Wed, 25 Oct 2023

16:00 - 17:00
L6

Alternating knots and branched double covers

Soheil Azarpendar
(University of Oxford)
Abstract

An old and challenging conjecture proposed by R.H. Fox in 1962 states that the absolute values of the coefficients of the Alexander polynomial of an alternating knot are trapezoidal i.e. strictly increase, possibly plateau, then strictly decrease. We give a survey of the known results and use them to motivate the study of branched double covers. The second part of the talk focuses on the properties of the branched double covers of alternating knots.

Wed, 18 Oct 2023

16:00 - 17:00
L6

Fibring in manifolds and groups

Monika Kudlinska
(University of Oxford)
Abstract

Algebraic fibring is the group-theoretic analogue of fibration over the circle for manifolds. Generalising the work of Agol on hyperbolic 3-manifolds, Kielak showed that many groups virtually fibre. In this talk we will discuss the geometry of groups which fibre, with some fun applications to Poincare duality groups - groups whose homology and cohomology invariants satisfy a Poincare-Lefschetz type duality, like those of manifolds - as well as to exotic subgroups of Gromov hyperbolic groups. No prior knowledge of these topics will be assumed.

Disclaimer: This talk will contain many manifolds.

Fri, 27 Oct 2023

12:00 - 13:00

Kaplansky's Zerodivisor Conjecture and embeddings into division rings

Sam Fisher
(University of Oxford)
Abstract

Kaplansky's Zerodivisor Conjecture predicts that the group algebra kG is a domain, where k is a field and G is a torsion-free group. Though the general sentiment is that the conjecture is false, it still remains wide open after more than 70 years. In this talk we will survey known positive results surrounding the Zerodivisor Conjecture, with a focus on the technique of embedding group algebras into division rings. We will also present some new results in this direction, which are joint with Pablo Sánchez Peralta.

Wed, 11 Oct 2023
16:00
L6

Reasons to be accessible

Joseph MacManus
(University of Oxford)
Abstract

If some structure, mathematical or otherwise, is giving you grief, then often the first thing to do is to attempt to break the offending object down into (finitely many) simpler pieces.

In group theory, when we speak of questions of *accessibility* we are referring to the ability to achieve precisely this. The idea of an 'accessible group' was first coined by Terry Wall in the 70s, and since then has left quite a mark on our field (and others). In this talk I will introduce the toolbox required to study accessibility, and walk you and your groups through some reasons to be accessible.

Tue, 24 Oct 2023

14:00 - 15:00
L3

Monochromatic products and sums in N and Q

Matt Bowen
(University of Oxford)
Abstract

We show that every 2-coloring of the natural numbers and any finite coloring of the rationals contains monochromatic sets of the form $\{x, y, xy, x+y\}$. We also discuss generalizations and obstructions to extending this result to arbitrary finite coloring of the naturals. This is partially based on joint work with Marcin Sabok.

Mon, 06 Nov 2023
16:00
L1

A Basic Problem in Analytic Number Theory

George Robinson
(University of Oxford)
Abstract

I will discuss a basic problem in analytic number theory which has appeared recently in my work. This will be a gentle introduction to the Gauss circle problem, hopefully with a discussion of some extensions and applications to understanding L-functions.

Mon, 30 Oct 2023
16:00
C2

Hodge theory in positive characteristic

Inés Borchers Arias
(University of Oxford)
Abstract

I will introduce the Hodge-de-Rham spectral sequence and formulate an algebraic Hodge decomposition theorem. Time permitting, I will sketch Deligne and Illusie’s proof of the Hodge decomposition using positive characteristic methods.

Mon, 09 Oct 2023
16:00
C3

Primes in arithmetic progressions to smooth moduli

Julia Stadlmann
(University of Oxford)
Abstract

The twin prime conjecture asserts that there are infinitely many primes p for which p+2 is also prime. This conjecture appears far out of reach of current mathematical techniques. However, in 2013 Zhang achieved a breakthrough, showing that there exists some positive integer h for which p and p+h are both prime infinitely often. Equidistribution estimates for primes in arithmetic progressions to smooth moduli were a key ingredient of his work. In this talk, I will sketch what role these estimates play in proofs of bounded gaps between primes. I will also show how a refinement of the q-van der Corput method can be used to improve on equidistribution estimates of the Polymath project for primes in APs to smooth moduli.

Tue, 07 Nov 2023

14:00 - 15:00
L5

A solution functor for D-cap-modules

Finn Wiersig
(University of Oxford)
Abstract

The theory of D-modules has found remarkable applications in various mathematical areas, for example, the representation theory of complex semi-simple Lie algebras. Two pivotal theorems in this field are the Beilinson-Bernstein Localisation Theorem and the Riemann-Hilbert Correspondence. This talk will explore a p-adic analogue. Ardakov-Wadsley introduced the sheaf D-cap of infinite order differential operators on a given smooth rigid-analytic variety to develop a p-adic counterpart for the Beilinson-Bernstein localisation. However, the classical approach to the Riemann-Hilbert Correspondence does not apply in the p-adic context. I will present an alternative approach, introducing a solution functor for D-cap-modules using new methods from p-adic Hodge theory.

Subscribe to University of Oxford