TBA

 Given a group acting on a metric space X, one is often interested in the kernel of the action, consisting of those elements that fix every point of X. From a coarse geometric perspective, however, this notion is unsatisfactory, as the kernel is generally not invariant under G-equivariant quasi-isometries. To address this, one can instead consider the coarse kernel, defined as the collection of group elements that move every point of X by a uniformly bounded amount. In this talk, we study this coarse kernel under various assumptions on the action. 

When the action is geometric, we give a purely algebraic characterisation of the coarse kernel as the FC-centre of the group. We then specialise to actions on CAT(0) spaces, where we investigate the coarse kernel via the curtain model, a hyperbolic space associated to a CAT(0) space introduced by Petyt, Spriano, and Zalloum. Along the way, we will meet centralisers, boundaries, and actions on hyperbolic spaces! This is based on my summer project supervised by Davide Spriano and Harry Petyt.

Random walks on graphs can mix slowly. To speed it up, imagine that at each step instead of choosing the neighbor at random, there is a small probability $\varepsilon > 0$ that we can choose it. We show that in this case, at least for graphs of bounded degree, there is a way to steer the walk so that we visit every vertex in $n^{1+o(1)}$ many steps. The key to this result is a way to decompose arbitrary graphs into small-diameter pieces.

The classical Leopoldt conjecture predicts that the global units of a number field (tensored with Qp) inject into the local units at p. In this talk, I'll discuss some non-abelian generalisations of this in the setting of Galois representations.

Joint with Atul Sharma arxiv:2512.04512.