The dynamic nature of first order methods can be interpreted by means of continuous time models. In this survey talk, we explain how physical concepts like acceleration, inertia or momentum have been used to improve the performance of convex optimization algorithms. 

We give special attention to the historical evolution of complexity results, especially in the form of convergence rates, under the light of this connection. We also discuss different ways in which acceleration schemes can be applied when the smoothness or strong convexity parameters are unknown, and how these ideas extend to saddle point and constrained problems. 

 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.