This talk will serve as an introduction to the outer automorphism group of a free group, its properties and the objects used to study it: especially train track maps (with various adjectives) and Culler--Vogtmann outer space. If time allows I will discuss recent work joint with Hillen, Lyman and Pfaff on stretch factors in rank 3, but the goal of the talk will be to introduce the topic well rather than to speedrun towards the theorem.

RFRS groups were introduced by Ian Agol in connection with virtual fibering of 3-manifolds. Notably, the class of RFRS groups contains all compact special groups, which are groups with particularly nice cocompact actions on cube complexes. In this talk, I will give an introduction to ℓ²-Betti numbers from an algebraic perspective and discuss what group theoretic properties we can conclude from the (non)vanishing of the ℓ²-Betti numbers of a RFRS group.

We will try to solve the isomorphism problem amongst Coxeter groups by looking at finite quotients.  Some success is found in the classes of affine and right-angled Coxeter groups.  Based on joint work with Samuel Corson, Philip Moeller, and Olga Varghese.

The study of counting point-hyperplane incidences in the $d$-dimensional space was initiated in the 1990's by Chazelle and became one of the central problems in discrete geometry. It has interesting connections to many other topics, such as additive combinatorics and theoretical computer science. Assuming a standard non-degeneracy condition, i.e., that no $s$ points are contained in the intersection of $s$ hyperplanes, the currently best known upper bound on the number of incidences of $m$ points and $n$ hyperplanes in $\mathbb{R}^d$ is $O((mn)^{1-1/(d+1)}+m+n)$. This bound by Apfelbaum and Sharir is based on geometrical space partitioning techniques, which apply only over the real numbers.

In this talk, we discuss a novel combinatorial approach to study such incidence problems over arbitrary fields. Perhaps surprisingly, this approach matches the best known bounds for point-hyperplane incidences in $\mathbb{R}^d$ for many interesting values of $m, n, d$. Moreover, in finite fields our bounds are sharp as a function of $m$ and $n$ in every dimension. This approach can also be used to study point-variety incidences and unit-distance problem in finite fields, giving tight bounds for both problems under a similar non-degeneracy assumption. Joint work with A. Milojevic and I. Tomon.