Thu, 07 Mar 2024

11:00 - 12:00
C3

Model theory of Booleanizations, products and sheaves of structures

Jamshid Derakhshan
(University of Oxford)
Abstract

I will talk about some model-theoretic properties of Booleanizations of theories, subdirect products of structures, and sheaves of structures. I will discuss a result of Macintyre from 1973 on model-completeness, and more recent results jointly with Ehud Hrushovski and with Angus Macintyre.

Thu, 29 Feb 2024

11:00 - 12:00
C3

Coherent group actions

Martin Bays
(University of Oxford)
Abstract

I will discuss aspects of some work in progress with Tingxiang Zou, in which we continue the investigation of pseudofinite sets coarsely respecting structures of algebraic geometry, focusing on algebraic group actions. Using a version of Balog-Szemerédi-Gowers-Tao for group actions, we find quite weak hypotheses which rule out non-abelian group actions, and we are applying this to obtain new Elekes-Szabó results in which the general position hypothesis is fully weakened in one co-ordinate.

Tue, 20 Feb 2024

14:00 - 15:00
L4

Hamiltonicity of expanders: optimal bounds and applications

Nemanja Draganić
(University of Oxford)
Abstract

An $n$-vertex graph $G$ is a $C$-expander if $|N(X)|\geq C|X|$ for every $X\subseteq V(G)$ with $|X|< n/2C$ and there is an edge between every two disjoint sets of at least $n/2C$ vertices.

We show that there is some constant $C>0$ for which every $C$-expander is Hamiltonian. In particular, this implies the well known conjecture of Krivelevich and Sudakov from 2003 on Hamilton cycles in $(n,d,\lambda)$-graphs. This completes a long line of research on the Hamiltonicity of sparse graphs, and has many applications.

Joint work with R. Montgomery, D. Munhá Correia, A. Pokrovskiy and B. Sudakov.

Thu, 08 Feb 2024

11:00 - 12:00
C3

Model companions of fields with no points in hyperbolic varieties

Michal Szachniewicz
(University of Oxford)
Abstract

This talk is based on a joint work with Vincent Jinhe Ye. I will define various classes of hyperbolic varieties (Broody hyperbolic, algebraically hyperbolic, bounded, groupless) and discuss existence of model companions of classes of fields that exclude them. This is related to moduli spaces of maps to hyperbolic varieties and to the (open) question whether the above mentioned hyperbolicity notions are in fact equivalent.

Thu, 07 Mar 2024

17:00 - 18:00

Some applications of motivic integration in group theory and arithmetic geometry

Itay Glazer
(University of Oxford)
Abstract
Let f:X-->Y be a polynomial map between smooth varieties, and let mu be a smooth, compactly supported measure on X(F), where F is a local field. An interesting phenomenon is that bad singularities of f manifest themselves in poor analytic behavior of the pushforward f_*(mu) of mu by f. 
I will discuss this phenomenon in two settings; the first is when f:A^n-->A^m is a polynomial map between affine spaces and mu is the Haar measure on Z_p^n, and the second is when f:G^2-->G is a word map (e.g. the commutator map (g,h)-->ghg^(-1)h^(-1)) between simple algebraic groups, and mu is a Haar measure on G(Z_p). 
In these cases (and in other "real life situations"), mu and consequently f_*(mu) are constructible measures in the sense of Cluckers-Loeser motivic integration. We utilize this fact to show that the analytic behavior of f_*(mu) cannot be too bad, leading to geometric and probabilistic applications.
 
Based on joint works with Yotam Hendel and Raf Cluckers.
Thu, 15 Feb 2024

17:00 - 18:00

On logical structure of physical theories and limits

Boris Zilber
(University of Oxford)
Abstract

I am going to discuss main results of my paper "Physics over a finite field and Wick rotation", arxiv 2306.15698. It introduces a structure over a pseudo-finite field which might be of interest in Foundations of Physics. The main theorem establishes an analogue of the polar co-ordinate system in the pseudo-finite field. A stability classification status of the structure is an open question.

Thu, 01 Feb 2024

11:00 - 12:00
C3

Non-archimedean equidistribution and L-polynomials of curves over finite fields

Francesco Ballini
(University of Oxford)
Abstract

Let q be a prime power and let C be a smooth curve defined over F_q. The number of points of C over the finite extensions of F_q are determined by the Zeta function of C, which can be written in the form P_C(t)/((1-t)(1-qt)), where P_C(t) is a polynomial of degree 2g and g is the genus of C; this is often called the L-polynomial of C. We use a Chebotarev-like statement (over function fields instead of Z) due to Katz in order to study the distribution, as C varies, of the coefficients of P_C(t) in a non-archimedean setting.

Tue, 06 Feb 2024

14:00 - 15:00
L4

Typical Ramsey properties of the primes and abelian groups

Robert Hancock
(University of Oxford)
Abstract

Given a matrix $A$ with integer entries, a subset $S$ of an abelian group and $r\in\mathbb N$, we say that $S$ is $(A,r)$-Rado if any $r$-colouring of $S$ yields a monochromatic solution to the system of equations $Ax=0$. A classical result of Rado characterises all those matrices $A$ such that $\mathbb N$ is $(A,r)$-Rado for all $r \in \mathbb N$. Rödl and Ruciński, and Friedgut, Rödl and Schacht proved a random version of Rado’s theorem where one considers a random subset of $[n]:=\{1,\dots,n\}$.

In this paper, we investigate the analogous random Ramsey problem in the more general setting of abelian groups. Given a sequence $(S_n)_{n\in\mathbb N}$ of finite subsets of abelian groups, let $S_{n,p}$ be a random subset of $S_n$ obtained by including each element of $S_n$ independently with probability $p$. We are interested in determining the probability threshold for $S_{n,p}$ being $(A,r)$-Rado.

Our main result is a general black box for hypergraphs which we use to tackle problems of this type. Using this tool in conjunction with a series of supersaturation results, we determine the probability threshold for a number of different cases. A consequence of the Green-Tao theorem is the van der Waerden theorem for the primes: every finite colouring of the primes contains arbitrarily long monochromatic arithmetic progressions. Using our machinery, we obtain a random version of this result. We also prove a novel supersaturation result for $[n]^d$ and use it to prove an integer lattice generalisation of the random version of Rado's theorem.

This is joint work with Andrea Freschi and Andrew Treglown (both University of Birmingham).

Fri, 09 Feb 2024

12:00 - 13:00

A (higher) categorical approach to analytic D-modules

Arun Soor
(University of Oxford)
Abstract

In this possibly speculative talk I will try to outline a way to define analytic D-modules, using (higher) category theory and the ``six operations" on quasicoherent sheaves as the main tools. The aim is to follow the successful approach of Andy Jiang in the algebraic setting, who obtained such a theory without using stacks or formal schemes (as in Gaitsgory-Rozenblyum's approach). By using local cohomology, Jiang was able to avoid enlarging the category of algebras beyond the usual ones. We believe that an analytic variant of local cohomology can be used to recover the Ardakov-Wadsley theory of D-cap modules ``on the nose". (Work in progress).

Fri, 08 Mar 2024

12:00 - 13:00
Quillen Room

Another Flavour of String Topology

Joe Davies
(University of Oxford)
Abstract

String topology is an umbrella under which lives a family of algebraic structures on the homology of the (compact-open) loop space of a closed smooth manifold, M. Of great interest are the string product and coproduct, in view of the failure of the latter to be a homotopy invariant. We will discuss some existing algebraic and geometric perspectives on these operations, and give some examples that probe the extent to which the string coproduct fails to be a homotopy invariant. We will sketch an alternative point of view on string topology as the study of the derived bornological smooth loop stack and explain why this is a promising model for the observed phenomena of string topology.

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