In recent years, there has been an increased interest in exploiting rank structure of matrices arising from the discretization of partial differential equations to develop fast direct solvers. In this talk, I will outline the fundamental ideas of this topic in the context of solving the integral equation formulation of the Helmholtz equation, known as the Lippmann-Schwinger equation, and will discuss some plans for future work to develop new, higher-order solvers. This is joint work with Gunnar Martinsson.

This talk will feature a brief introduction to persistent homology, the vanguard technique in topological data analysis. Nothing will be required of the audience beyond a willingness to row-reduce enormous matrices (by hand if we can, by machine if we must).

For decades, researchers have been studying efficient numerical methods to solve differential equations, most of them optimised for one-core processors. However, we are about to reach the limit in the amount of processing power we can squeeze into a single processor. This explains the trend in today's computing industry to design high-performance processors looking at parallel architectures. As a result, there is a need to develop low-complexity parallel algorithms capable of running efficiently in terms of computational time and electric power.

Parallelisation across time appears to be a promising way to provide more parallelism. In this talk, we will introduce the main algorithms, following (Gander, 2015), with a particular focus on the parareal algorithm.

Buildings are geometric objects, originally introduced by Tits to study Lie groups that act on their corresponding building. Apart from their significance for Lie groups, buidings and their automorphism groups are a rich source of examples for groups with interesting properties (for example, it is a result of Caprace that some buildings admit an automorphism group which is compactly generated, abstractly simple and locally compact). Right Angled Buildings (RABs) are a specific kind of building whose geometry can be well understood as it resembles the geometry of a tree. This allows one to generalise ideas like the Burger-Mozes universal groups to the setting of RABs.
I plan to give an introduction to RABs. As a complete formal introduction to buildings would take more than an hour, I will instead present various illustrative examples to give you an idea of what you should have in mind when you think of a (right-angled) building. I will be as formal as I can in presenting the basic features of buildings - Coxeter complexes, chambers, apartments, retractions and residues.  In the remaining time I will say as much as I can about the geometry of RABs, and explain how to use this geometry to derive a structure theorem for the automorphism group of a RAB, towards a definition of Burger-Mozes universal groups for RABs.