Fri, 15 Nov 2024

12:00 - 13:00
Quillen Room

Ring-theoretic properties of affine and graded Hecke algebras

Max Mackie
(University of Oxford)
Abstract

After recalling how Hecke algebras occur in the representation theory of reductive groups, we will introduce affine Hecke algebras through a combinatorial object called a root datum. Through a worked example we will construct a filtration on the affine Hecke algebra from which we obtain the graded Hecke algebra. This has a role analogous to the Lie algebra of an algebraic group.

We will discuss star operations on these rings, with a view towards the classical problem of studying unitary representations of reductive groups.

Wed, 06 Nov 2024
16:00
L6

Presentations of Bordism Categories

Filippos Sytilidis
(University of Oxford)
Abstract

A topological quantum field theory (TQFT) is a functor from a category of bordisms to a category of vector spaces. Classifying low-dimensional TQFTs often involves considering presentations of bordism categories in terms of generators and relations. In this talk, we will introduce these concepts and outline a program for obtaining such presentations using Morse–Cerf theory.

Mon, 04 Nov 2024
16:00
C3

Approximating Primes

Lasse Grimmelt
(University of Oxford)
Abstract

A successful strategy to handle problems involving primes is to approximate them by a more 'simple' function. Two aspects need to be balanced. On the one hand, the approximant should be simple enough so that the considered problem can be solved for it. On the other hand, it needs to be close enough to the primes in order to make it an admissible to replacement. In this talk I will present how one can construct general approximants in the context of the Circle Method and will use this to give a different perspective on Goldbach type applications.

Fri, 01 Nov 2024

12:00 - 13:00
Quillen Room

The Bruhat-Tits building

Mick Gielen
(University of Oxford)
Abstract

The Bruhat-Tits building is a crucial combinatorial tool in the study of reductive p-adic groups and their representation theory. Given a p-adic group, its Bruhat-Tits building is a simplicial complex upon which it acts with remarkable properties. In this talk I will give an introduction to the Bruhat-Tits building by sketching its definition and going over some of its basic properties. I will then show the usefulness of the Bruhat-Tits by determining the maximal compact subgroups of a p-adic group up to conjugacy by using the Bruhat-Tits building.

Tue, 03 Dec 2024
16:00
C3

The space of traces of certain discrete groups

Raz Slutsky
(University of Oxford)
Abstract

A trace on a group is a positive-definite conjugation-invariant function on it. These traces correspond to tracial states on the group's maximal  C*-algebra. In the past couple of decades, the study of traces has led to exciting connections to the rigidity, stability, and dynamics of groups. In this talk, I will explain these connections and focus on the topological structure of the space of traces of some groups. We will see the different behaviours of these spaces for free groups vs. higher-rank lattices, and how our strategy for the free group can be used to answer a question of Musat and Rørdam regarding free products of matrix algebras. This is based on joint works with Arie Levit, Joav Orovitz, and Itamar Vigdorovich.

Wed, 23 Oct 2024
16:00
L6

Coherence in Dimension 2

Sam Fisher
(University of Oxford)
Abstract

A group is coherent if all its finitely generated subgroups are finitely presented. Aside from some easy cases, it appears that coherence is a phenomenon that occurs only among groups of cohomological dimension 2. In this talk, we will give many examples of coherent and incoherent groups, discuss techniques to prove a group is coherent, and mention some open problems in the area.

Wed, 20 Nov 2024
11:00
L4

Quadratic and $p^\mathrm{th}$ variation of stochastic processes through Schauder expansions

Yuchen Fan
(University of Oxford)
Abstract
We present a class of stochastic processes which admit a unique quadratic variation along any sequence of partitions $(\pi^n)_{n\geq 1}$ with $\sum_{n\geq 1}|\pi^n|<\infty$, which generalizes the previous results for finitely refining partitions. This class of processes contains some signed Takagi-Landsberg functions with random coefficients and standard Brownian motions, and these processes admit $\frac{1}{4}$-Hölder continuous version. We study the quadratic and $p^\mathrm{th}$ variation of signed Takagi-Landsberg functions with random coefficients. Finally, we seek some generalizations and applications of our results.


 

Tue, 23 Jan 2024

11:00 - 12:00

[Rescheduled] A new axiom for Q_p^ab and non-standard methods for perfectoid fields

Leo Gitin
(University of Oxford)
Abstract

The class of henselian valued fields with non-discrete value group is not well-understood. In 2018, Koenigsmann conjectured that a list of seven natural axioms describes a complete axiomatisation of Q_p^ab, the maximal extension of the p-adic numbers Q_p with abelian Galois group, which is an example of such a valued field. Informed by the recent work of Jahnke-Kartas on the model theory of perfectoid fields, we formulate an eighth axiom (the discriminant property) that is not a consequence of the other seven. Revisiting work by Koenigsmann (the Galois characterisation of Q_p) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method can yield short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).

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