University of Oxford

Condensed Mathematics is a tool recently developed by Clausen and Scholze and it is proving fruitful in many areas of algebra and geometry. In this talk, we will cover the definition of condensed sets, the analogues of topological spaces in the condensed setting. We will also talk about condensed modules over a ring and some of their nice properties like forming an abelian category. Finally, we'll discuss some recent results that have been obtained through the application of Condensed Mathematics.

Correlation operators are used in data assimilation algorithms
to weight the contribution of prior and observation information.
Efficient implementation of these operators is therefore crucial for
operational implementations. Diffusion-based correlation operators are popular in ocean data assimilation, but can require a large number of serial matrix-vector products. An all-at-once formulation removes this requirement, and offers the opportunity to exploit modern computer architectures. High quality preconditioners for the all-at-once approach are well-known, but impossible to apply in practice for the
high-dimensional problems that occur in oceanography. In this talk we
consider a nested preconditioning approach which retains many of the
beneficial properties of the ideal analytic preconditioner while
remaining affordable in terms of memory and computational resource.

The Riemann hypothesis (RH) is one of the great open problems in
mathematics. It arose from the study of prime numbers in an analytic
context, and—as often occurs in mathematics—developed analogies in an
algebraic setting, leading to the influential Weil conjectures. RH for
curves over finite fields was proven in the 1940’s by Weil using
algebraic-geometric methods, and later reproven by Stepanov and
Bombieri by elementary means. In this talk, we use RH for curves to
prove the Weil bound for certain (Kloosterman) exponential sums, which
in turn is a fundamental tool in the study of prime numbers.

Persistent homology is both a powerful framework for data science and a fruitful source of mathematical questions. Here, we will give an introduction to both single-parameter and multiparameter persistent homology. We will see some examples of how topology has been successfully applied to the real world, and also explore some of the pure-mathematical ideas that arise from this new perspective.

We review a few instances in which the first $\ell^2$ Betti number of a group is a profinite invariant and we discuss some applications and open problems.

We survey the theory of $\ell^2$-invariants, their applications in group theory and topology, and introduce a positive characteristic version of $\ell^2$-theory. We also discuss the Atiyah and Lück approximation conjectures, two of the central problems in this area.

Profinite rigidity is essentially the study of which groups can be distinguished from each other by their finite quotients. This talk is meant to give a gentle introduction to the area - I will explain which questions are the right ones to ask and give an overview of some of the main results in the field. I will assume knowledge of what a group presentation is.

Since Segal's formulation of axioms for 2d CFTs in the 80s, it has remained a major problem to construct examples of CFTs that satisfy the axioms.

I will report on ongoing joint work with James Tener in that direction.

Let $X$ denote an open subset of $\mathbb{C}^d$, and $\mathcal{O}$ its sheaf of holomorphic functions. In the 1970’s, Ishimura studied the morphisms of sheaves $P\colon\mathcal{O}\to\mathcal{O}$ of $\mathbb{C}$-vector spaces which are continuous, that is the maps $P(U)\colon\mathcal{O}(U)\to\mathcal{O}(U)$ on the sections are continuous. In this talk, we explain his result, and explore its analogues in the non-Archimedean world.

The canonical dimension is a fundamental integer-valued invariant that is attached to mod p representations of p-adic Lie groups. I will explain why it is both an asymptotic measure of growth, and an algebraic quantity strongly related to Krull dimension. We will survey algebraic tools that can be applied in its calculation, and describe results spanning the last twenty years. I'll present a new theorem and suggest its possible significance for the mod p local Langlands programme.