L4

A temporal graph $G$ is a sequence of graphs $G_1, G_2, \ldots, G_t$ on the same vertex set. In this talk, we are interested in the analogue of the Travelling Salesman Problem for temporal graphs. It is referred to in the literature as the Temporal Exploration Problem, and asks for the minimum length of an exploration of the graph, that is, a sequence of vertices such that at each time step $t$, one either stays at the same vertex or moves along a single edge of $G_t$.

One natural and still open case is when each graph $G_t$ is connected and has bounded maximum degree. We present a short proof that any such graph admits an exploration in $O(n^{3/2}\sqrt{\log n})$ time steps. In fact, we deduce this result from a more general statement by introducing the notion of average temporal maximum degree. This more general statement improves the previous best bounds, under a unified approach, for several studied exploration problems.

This is based on joint work with Carla Groenland, Lukas Michel and Clément Rambaud.

I will report on ongoing joint work with Sam Fisher on showing that the mapping class group has a finite index subgroup whose group ring embeds in a division ring. Our methods involve p-adic analytic groups, but no prior knowledge of this will be assumed and much of the talk will be devoted to explaining some of the underlying theory. Time permitting, I will also discuss some consequences for the profinite topology for the mapping class group and potential extensions to Out(RAAG).

I will discuss recent and ongoing work (mostly with J. Zou).

In this talk, I will present a result proved in my recent paper arXiv:2502.13580. I will discuss biharmonic maps between (and submanifolds of) conformally compact manifolds, a large class of complete manifolds generalizing hyperbolic space. After an introduction to conformally compact geometry, I will discuss one of the main results of the paper: if S is a properly embedded sub-manifold of a conformally compact manifold (N,h), and moreover S is transverse to the boundary and (N,h) has non-positive curvature, then S must be minimal. This result confirms a conjecture known as the Generalized Chen’s Conjecture, in the conformally compact context.