Optimal transport (OT) has recently gained a lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability distributions. It finds applications in both supervised learning (using geometric loss functions) and unsupervised learning (to perform generative model fitting). OT is however plagued by the curse of dimensionality, since it might require a number of samples which grows exponentially with the dimension. In this talk, I will explain how to leverage entropic regularization methods to define computationally efficient loss functions, approximating OT with a better sample complexity. More information and references can be found on the website of our book "Computational Optimal Transport".

In work of Haiden-Katzarkov-Konsevich-Pandit (HKKP), a canonical filtration, labeled by sequences of real numbers, of a semistable quiver representation or vector bundle on a curve is defined. The HKKP filtration is a purely algebraic object that depends only on a lattice, yet it governs the asymptotic behaviour of a natural gradient flow in the space of metrics of the object. In this talk, we show that the HKKP filtration can be recovered from the stack of semistable objects and a so called norm on graded points, thereby generalising the HKKP filtration to other moduli problems of non-linear origin.

Asymptotic dimension was introduced by Gromov as an invariant of finitely generated groups. It can be shown that if two metric spaces are quasi-isometric then they have the same asymptotic dimension. In 1998, the asymptotic dimension achieved particular prominence in geometric group theory after a paper of Guoliang Yu, which proved the Novikov conjecture for groups with finite asymptotic dimension. Unfortunately, not all finitely generated groups have finite asymptotic dimension. 

In this talk, we will introduce some basic tools to compute the asymptotic dimension of groups. We will also find upper bounds for the asymptotic dimension of a few well-known classes of finitely generated groups, such as hyperbolic groups, and if time permits, we will see why one-relator groups have asymptotic dimension at most two.

A common pastime of geometric group theorists is to try and derive algebraic information about a group from the geometric properties of its Cayley graphs. One of the most classical demonstrations of this can be seen in the work of Maschke (1896) in characterising those finite groups with planar Cayley graphs. Since then, much work has been done on this topic. In this talk, I will attempt to survey some results in this area, and show that the class group with planar Cayley graphs is QI-rigid.