Thu, 16 May 2024

11:00 - 12:00
C3

Basics of Globally Valued Fields and density of norms

Michał Szachniewicz
(University of Oxford)
Abstract

I will report on a joint work with Pablo Destic and Nuno Hultberg, about some applications of Globally Valued Fields (GVFs) and I will describe a density result that we needed, which turns out to be connected to Riemann-Zariski and Berkovich spaces.

Thu, 09 May 2024

11:00 - 12:00
C3

Skolem problem for several matrices

Emmanuel Breuillard
(University of Oxford)
Abstract

I will present a recent work with G. Kocharyan, where we show the undecidability of the following two problems: given a finitely generated subgroup G of GL(n,Q), a) determine whether G has a non-identity element whose (i,j) entry is equal to zero, and b) determine whether the stabilizer of a given vector in G is non-trivial. Undecidability of problem b) answers a question of Dixon from 1985. The proofs reduce to the undecidability of the word problem for finitely presented groups.

Wed, 08 May 2024

16:00 - 17:00
L6

The Morse local-to-global property

Davide Spriano
(University of Oxford)
Abstract

I'll talk about the Morse local-to-global property and try to convince you that is a good property. There are three reasons. Firstly, it is satisfied by many examples of interest. Secondly, it allows to prove many theorems. Thirdly, it sits nicely in the larger program of classifying groups up to quasi-isometry and it has connections with open questions.

Wed, 15 May 2024

16:00 - 17:00
L6

Out(Fₙ) and friends

Naomi Andrew
(University of Oxford)
Abstract

This talk will serve as an introduction to the outer automorphism group of a free group, its properties and the objects used to study it: especially train track maps (with various adjectives) and Culler--Vogtmann outer space. If time allows I will discuss recent work joint with Hillen, Lyman and Pfaff on stretch factors in rank 3, but the goal of the talk will be to introduce the topic well rather than to speedrun towards the theorem.

Wed, 01 May 2024

16:00 - 17:00
L6

ℓ²-Betti numbers of RFRS groups

Sam Fisher
(University of Oxford)
Abstract

RFRS groups were introduced by Ian Agol in connection with virtual fibering of 3-manifolds. Notably, the class of RFRS groups contains all compact special groups, which are groups with particularly nice cocompact actions on cube complexes. In this talk, I will give an introduction to ℓ²-Betti numbers from an algebraic perspective and discuss what group theoretic properties we can conclude from the (non)vanishing of the ℓ²-Betti numbers of a RFRS group.

Wed, 22 May 2024

16:00 - 17:00
L6

Finite quotients of Coxeter groups

Sam Hughes
(University of Oxford)
Abstract

We will try to solve the isomorphism problem amongst Coxeter groups by looking at finite quotients.  Some success is found in the classes of affine and right-angled Coxeter groups.  Based on joint work with Samuel Corson, Philip Moeller, and Olga Varghese.

Thu, 23 May 2024
12:00
L5

Cancelled

Andrea Clini
(University of Oxford)
Abstract

Cancelled

Thu, 02 May 2024
12:00
L5

Gradient Flow Approach to Minimal Surfaces

Christopher Wright
(University of Oxford)
Abstract

Minimal surfaces, which are critical points of the area functional, have long been a source of fruitful problems in geometry. In this talk, I will introduce a new approach, primarily coming from a recent paper of M. Struwe, to constructing free boundary minimal discs using a gradient flow of a suitable energy functional. I will discuss the uniqueness of solutions to the gradient flow, including recent work on the uniqueness of weak solutions, and also what is known about the qualitative behaviour of the flow, especially regarding the interpretation of singularities which arise. Time permitting, I will also mention ongoing joint work with M. Rupflin and M. Struwe on extending this theory to general surfaces with boundary.

Thu, 06 Jun 2024

17:00 - 18:00
L3

Model theory of limits

Leo Gitin
(University of Oxford)
Abstract

Does the limit construction for inverse systems of first-order structures preserve elementary equivalence? I will give sufficient conditions for when this is the case. Using Karp's theorem, we explain the connection between a syntactic and formal-semantic approach to inverse limits of structures. We use this to give a simple proof of van den Dries' AKE theorem (in ZFC), a general AKE theorem for mixed characteristic henselian valued fields with no assumptions on ramification. We also recall a seemingly forgotten result of Feferman, that can be interpreted as a "saturated" AKE theorem in positive characteristic: given two elementarily equivalent $\aleph_1$-saturated fields $k$ and $k'$, the formal power series rings $k[[t]]$ and $k'[[t]]$ are elementarily equivalent as well. We thus hope to popularise some ideas from categorical logic.

Mon, 03 Jun 2024
16:00
L2

Upper bounds on large deviations of Dirichlet L-functions in the Q-aspect

Nathan Creighton
(University of Oxford)
Abstract

Congruent numbers are natural numbers which are the area of right angled triangles with all rational sides. This talk will investigate conjectures for the density of congruent numbers up to some value $X$. One can phrase the question of whether a natural number is congruent in terms of whether an elliptic curve has non−zero rank. A theorem of Coates and Wiles connects this to whether the $L$-function associated to this elliptic curve vanishes at $1$. We will mention the conjecture of Keating on the asymptotic density based on random matrix considerations, and prove Tunnell’s Theorem, which connects the question of whether a natural number is a congruent number to counting integral points on varieties. Finally, I will hint at some future work I hope to do on non-vanishing of the $L$-functions.

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