Mon, 03 Nov 2025
16:00
C3

Abelian number fields with restricted ramification and rational points on stacks

Julie Tavernier
(University of Bath)
Abstract

A conjecture by Malle gives a prediction for the number of number fields of bounded discriminant. In this talk I will give an asymptotic formula for the number of abelian number fields of bounded height whose ramification type has been restricted to lie in a given subset of the Galois group and provide an explicit formula for the leading constant. I will then describe how counting these number fields can be viewed as a problem of counting rational points on the stack BG and how the existence of such number fields is controlled by a Brauer-Manin obstruction. No prior knowledge of stacks is needed for this talk!

Mon, 03 Nov 2025
16:00
C3

TBC

Julie Tavernier
(University of Bath)
Abstract

TBC

Mon, 27 Oct 2025
16:00
C3

On the distribution of very short character sums

Paweł Nosal
(University of Warwick)
Abstract
In their paper concerning quadratic residues Davenport and Erdős show that normalized sums of Legendre symbols $(\tfrac{n}{p})$ of suitable length $H(p) = p^{o(1)}$, with uniformly random starting point $X \in [0,...,p-1]$ obey the Central Limit Theorem, as the size of prime conductor goes to infinity.  
 
Recently, Basak, Nath and Zaharescu proved that the CLT still holds, if we pick $X$ uniformly at random from $[0,...,(\log p)^A], A>1$ , set $H(p) = (\log p)^{o(1)}$ and take the limit along full density subset of primes.  
 
In this talk, I will present a modification of their approach, inspired by the work of Harper on short character sums over moving intervals. This allows us to obtain the CLT of this type with $X$ uniformly random from $[0,...,g(p)]$ with practically arbitrary $g(p) \ll p^{\epsilon}$ for all $\epsilon >0$.
Mon, 20 Oct 2025
16:00
C3

An application of Goursat’s Lemma to the irreducibility of Galois representations

Zachary Feng
(University of Oxford)
Abstract
Goursat’s Lemma is an elementary, but perhaps niche, result in group theory classifying subdirect products of the product of two groups. In this talk, I will review what this lemma says, and describe how it can be used to deduce the irreducibility of Galois representations.

 
Mon, 13 Oct 2025
16:00
C3

Eigenvalues of non-backtracking matrices

Cedric Pilatte
(Mathematical Insitute, Oxford)
Abstract
Understanding the eigenvalues of the adjacency matrix of a (possibly weighted) graph is a problem arising in various fields of mathematics. Since a direct computation of the spectrum is often too difficult, a common strategy is to instead study the trace of a high power of the matrix, which corresponds to a high moment of the eigenvalues. The utility of this method comes from its combinatorial interpretation: the trace counts the weighted, closed walks of a given length within the graph.
 
However, a common obstacle arises when these walk-counts are dominated by trivial "backtracking" walks—walks that travel along an edge and immediately return. Such paths can mask the more meaningful structural properties of the graph, yielding only trivial bounds.
 
This talk will introduce a powerful tool for resolving this issue: the non-backtracking matrix. We will explore the fundamental relationship between its spectrum and that of the original matrix. This technique has been successfully applied in computer science and random graph theory, and it is a key ingredient in upcoming work on the 2-point logarithmic Chowla conjecture.
Tue, 04 Nov 2025
12:30
C3

How General Relativity shapes our universe

Alice Luscher, Mathematical Physics
Abstract

Einstein’s theory of general relativity reshaped our understanding of the universe. Instead of thinking of gravity as a force, Einstein showed it is the bending and warping of space and time caused by mass and energy. This radical idea not only explained how planets orbit stars, but also opened the door to astonishing predictions. In this seminar we will explore some of its most fascinating consequences from the expansion of the universe, to gravitational waves, and the existence of black holes.

Tue, 21 Oct 2025
12:30
C3

Mathematical modelling of a mass-conserving electrolytic cell

Georgina Ryan, OCIAM
Abstract

The electrochemical processes in electrolytic cells are the basis for modern energy technology such as batteries. Electrolytic cells consist of an electrolyte (an salt dissolved in solution), two electrodes, and a battery. The Poisson–Nernst–Planck equations are the simplest mathematical model of steady state ionic transport in an electrolytic cell. We find the matched asymptotic solutions for the ionic concentrations and electric potential inside the electrolytic cell with mass conservation and known flux boundary conditions. The mass conservation condition necessitates solving for a higher order solution in the outer region. Our results provide insight into the behaviour of an electrochemical system with a known voltage and current, which are both experimentally measurable quantities.

Mon, 01 Dec 2025
16:00
C3

TBC

Søren Eilers
(Unviersity of Copenhagen)
Abstract

to follow

Thu, 20 Nov 2025
16:00
C3

TBC

Marius Dadarlat
(Purdue)
Abstract

to follow

Tue, 30 Sep 2025
15:00
C3

Spacetime reconstruction and measured Lorentz-Gromov-Hausdorff convergence

Mathias Braun
(École Polytechnique Fédérale de Lausanne (EPFL))
Abstract

We present Gromov's celebrated reconstruction theorem in Lorentzian geometry and show two applications. First, we introduce several notions of convergence of (isomorphism classes of) normalized bounded Lorentzian metric measure spaces, for which we describe several fundamental properties. Second, we state a version within the spacetime reconstruction problem from quantum gravity. Partly in collaboration with Clemens Sämann (University of Vienna).

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