Your brain has its own waterscape: whether you are reading or sleeping, fluid flows around or through the brain tissue and clears waste in the process. These physiological processes are crucial for the well-being of the brain. In spite of their importance we understand them but little. Mathematics and numerics could play a crucial role in gaining new insight. Indeed, medical doctors express an urgent need for modeling of water transport through the brain, to overcome limitations in traditional techniques. Surprisingly little attention has been paid to the numerics of the brain’s waterscape however, and fundamental knowledge is missing. In this talk, I will discuss mathematical models and numerical methods for the brain's waterscape across scales - from viewing the brain as a poroelastic medium at the macroscale and zooming in to studying electrical, chemical and mechanical interactions between brain cells at the microscale.
 

A group splits as an HNN-extension if and only if the rank of its abelianisation is strictly positive. If we fix a class of groups one may ask a few questions about these splittings: How distorted are the vertex and edge groups? What form can the vertex and edge groups take? If they remain in our fixed class, do they also split? If so, under iteration will we terminate at something nice? In this talk we will answer all these questions for the class of one-relator groups and go through an example or two. Time permitting, we will also discuss possible generalisations to groups with staggered presentations.

We will answer the following question: given a finite simplicial complex X acted on by a finite group G, which object stores the minimal amount of information about the symmetries of X in such a way that we can reconstruct both X and the group action? The natural first guess would be the quotient X/G, which remembers one representative from each orbit. However, it does not tell us the size of each orbit or how to glue together simplices to recover X. Our desired object is, in fact, a complex of groups. We will understand two processes: compression and reconstruction and see primarily through an example how to answer our initial question.

The mapping class group of a surface is a group of homeomorphisms of that surface, and these groups have been very well studied in the last 50 years. The talk will be focused on a way to understand such a group by looking at the subsurfaces of the corresponding surface; this is the so-called "Masur-Minsky hierarchy machinery". We'll finish with a non-technical discussion of hierarchically hyperbolic groups, which are a popular area of current research, and of which mapping class groups are important motivating examples. No prior knowledge of the objects involved will be assumed.

Every Cayley graph of a finitely generated group has some basic properties: they are locally finite, connected, and vertex-transitive. These are not sufficient conditions, there are some well known examples of graphs that have all these properties but are non-Cayley. These examples do however "look like" Cayley graphs, which leads to the natural question of if there exist any vertex-transitive graphs that are completely unlike any Cayley graph. I plan to give some of the history of this question, as well as the construction of the example that finally answered it.

Mirror symmetry, in a crude formulation, is usually presented as a correspondence between curve counting on a Calabi-Yau variety X, and some invariants extracted from a mirror family of Calabi-Yau varieties. After the physicists Bershadsky-Cecotti-Ooguri-Vafa (henceforth BCOV), this is organised according to the genus of the curves in X we wish to enumerate, and gives rise to an infinite recurrence of differential equations. In this talk, I will give a general introduction to these problems, and present a rigorous mathematical formulation of the BCOV conjecture at genus one, in terms of a lifting of the Grothendieck-Riemann-Roch. I will explain the main ideas of the proof of the conjecture for Calabi-Yau hypersurfaces in projective space, based on the Riemann-Roch theorem in Arakelov geometry. Our results generalise from dimension 3 to arbitrary dimensions previous work of Fang-Lu-Yoshikawa.
 

This is joint work with G. Freixas and C. Mourougane.