Thu, 19 Jun 2025
12:00
C6

Local behaviour of solutions to non-local kinetic equations

Amélie Loher
(University of Cambridge)
Abstract

We will discuss local regularity properties for solutions to non-local equations naturally arising in kinetic theory. We will focus on the Strong Harnack inequality for global solutions to a non-local kinetic equation in divergence form. We will explain the connection to the Boltzmann equation and we will mention a few consequences on the asymptotic behaviour of the solutions.

Tue, 03 Jun 2025

14:00 - 15:00
L4

A new lower bound for the Ramsey numbers $R(3,k)$

Julian Sahasrabudhe
(University of Cambridge)
Abstract

In this talk I will discuss a new lower bound for the off-diagonal Ramsey numbers $R(3,k)$. For this, we develop a version of the triangle-free process that is significantly easier to analyse than the original process. We then 'seed' this process with a carefully chosen graph and show that it results in a denser graph that is still sufficiently pseudo-random to have small independence number.

This is joint work with Marcelo Campos, Matthew Jenssen and Marcus Michelen.

Tue, 17 Jun 2025

14:00 - 15:00
L4

The Maze Problem

Imre Leader
(University of Cambridge)
Abstract

Do there exist universal sequences for all mazes on the two-dimensional integer lattice? We will give background on this question, as well as some recent results. Joint work with Mariaclara Ragosta.

Tue, 13 May 2025

14:00 - 15:00
L4

Frame matroids with a distinguished frame element

James Davies
(University of Cambridge)
Abstract

A matroid is frame if it can be extended such that it possesses a basis $B$ (a frame) such that every element is spanned by at most two elements of $B$. Frame matroids extend the class of graphic matroids and also have natural graphical representations. We characterise the inequivalent graphical representations of 3-connected frame matroids that have a fixed element $\ell$ in their frame $B$. One consequence is a polynomial time recognition algorithm for frame matroids with a distinguished frame element.

Joint work with Jim Geelen and Cynthia Rodríquez.

Wed, 18 Jun 2025
16:00
L6

Profinite Rigidity: Then and Now

Julian Wykowski
(University of Cambridge)
Abstract

Is it possible to tell the isomorphism type of an infinite group from its collection of finite quotients? This question, known as profinite rigidity, has deep roots in various areas of mathematics, ranging from arithmetic geometry to group theory. In this talk, I will introduce the question, its history and context. I will explain how profinite rigidity is studied using the machinery of profinite completions, including elementary proofs and counterexamples. Then I will outline some of the key results in the field, ranging from 1970 to the present day. Time permitting, I will elaborate on recent results of myself on the profinite rigidity of certain classes of solvable groups. 

Thu, 19 Jun 2025
16:00
Lecture Room 4

Crystalline liftability of irregular weights and partial weight one modularity

Hanneke Wiersema
(University of Cambridge)
Abstract

Let $p$ be an odd prime. Let $K/\mathbf{Q}_p$ be a finite unramified extension. Let $\rho: G_K \to \mathrm{GL}_2(\overline{\mathbf{F}}_p)$ be a continuous representation. We prove that $\rho$ has a crystalline lift of small irregular weight if and only if it has multiple crystalline lifts of certain specified regular weights. The inspiration for this result comes from recent work of Diamond and Sasaki on geometric Serre weight conjectures. We also discuss applications to partial weight one modularity.

Thu, 23 Jan 2025

12:00 - 13:00
L3

Optimal design of odd active solids

Anton Souslov
(University of Cambridge)
Further Information

Anton Souslov is an Associate Professor of Theoretical Statistical Physics working on the theory of soft materials, including mechanical metamaterials, active matter, topological states, and polymer physics.

Abstract

Active solids consume energy to allow for actuation and shape change not possible in equilibrium. I will first introduce active solids in comparison with their active fluid counterparts. I will then focus on active solids composed of non-reciprocal springs and show how so-called odd elastic moduli arise in these materials. Odd active solids have counter-intuitive elastic properties and require new design principles for optimal response. For example, in floppy lattices, zero modes couple to microscopic non-reciprocity, which destroys odd moduli entirely in a phenomenon reminiscent of rigidity percolation. Instead, an optimal odd lattice will be sufficiently soft to activate elastic deformations, but not too soft. These results provide a theoretical underpinning for recent experiments and point to the design of novel soft machines.

 

 

Thu, 07 Nov 2024
12:00
C6

Ant lane formation: particle system and mean-field limit PDE

Oscar De Wit
(University of Cambridge)
Abstract

We investigate an interacting particle model to simulate a foraging colony of ants, where each ant is represented as a so-called active Brownian particle. Interactions among ants are mediated through chemotaxis, aligning their orientations with the upward gradient of the pheromone field. We show how the empirical measure of the interacting particle system converges to a solution of a mean-field limit (MFL) PDE for some subset of the model parameters. We situate the MFL PDE as a non-gradient flow continuity equation with some other recent examples. We then demonstrate that the MFL PDE for the ant model has two distinctive behaviors: the well-known Keller--Segel aggregation into spots and the formation of lanes along which the ants travel. Using linear and nonlinear analysis and numerical methods we provide the foundations for understanding these particle behaviors at the mean-field level. We conclude with long-time estimates that imply that there is no infinite time blow-up for the MFL PDE.

Wed, 20 Nov 2024
17:00
Lecture Theatre 1, Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, OX2 6GG

Chance, luck, and ignorance: how to put our uncertainty into numbers - David Spiegelhalter

David Spiegelhalter
(University of Cambridge)
Further Information

We all have to live with uncertainty about what is going to happen, what has happened, and why things turned out how they did.  We attribute good and bad events as ‘due to chance’, label people as ‘lucky’, and (sometimes) admit our ignorance.  I will show how to use the theory of probability to take apart all these ideas, and demonstrate how you can put numbers on your ignorance, and then measure how good those numbers are. Along the way we will look at three types of luck, and judge whether Derren Brown was lucky or unlucky when he was filmed flipping ten Heads in a row.

David Spiegelhalter was Cambridge University's first Winton Professor of the Public Understanding of Risk. He has appeared regularly on television and radio and is the author of several books, the latest of which is The Art of Uncertainty: How to Navigate Chance, Ignorance, Risk and Luck (Penguin, September 2024).

Please email @email to register to attend in person.

The lecture will be broadcast on the Oxford Mathematics YouTube Channel on Wednesday 11 December at 5-6pm and any time after (no need to register for the online version).

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

Thu, 13 Feb 2025
16:00
Lecture Room 4

On the exceptional set in the abc conjecture

Joni Teräväinen
(University of Cambridge)
Abstract
The well known abc conjecture asserts that for any coprime triple of positive integers satisfying $a+b=c$, we have $c<K_{\varepsilon} \mathrm{rad}(abc)^{1+\varepsilon}$, where $\mathrm{rad}$ is the squarefree radical function. 
 
In this talk, I will discuss a proof giving the first power-saving improvement over the trivial bound for the number of exceptions to this conjecture. The proof is based on a combination of various methods for counting rational points on curves, and a combinatorial analysis to patch these cases together.
 
This is joint work with Tim Browning and Jared Lichtman.
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