichlet L functions. Joint work with Yung Chi Li. 

Mon, 19 May 2025
16:00
L6

On derived deformations of Galois representations (after Galatius-Venkatesh)

Samuel Moore
(University of Oxford)
Abstract


Given a mod $p$ Galois representation, one often wonders whether it arises by reducing a $p$-adic one, and whether these lifts are suitably 'well-behaved'. In this talk, we discuss how ideas from homotopy theory aid the study of Galois deformations, reviewing work of Galatius-Venkatesh.

Thu, 08 May 2025
16:00
Lecture Room 4, Mathematical Institute

Uniform Equidistribution of Quadratic Polynomials via Averages of $\mathrm{SL}_2(\mathbb{R})$ Automorphic Kernels

Lasse Grimmelt
(University of Oxford)
Abstract

In recent joint work with J. Merikoski, we developed a new way to employ $\mathrm{SL}_2(\mathbb{R})$  spectral methods to number-theoretical counting problems, entirely avoiding Kloosterman sums and the Kuznetsov formula. The main result is an asymptotic formula for an automorphic kernel, with error terms controlled by two new kernels. This framework proves particularly effective when averaging over the level and leads to improvements in equidistribution results involving quadratic polynomials. In particular, we show that the largest prime divisor of $n^2 + h$ is infinitely often larger than $n^{1.312}$, recovering earlier results that had relied on the Selberg eigenvalue conjecture. Furthermore, we obtain, for the first time in this setting, strong uniformity in the parameter $h$.
 

Tue, 13 May 2025
14:00
L6

Stacky interpretation of D-cap modules

Arun Soor
(University of Oxford)
Abstract

I will construct a fully-faithful functor from the category of co-admissible D-cap modules of Ardakov—Wadsley, to the category of quasi-coherent sheaves on the "analytic de Rham space”, at least in the case when the rigid variety is affinoid and étale over a polydisk. 

Wed, 04 Jun 2025
16:00
L6

Even the Loch Ness monster deserves a curve graph

Filippo Baroni
(University of Oxford)
Abstract
Every topologist knows that a mug is a doughnut, but did you know that the Loch Ness monster is a baguette?
 
This talk is meant as a gentle introduction to the theory of big surfaces and their mapping class groups. This is a topic that has gained significant traction in the last few years, and is undergoing an exciting phase of explosive expansion.
 
We will start by giving lots of examples of surfaces of infinite type, working our way towards a general classification theorem. We will then introduce big mapping class groups, and outline some of their topological properties that are reminiscent of classical geometric group theory. Finally, following a programme proposed by Calegari in 2009,  we will investigate to what extent the classical theory of curve and arc graphs of finite-type surfaces generalises to the infinite-type setting. 
 
The level of prior required knowledge on the topic of big mapping class groups will be the same as that of the speaker one week before the talk — that is, none.
Wed, 21 May 2025
16:00
L2

Fat minors and where to find them

Joseph MacManus
(University of Oxford)
Abstract

Recently, much attention has been paid to the intersection between coarse geometry and graph theory, giving rise to the fresh, exciting new field aptly known as ‘coarse graph theory’. One aspect of this area is the study of so-called ‘fat minors’, a large-scale analogue of the usual idea of a graph minor.

In this talk, I will introduce this area and motivate some interesting questions and conjectures. I will then sketch a proof that a finitely presented group is either virtually planar or contains arbitrarily ‘fat’ copies of every finite graph.

No prior knowledge or passion for graph theory will be assumed in this talk.

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