Confined subgroups of a Lie group or of a discrete group are generalizations of a Fuchsian group G that gives rise to a hyperbolic surface M of bounded injectivity radius. When the surface M has finite volume, the growth rate of G equals the volume growth rate of the hyperbolic plane H^2. Gekhtman and Levit established an inequality between the growth rates of G and the ambient space for a general confined subgroup of a rank-1 Lie group. In this talk, I will explain an analogous result for a confined subgroup G of a hyperbolic group, rank-1 CAT(0) group, or the mapping class group. This is established by studying the single (as opposed to the double) boundary action of G. Joint work with Ilya Gekhtman, Wenyuan Yang, and Tianyi Zheng.

In this talk, I will explain a recent result that pseudo-Anosov mapping classes are generic in every Cayley graph of mapping class groups. If time permits, I will also explain why this strategy goes well with quasi-isometries and implies genericity of Morse elements for groups quasi-isometric to (many) 3-manifold groups and special cubical groups.

Hilbert’s sixth problem asks for the axiomatic derivation of the laws of physics. Within this broad question, Hilbert singled out the derivation of the laws of fluid mechanics (like Euler’s or Navier-Stokes’ equations) by way of rigorously justifying Boltzmann’s kinetic theory for particle systems. This will be the subject of our talk. The rigorous derivation of Boltzmann’s kinetic equation was established for short times in a classic work of Lanford (1975), following inputs from Grad and Cercignani. However, Hilbert’s sixth problem requires a long time version of this result. In a recent work with Yu Deng (Chicago) and Xiao Ma (Michigan), we extend Lanford’s theorem to long times; more precisely, for as long as the solution of Boltzmann’s equation exists. The proof relies on a novel methodology that allows us to obtain effective estimates on the probability of many recollisions.