University of Oxford

We will give a fairly self contained introduction to acylindrically hyperbolic groups, using mapping class groups as a motivating example. This will be a mainly expository talk, the aim is to make my topology seminar talk in week 5 more accessible to people who are less familiar with these topics.

The Skolem Problem asks to decide whether a linearly recurrent sequence (LRS) over the rationals has a zero term.  It is sometimes considered as the halting problem for linear loops.   In this talk we will give an overview of two current approaches to establishing decidability of this problem.  First, we observe that the Skolem Problem for LRS with simple characteristic roots is decidable subject to the $p$-adic Schanuel conjecture and the exponential-local-global principle.  Next, we define a set $S$ of positive integers such that (i) $S$ has positive lower density and (ii) The Skolem Problem is decidable relative to $S$, i.e., one can effectively determine the set of all zeros of a given LRS that lie in $S$.

The talk is based on joint work with Y. Bilu, F. Luca, J. Ouaknine, D. Pursar, and J. Nieuwveld.  

We will look at the branching of irreducible representations of symmetric groups from the perspective of Okounkov-Vershik, and then look at Hecke algebras, affine Hecke algebras and cyclotomic Hecke algebras, in particular how the graded Grothendieck groups of their module categories “are” irreducible highest weight modules for affine $sl_l$, where $l$ is the “quantum characteristic”, and the branching graph is a highest weight crystal (for affine $sl_l$). The Fock space realisation of the highest weight crystal will get us back to  the Young graph for in the case of the symmetric group that we considered at the beginning.

The question of when you can quasiisometrically embed a solvable Baumslag-Solitar group into another turns out to be equivalent to the question of when you can (1,A)-quasiisometrically embed a rooted tree into another rooted tree. We will briefly describe the geometry of the solvable Baumslag-Solitar groups before attacking the problem of embedding trees. We will find that the existence of (1,A)-quasiisometric embeddings between trees is intimately related to the boundedness of a family of integer sequences. 

Combining the classical theory of optimal transport with modern operator splitting techniques, I will present a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media,materials science, and biological swarming. Using the JKO scheme, along with the Benamou-Brenier dynamical characterization of the Wasserstein distance, we reduce computing the solution of these evolutionary PDEs to solving a sequence of fully discrete minimization problems, with strictly convex objective function and linear constraint. We compute the minimizer of these fully discrete problems by applying a recent, provably convergent primal dual splitting scheme for three operators. By leveraging the PDE’s underlying variational structure, ourmethod overcomes traditional stability issues arising from the strong nonlinearity and degeneracy, and it is also naturally positivity preserving and entropy decreasing. Furthermore, by transforming the traditional linear equality constraint, as has appeared in previous work, into a linear inequality constraint, our method converges in fewer iterations without sacrificing any accuracy. We prove that minimizers of the fully discrete problem converge to minimizers of the continuum JKO problem as the discretization is refined, and in the process, we recover convergence results for existing numerical methods for computing Wasserstein geodesics. Simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our numerical method will be shown.

Localisation is a powerful tool in proving and analysing various geometric inequalities, including isoperimertic inequality in the context of metric measure spaces. Its multi-dimensional generalisation is linked to optimal transport of vector measures and vector-valued Lipschitz maps. I shall present recent developments in this area: a partial affirmative answer to a conjecture of Klartag concerning partitions associated to Lipschitz maps on Euclidean space, and a negative answer to another conjecture of his concerning mass-balance condition for absolutely continuous vector measures. During the course of the talk I shall also discuss an intriguing notion of ghost subspaces related to the above mentioned partitions. 

In 2004, Diaconis and Gamburd computed statistics of secular coefficients in the circular unitary ensemble. They expressed the moments of the secular coefficients in terms of counts of magic squares. Their proof relied on the RSK correspondence. We'll present a combinatorial proof of their result, involving the characteristic map. The combinatorial proof is quite flexible and can handle other statistics as well. We'll connect the result and its proof to old and new questions in number theory, by formulating integer and function field analogues of the result, inspired by the Random Matrix Theory model for L-functions.

Partly based on the arXiv preprint https://arxiv.org/abs/2102.11966

In this talk we will discuss the correlations of the Riemann Zeta in various ranges, and prove a new result for correlations of squares. This problem is closely related to correlations of the characteristic polynomial of CUE with a very subtle difference. We will explain where this difference comes from, and what it means for the moments of moments of the Riemann Zeta, and its maximum in short intervals.

I will explain in what sense the theory of finite tensor categories is a categorification of the theory of finite dimensional algebras. In particular, I will introduce finite module categories, review a key result of Ostrik, and present Morita theory for finite categories. I will give many examples to illustrate these ideas. Then, I will explain the elementary properties of finite braided tensor categories. If time permits, I will also mention my own work, which consists in categorifying these ideas once more!

Understanding the impact of societal and economic change on the labour market is important for many causes, such as automation or the post-carbon transition. Occupational mobility plays a role in how these changes impact the labour market because of indirect effects, brought on by the different levels of direct impact felt by individual occupations. We develop an agent-based model which uses a network representation of the labour market to understand these impacts. This network connects occupations that workers have transitioned between in the past, and captures the complex structure of relationships between occupations within the labour market. We develop these networks in both space and time using rich survey data to compare occupational mobility across the United States and through economic upturns and downturns to start understanding the factors that influence differences in occupational mobility.