A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs.  Given a hereditary graph property H, the speed of H is the function which sends an integer n to the number of distinct elements in H with underlying set {1,...,n}.  Not just any function can occur as the speed of hereditary graph property.  Specifically, there are discrete ``jumps" in the possible speeds.  Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollob\'{a}s, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized.  In contrast to this, many aspects of this problem in the hypergraph setting remained unknown.  In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds.  The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss.  This is joint work with Chris Laskowski.

We take a look at difference fields with several commuting automorphisms. The theory of difference fields with one distinguished automorphism has a model companion known as ACFA, which Zoe Chatzidakis and Ehud Hrushovski have studied in depth. However, Hrushovski has proved that if you look at fields with two or more commuting automorphisms, then the existentially closed models of the theory do not form a first order model class. We introduce a non-elementary framework for studying them. We then discuss how to generalise a result of Kowalski and Pillay that every definable group (in ACFA) virtually embeds into an algebraic group. This is joint work in progress with Zoe Chatzidakis and Nick Ramsey.

We will be discussing a Fourier-analytic approach
to optimal matching between independent samples, with
an elementary proof of the Ajtai-Komlos-Tusnady theorem.
The talk is based on a joint work with Michel Ledoux.

Consider a random N by N unitary matrix chosen according to Haar measure. A classical result of Diaconis and Shashahani shows that traces of low powers of this matrix tend in distribution to independent centered gaussians as N grows. A result of Johansson shows that this convergence is very fast -- superexponential in fact. Similar results hold for other classical compact groups. This talk will discuss analogues of these results for N by N matrices taken from a classical group over a finite field, showing that as N grows, traces of powers of these matrices equidistribute superexponentially. A little surprisingly, the proof is connected to the distribution in short intervals of certain arithmetic functions in F_q[T]. This is joint work with O. Gorodetsky.

Background: The traditional business models for B2B freight and distribution are struggling with underutilised transport capacities resulting in higher costs, excessive environmental damage and unnecessary congestion. The scale of the problem is captured by the European Environmental Agency: only 63% of journeys carry useful load and the average vehicle utilisation is under 60% (by weight or volume). Decarbonisation of vehicles would address only part of the problem. That is why leading sector researchers estimate that freight collaboration (co-shipment) will deliver a step change improvement in vehicle fill and thus remove unproductive journeys delivering over 20% of cost savings and >25% reduction in environmental footprint. However, these benefits can only be achieved at a scale that involves 100’s of players collaborating at a national or pan-regional level. Such scale and level of complexity creates a massive optimisation challenge that current market solutions are unable to handle (modern route planning solutions optimise deliveries only within the “4 walls” of a single business).

Maths challenge: The mentioned above optimisation challenge could be expressed as an extended version of the TSP, but with multiple optimisation objectives (other than distance). Moreover, besides the scale and multi-agent setup (many shippers, carriers and recipients engaged simultaneously) the model would have to operate a number of variables and constraints, which in addition to the obvious ones also include: time (despatch/delivery dates/slots and journey durations), volume (items to be delivered), transport equipment with respective rate-cards from different carriers, et al. With the possible variability of despatch locations (when clients have multi-warehouse setup) this potentially creates a very-large non-convex optimisation problem that would require development of new, much faster algorithms and approaches. Such algorithm should be capable of finding “local” optimums and subsequently improve them within a very short window i.e. in minutes, which would be required to drive and manage effective inter-company collaboration across many parties involved. We tried a few different approaches eg used Gurobi solver, which even with clustering was still too slow and lacked scalability, only to realise that we need to build such an algorithm in-house.

Ask: We started to investigate other approaches like Simulated Annealing or Gravitational Emulation Local Search but this work is preliminary and new and better ideas are of interest. So in support of our Technical Feasibility study we are looking for support in identification of the best approach and design of the actual algorithm that we’ll use in the development of our Proof of Concept.  

I will explain the basics of 2-representation theory and will explain an approach to classifying 'simple' 2-representations of the Hecke 2-category (aka Soergel bimodules) for finite Coxeter types.