One of many overlaps between logic and topology is duality: Stone duality links Boolean algebras with zero-dimensional compact Hausdorff spaces, and gives a useful topological way of describing certain phenomena in first order logic; and there are generalisations that allow one to study infinitary logics also. We will look at a couple of ways in which this duality theory is useful.'

The smooth representation theory of a p-adic reductive group G with characteristic zero coefficients is very closely connected to the module theory of its (pro-p) Iwahori-Hecke algebra H(G). In the modular case, where the coefficients have characteristic p, this connection breaks down to a large extent. I will first explain how this connection can be reinstated by passing to a derived setting. It involves a certain differential graded algebra whose zeroth cohomology is H(G). Then I will report on a joint project with R. Ollivier in which we analyze the higher cohomology groups of this dg algebra for the group G = SL_2.

Inverse problems are ubiquitous in many areas of Science and Engineering and, once discretised, they lead to ill-conditioned linear systems, often of huge dimensions: regularisation consists in replacing the original system by a nearby problem with better numerical properties, in order to find meaningful approximations of its solution. In this talk we will explore the regularisation properties of many iterative methods based on Krylov subspaces. After surveying some basic methods such as CGLS and GMRES, innovative approaches based on flexible variants of CGLS and GMRES will be presented, in order to efficiently enforce nonnegativity and sparsity into the solution.