I will talk about the behaviour of random maps on surfaces (for example, random triangulations) of given genus, when their size tends to infinity. Such questions can be asked from the viewpoint of the local behaviour (Benjamini-Schramm convergence) or global behaviour (diameter, Gromov Hausdorff convergence), and in both cases, much combinatorics is involved. I will survey the landmark results for the case of fixed genus, and state very recent results in which we manage to address the "high genus" regime, when the genus grows proportionally to the size – for this regime we establish isoperimetric inequalities and prove the long-suspected fact that the diameter is logarithmic with high probability.

Based on joint work with Thomas Budzinski and Baptiste Louf.

Coboundary expansion is a notion introduced by Linial and Meshulam, and by Gromov that combines combinatorics topology and linear algebra. Kaufman and Lubotzky observed its relation to "Property testing", and in recent years it has found several applications in theoretical computer science, including for error correcting codes (both classical and quantum), for PCP agreement tests, and even for studying polarization in social networks.

In the talk I will introduce this notion and some of its applications. No prior knowledge is assumed, of course.

Does the limit construction for inverse systems of first-order structures preserve elementary equivalence? I will give sufficient conditions for when this is the case. Using Karp's theorem, we explain the connection between a syntactic and formal-semantic approach to inverse limits of structures. We use this to give a simple proof of van den Dries' AKE theorem (in ZFC), a general AKE theorem for mixed characteristic henselian valued fields with no assumptions on ramification. We also recall a seemingly forgotten result of Feferman, that can be interpreted as a "saturated" AKE theorem in positive characteristic: given two elementarily equivalent $\aleph_1$-saturated fields $k$ and $k'$, the formal power series rings $k[[t]]$ and $k'[[t]]$ are elementarily equivalent as well. We thus hope to popularise some ideas from categorical logic.

Motivated by work of Hrushovski on pseudofinite fields with an additive character we investigate the theory ACFA+ which is the model companion of the theory of difference fields with an additive character on the fixed field. Building on results by Hrushovski we can recover it as the characteristic 0-asymptotic theory of the algebraic closure of finite fields with the Frobenius-automorphism and the standard character on the fixed field. We characterise 3-amalgamation in ACFA+. As cosequences we obtain that ACFA+ is a simple theory, an explicit description of the connected component of the Kim-Pillay group and (weak) elimination of imaginaries. If time permits we present some results on higher amalgamation.

There will be free pizza provided for all attendees directly after the event just outside L1, so please do come along!

North Wing
Speaker: Alexandru Pascadi 
Title: Points on modular hyperbolas and sums of Kloosterman sums
Abstract: Given a positive integer c, how many integer points (x, y) with xy = 1 (mod c) can we find in a small box? The dual of this problem concerns bounding certain exponential sums, which show up in methods from the spectral theory of automorphic forms. We'll explore how a simple combinatorial trick of Cilleruelo-Garaev leads to good bounds for these sums; following recent work of the speaker, this ultimately has consequences about multiple problems in analytic number theory (such as counting primes in arithmetic progressions to large moduli, and studying the greatest prime factors of quadratic polynomials).

South Wing
Speaker: Tim LaRock
Title: Encapsulation Structure and Dynamics in Hypergraphs
Abstract: Within the field of Network Science, hypergraphs are a powerful modelling framework used to represent systems where interactions may involve an arbitrary number of agents, rather than exactly two agents at a time as in traditional network models. As part of a recent push to understand the structure of these group interactions, in this talk we will explore the extent to which smaller hyperedges are subsets of larger hyperedges in real-world and synthetic hypergraphs, a property that we call encapsulation. Building on the concept of line graphs, we develop measures to quantify the relations existing between hyperedges of different sizes and, as a byproduct, the compatibility of the data with a simplicial complex representation–whose encapsulation would be maximum. Finally, we will turn to the impact of the observed structural patterns on diffusive dynamics, focusing on a variant of threshold models, called encapsulation dynamics, and demonstrate that non-random patterns can accelerate spreading through the system.

Speaker: Mattia Magnabosco (Newton Fellow, Maths)
Title: Synthetic Ricci curvature bounds in sub-Riemannian manifolds
Abstract: In Riemannian manifolds, a uniform bound on the Ricci curvature tensor allows to control the volume growth along the geodesic flow. Building upon this observation, Lott, Sturm and Villani introduced a synthetic notion of curvature-dimension bounds in the non-smooth setting of metric measure spaces. This condition, called CD(K,N), is formulated in terms of the optimal transport interpolation of measures and consists in a convexity property of the Rényi entropy functionals along Wasserstein geodesics. The CD(K,N) condition represents a lower Ricci curvature bound by K and an upper bound on the dimension by N, and it is coherent with the smooth setting, as in a Riemannian manifold it is equivalent to a lower bound on the Ricci curvature tensor. However, the same relation between curvature and CD(K,N) condition does not hold for sub-Riemannian (and sub-Finsler) manifolds. 

Speaker: Rebecca Lewis (Florence Nightingale Bicentenary Fellow, Stats)
Title: High-dimensional statistics
Abstract: Due to the increasing ease with which we collect and store information, modern data sets have grown in size. Whilst these datasets have the potential to yield new insights in a variety of areas, extracting useful information from them can be difficult. In this talk, we will discuss these challenges.

The goal of Post-Quantum Cryptography (PQC) is to design cryptosystems which are secure against classical and quantum adversaries. A topic of fundamental research for decades, the status of PQC drastically changed with the NIST PQC standardization process. Recently there have been AI attacks on some of the proposed systems to PQC. In this talk, we will give an overview of the progress of quantum computing and how it will affect the security landscape. 

Group-based cryptography is a relatively new family in post-quantum cryptography, with high potential. I will give a general survey of the status of post-quantum group-based cryptography and present some recent results.

In the second part of my talk, I speak about Post-quantum hash functions using special linear groups with implication to post-quantum blockchain technologies.