The talks will present some of the research topics that will be supported by the IRL FAMSI. The presentations will be colloquium style, i.e. accessible to people trained at Honours level (or above!) in any field of Mathematics.

1-1:30pm: Andrew Hassell. A new approach to the nonlinear Schrödinger equation.

1:30-2pm: Kate Turner. A new computation paradigm for computing the persistent homology of Rips filtrations.

Abstract: Given a point cloud in Euclidean space, and a fixed length scale, we can create simplicial complexes (called Rips complexes) to represent that point cloud using the pairwise distances between the points. By tracking how the homology classes as we increase that length scale we summarise the topology and the geometry of the “shape” of the point cloud, in what is called the persistent homology of its Rips filtration. A major obstacle to more widespread take up of persistent homology as a data analysis tool is the long computation time and, more importantly, the large memory requirements needed to store the filtrations of Rips complexes and compute its persistent homology. We bypass these issues by finding a "Reduced Rips Filtration" which has the same degree-1 persistent homology but with dramatically fewer simplices. The talk is based off joint work is with Musashi Koyama, Facundo Memoli and Vanessa Robins.

2-2:30pm: Asilata Bapat. Faithfulness questions about generalised Burau representations.

Abstract: The n-strand braid group is an infinite group that plays a significant role in many areas of mathematics. Understanding "representations" — or matrix realisations — of the braid group is thus an important problem. A representation that gives an injective matrix realisation, also known as a "faithful representation", is particularly useful because it does not lose information. Indeed, n-strand braid groups are known to have faithful representations into matrix groups of sufficiently large dimension.

The Burau representation is a smaller dimensional representation of the n-strand braid group that was found almost 90 years ago. It is known to be faithful when n is 3, and known to be unfaithful when n is at least 5.
However, faithfulness remains unknown when n is 4. In this work we consider a zoo of parallel questions about the Burau representation for Artin--Tits groups, which generalise the classical braid group. We develop techniques to produce counterexamples to faithfulness for such representations. We use these techniques to prove that the Burau representation is unfaithful for Artin--Tits groups of type "affine A3", which is closely related to the classical n = 5 case.

This talk is based on joint work with Hoel Queffelec.

2:30-3pm: Afternoon Tea at Level 3 

3-3:30pm: Tony Licata. Group theory and dynamics in triangulated categories.

Abstract:  In the last twenty years, representation theorists have constructed a number of interesting examples of group actions on triangulated categories.  These actions are extremely rich, and several features of these actions suggest that they should be closely related to other, more classical parts of geometric group theory.  I'll try to give a sense of both what is interesting about group actions on triangulated categories and also describe some concrete goals for the next several years.

 3:30-4pm: Ben Andrews. Mean curvature flow of spatial regions in a spacetime.

Abstract:  I will describe some current work with my current PhD student Qiyu Zhou, in which we try to deform “space-like convex” submanifolds with one spacelike codimension in a Pseudo-Riemannian spacetime.  These submanifolds are spacelike (meaning that tangent vectors are all spacelike), and also have spacelike second fundamental form at each point.  I will explain some of the known results for convex hypersurfaces evolving under mean curvature flow in Euclidean space, and our new results for spacelike-convex submanifolds in Minkowski space times.  At the end I will mention some more speculative ideas about the situation in physical reasonable relativistic spacetimes.
 

4-4:30pm: Pierre Portal. Diffusion equations with rough data and coefficients. 

Abstract: Surely linear diffusion equations with $L^p$ initial data and coefficients that are bounded and measurable in all (space and time) variables are well posed, right? If $p=2$, yes: this goes back to J.L. Lions in 1957. For $p \neq 0$, not so much, but fear not: Pascal Auscher, Sylvie Monniaux and I have developed a satisfying well posedness theory over the past ten years. The key idea is to understand what an appropriate "$L^p$ energy space" should be.   

4:30-6pm: research discussions with Frederic Herau (CNRS-Maths Deputy Director for International Relations), and Sylvie Monniaux (IRL FAMSI Director).

6-8pm: join us for drinks and dinner at Badger&Co.