University of Oxford

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).

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.

We will give a relaxed introduction to some of the most classical dynamical systems - Anosov flows. These flows were highly influential in the development of ideas which the audience might be more familiar with. For example, Anosov flows give rise to exponential group growth and taut foliations, both of which we will discuss. Finally, we will talk about some recent work obstructing Anosov flows and their combinatorial analogs - veering triangulations

Property (T) was introduced by Kazhdan in the sixties to show that lattices in higher rank semisimple Lie groups are finitely generated. We will discuss some classical examples of groups that satisfy this property, with a particular focus on SL(3, R).

For a hyperbolic knot there are two types of invariants, the hyperbolic invariants coming from the geometric structure and the classical invariants coming from the topology or combinatorics. It has been observed in many different cases that these seemingly different types of invariants are in fact related. I will give examples of these relationships and discuss in particular a link by Stoimenow between the determinant and volume.  

The Ivanov conjecture is equivalent to the statement that every covering map of surfaces has the so-called Putman-Wieland property. I will discuss my recent work with Vlad Marković, where we prove it for asymptotically all coverings as the degree grows. I will give some overview of our main tool: spectral geometry, which is related to objects like the heat kernel of a hyperbolic surface, or Cheeger connectivity constant.

Let K be a non-archimedean field of mixed characteristic (0,p), and let L be a finite extension of
the p-adic numbers contained in K. The speaker is interested in the continuous representations of a
given L-analytic group G in locally convex (usually infinite dimensional) topological vector spaces over K.
This is, up to technicalities, equivalent to studying certain topological modules over the locally
analytic distribution algebra D(G,K) of G. But doing algebra with topological objects is hard!
In this talk, we present an excellent remedy, found by Schneider and Teitelbaum in the early 2000s.

In the 1960's Langlands proposed a generalisation of Class Field Theory. I will review this and describe a new approach using the trace formua as well as some analytic arguments reminiscent of those used in the classical case. In more concrete terms the problem is to prove general modularity theorems, and I will explain the progress I have made on this problem.