Wed, 14 May 2025
16:00
L6

Coarse cohomology of metric spaces and quasimorphisms

William Thomas
(University of Oxford)
Abstract

In this talk, we give an accessible introduction to the theory of coarse cohomology of metric spaces in the sense of Margolis, which we present in direct analogy with group cohomology for discrete groups. We explain how this yields the robust notion of coarse cohomological dimension (due to Margolis), which is a genuine quasi-isometry invariant of metric spaces generalising the cohomological dimension of groups when the latter is finite. We then give applications to geometric properties of quasimorphisms and motivate how such considerations might be useful in the setting of non-positively curved groups. This is joint reading/work with Paula Heim.

Wed, 30 Apr 2025
16:00
L3

Property (T) via Sum of Squares

Gargi Biswas
(University of Oxford)
Abstract

Property (T) is a rigidity property for group representations. It is generally very difficult to determine whether an infinite group has property (T) or not. It has long been known that a discrete group with a finite symmetric generating set has property (T) if and only if the group Laplacian is a positive element in the maximal group C*-algebra. However, this characterization has not been useful in addressing the question for automorphism groups of (non-abelian) free groups. In his 2016 paper, Ozawa proved that the phenomenon of 'positivity' of the group Laplacian is observed in the real group algebra, meaning that the Laplacian can be decomposed into a 'sum of squares'. This result transformed checking property (T) into a finite-dimensional condition that can be performed with the assistance of computers. In this talk, we will introduce property (T) and discuss Ozawa's result in detail.

Tue, 11 Mar 2025

14:00 - 15:00
L4

A 200000-colour theorem

Jane Tan
(University of Oxford)
Abstract

The class of $t$-perfect graphs consists of graphs whose stable set polytopes are defined by their non-negativity, edge inequalities, and odd circuit inequalities. These were first studied by Chvátal in 1975, motivated by the related and well-studied class of perfect graphs. While perfect graphs are easy to colour, the same is not true for $t$-perfect graphs; numerous questions and conjectures have been posed, and even the most basic, on whether there exists some $k$ such that every $t$-perfect graph is $k$-colourable, has remained open since 1994. I will talk about joint work with Maria Chudnovsky, Linda Cook, James Davies, and Sang-il Oum in which we establish the first finite bound and show that a little less than 200 000 colours suffice.

Thu, 13 Mar 2025
16:00
L6

Parametrising complete intersections

Jakub Wiaterek
(University of Oxford)
Abstract

For some values of degrees d=(d_1,...,d_c), we construct a compactification of a Hilbert scheme of complete intersections of type d. We present both a quotient and a direct construction. Then we work towards the construction of a quasiprojective coarse moduli space of smooth complete intersections via Geometric Invariant Theory.

Tue, 20 May 2025
16:00
L6

Approaching the two-point Chowla conjecture via matrices

Cedric Pilatte
(University of Oxford)
Abstract

The two-point Chowla conjecture predicts that $\sum_{x<n<2x} \lambda(n)\lambda(n+1) = o(x)$ as $x\to \infty$, where $\lambda$ is the Liouville function (a $\{\pm 1\}$-valued multiplicative function encoding the parity of the number of prime factors). While this remains an open problem, weaker versions of this conjecture are known. In this talk, we outline an approach initiated by Helfgott and Radziwill, which reformulates the problem in terms of bounding the eigenvalues of a certain matrix.

Thu, 27 Feb 2025
12:00
C6

Aggregation-diffusion equations with saturation

Alejandro Fernández-Jiménez
(University of Oxford)
Abstract

On this talk we will focus on the family of aggregation-diffusion equations

 

$$\frac{\partial \rho}{\partial t} = \mathrm{div}\left(\mathrm{m}(\rho)\nabla (U'(\rho) + V) \right).$$

 

Here, $\mathrm{m}(s)$ represents a continuous and compactly supported nonlinear mobility (saturation) not necessarily concave. $U$ corresponds to the diffusive potential and includes all the porous medium cases, i.e. $U(s) = \frac{1}{m-1} s^m$ for $m > 0$ or $U(s) = s \log (s)$ if $m = 1$. $V$ corresponds to the attractive potential and it is such that $V \geq 0$, $V \in W^{2, \infty}$.

 

Taking advantage of a family of approximating problems, we show the existence of $C_0$-semigroups of $L^1$ contractions. We study the $\omega$-limit of the problem, its most relevant properties, and the appearance of free boundaries in the long-time behaviour. Furthermore, since this problem has a formal gradient-flow structure, we discuss the local/global minimisers of the corresponding free energy in the natural topology related to the set of initial data for the $L^\infty$-constrained gradient flow of probability densities. Finally, we explore the properties of a corresponding implicit finite volume scheme introduced by Bailo, Carrillo and Hu.

 

The talk presents joint work with Prof. J.A. Carrillo and Prof. D.  Gómez-Castro.

Tue, 18 Feb 2025

14:00 - 15:00
L4

Cube-root concentration of the chromatic number of $G(n,1/2)$ – sometimes

Oliver Riordan
(University of Oxford)
Abstract
A classical question in the theory of random graphs is 'how much does the chromatic number of $G(n,1/2)$ vary?' For example, roughly what is its standard deviation $\sigma_n$? An old argument of Shamir and Spencer gives an upper bound of $O(\sqrt{n})$, improved by a logarithmic factor by Alon. For general $n$, a result with Annika Heckel implies that $n^{1/2}$ is tight up to log factors. However, according to the 'zig-zag' conjecture $\sigma_n$ is expected to vary between $n^{1/4+o(1)}$ and $n^{1/2+o(1)}$ as $n$ varies. I will describe recent work with Rob Morris, building on work of Bollobás, Morris and Smith, giving an $O^*(n^{1/3})$ upper bound for certain values of $n$, the first bound beating $n^{1/2-o(1)}$, and almost matching the zig-zag conjecture for these $n$. The proof uses martingale methods, the entropy approach of Johansson, Kahn and Vu, the second moment method, and a new (we believe) way of thinking about the distribution of the independent sets in $G(n,1/2)$.
Thu, 06 Mar 2025

11:00 - 12:00
L5

Translation varieties (part 2)

Ehud Hrushovski
(University of Oxford)
Abstract

In algebraic geometry, the technique of dévissage reduces many questions to the case of curves. In difference and differential algebra, this is not the case, but the obstructions can be closely analysed. In difference algebra, they are difference varieties defined by equations of the form \si(𝑥)=𝑔𝑥\si(x)=gx, determined by an action of an algebraic group and an element g of this group. This is joint work with Zoé Chatzidakis.

Thu, 13 Feb 2025

11:00 - 12:00
C5

Around Siu inequality

Michał Szachniewicz
(University of Oxford)
Abstract

I will talk about the connections between the Siu inequality and existence of the model companion for GVFs. The talk will be partially based on a joint work with Antoine Sedillot.

Thu, 20 Feb 2025

11:00 - 12:00
C6

Translation varieties

Ehud Hrushovski
(University of Oxford)
Abstract

In algebraic geometry, the technique of dévissage reduces many questions to the case of curves. In difference and differential algebra, this is not the case, but the obstructions can be closely analysed. In difference algebra, they are difference varieties defined by equations of the form $\si(x)=g x$, determined by an action of an algebraic group and an element g of this group. This is joint work with Zoé Chatzidakis.

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