Thu, 24 Oct 2024
12:00
C5

A splitting theorem for manifolds with a convex boundary component.

Alessandro Cucinotta
(University of Oxford)
Abstract

The celebrated Splitting Theorem by Cheeger-Gromoll states that a manifold with non-negative Ricci curvature which contains a line is isometric to a product, where one of the factors is the real line. A related result was later proved by Kasue. He showed that a manifold with non-negative Ricci curvature and two mean convex boundary components, one of which is compact, is also isometric to a product. In this talk, I will present a variant of Kasue’s result based on joint work with Andrea Mondino. We consider manifolds with non-negative Ricci curvature and disconnected mean convex boundary. We show that if one boundary component is parabolic and convex, then the manifold is a product, where one of the factors is an interval of the real line. The result is an application of recently developed tools in synthetic geometry and exploits the interplay between Ricci curvature and optimal transport.

Thu, 05 Dec 2024

11:00 - 12:00
C1

Local-Global Principles and Fields Elementarily Characterised by Their Absolute Galois Groups

Benedikt Stock
(University of Oxford)
Abstract

Jochen Koenigsmann’s Habilitation introduced a classification of fields elementarily characterised by their absolute Galois groups, including two conjecturally empty families. The emptiness of one of these families would follow from a Galois cohomological conjecture concerning radically closed fields formulated by Koenigsmann. A promising approach to resolving this conjecture involves the use of local-global principles in Galois cohomology. This talk examines the conceptual foundations of this method, highlights its relevance to Koenigsmann’s classification, and evaluates existing local-global principles with regard to their applicability to this conjecture.

Thu, 21 Nov 2024

11:00 - 12:00
C3

Almost sure convergence to a constant for a mean-aggregated term language

Sam Adam-Day
(University of Oxford)
Abstract
With motivation coming from machine learning, we define a term language on graphs generalising many graph neural networks. Our main result is that the closed terms of this language converge almost surely to constants. This probabilistic result holds for Erdős–Rényi graphs for a variety of sparsity levels, as well as the Barabási–Albert preferential attachment graph distribution. The key technique is a kind of almost sure quantifier elimination. A natural extension of this language generalises first-order logic, and a similar convergence result can be obtained there.
 
Thu, 05 Dec 2024
16:00
Lecture Room 3

Zeros of polynomials with restricted coefficients: a problem of Littlewood

Benjamin Bedert
(University of Oxford)
Abstract

The study of polynomials whose coefficients lie in a given set $S$ (the most notable examples being $S=\{0,1\}$ or $\{-1,1\}$) has a long history leading to many interesting results and open problems. We begin with a brief general overview of this topic and then focus on the following old problem of Littlewood. Let $A$ be a set of positive integers, let $f_A(x)=\sum_{n\in A}\cos(nx)$ and define $Z(f_A)$ to be the number of zeros of $f_A$ in $[0,2\pi]$. The problem is to estimate the quantity $Z(N)$ which is defined to be the minimum of $Z(f_A)$ over all sets $A$ of size $N$. We discuss recent progress showing that $Z(N)\geqslant (\log \log N)^{1-o(1)}$ which provides an exponential improvement over the previous lower bound. 

A closely related question due to Borwein, Erd\'elyi and Littmann asks about the minimum number of zeros of a cosine polynomial with $\pm 1$-coefficients. Until recently it was unknown whether this even tends to infinity with the degree $N$. We also discuss work confirming this conjecture.

 

Thu, 28 Nov 2024
16:00
Lecture Room 3

Large sieve inequalities for exceptional Maass forms and applications

Alexandru Pascadi
(University of Oxford)
Abstract

A number of results on classical problems in analytic number theory rely on bounds for multilinear forms of Kloosterman sums, which in turn use deep inputs from the spectral theory of automorphic forms. We’ll discuss our recent work available at arxiv.org/abs/2404.04239, which uses this interplay between counting problems, exponential sums, and automorphic forms to improve results on the greatest prime factor of $n^2+1$, and on the exponents of distribution of primes and smooth numbers in arithmetic progressions.
The key ingredient in this work are certain “large sieve inequalities” for exceptional Maass forms, which improve classical results of Deshouillers-Iwaniec in special settings. These act as on-average substitutes for Selberg’s eigenvalue conjecture, narrowing (and sometimes completely closing) the gap between previous conditional and unconditional results.

Wed, 16 Oct 2024
16:00
L6

Solvability and Order Type for Finite Groups

Pawel Piwek
(University of Oxford)
Abstract

How much can the order type - the list of element orders (with multiplicities)—reveal about the structure of a finite group G? Can it tell us whether G is abelian, nilpotent? Can it always determine whether G is solvable? 

This last question was posed in 1987 by John G. Thompson and I answered it negatively this year. The search for a counterexample was quite a puzzle hunt! It involved turning the problem into linear algebra and solving an integer matrix equation Ax=b. This would be easy if not for the fact that the size of A was 100,000 by 10,000…

Mon, 02 Dec 2024
16:00
C3

TBC

Leo Gitin
(University of Oxford)
Abstract

TBC

Mon, 21 Oct 2024
16:00
C3

Monochromatic non-commuting products

Matt Bowen
(University of Oxford)
Abstract

We show that any finite coloring of an amenable group contains 'many' monochromatic sets of the form $\{x,y,xy,yx\},$ and natural extensions with more variables.  This gives the first combinatorial proof and extensions of Bergelson and McCutcheon's non-commutative Schur theorem.  Our main new tool is the introduction of what we call `quasirandom colorings,' a condition that is automatically satisfied by colorings of quasirandom groups, and a reduction to this case.

Mon, 14 Oct 2024
16:00
C3

Self-Similar Sets and Self-Similar Measures

Constantin Kogler
(University of Oxford)
Abstract

We give a gentle introduction to the theory of self-similar sets and self-similar measures. Connections of this topic to Diophantine approximation on Lie groups as well as to additive combinatorics will be exposed. In particular, we will discuss recent progress on Bernoulli convolutions. If time permits, we mention recent joint work with Samuel Kittle on absolutely continuous self-similar measures. 
 

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