**Date:**2023-03-28 14**:**00 ~ 15:00-
**Place:**129-301 (SNU) **Speaker**: Homin Lee (Northwestern University)**Title**: Higher rank lattice actions with positive entropy**Abstract**:

We discuss about smooth actions on manifold by higher rank lattices. We mainly focus on lattices in SLnR (n is at least 3). Recently, Brown-Fisher-Hurtado and Brown-Rodriguez Hertz-Wang showed that if the manifold has dimension at most (n-1), the action is either isometric or projective. Both cases, we don’t have chaotic dynamics from the action (zero entropy). We focus on the case when one element of the action acts with positive topological entropy. These dynamical properties (positive entropy element) significantly constrains the action. Especially, we deduce that if there is a smooth action with positive entropy element on a closed n-manifold by a lattice in SLnR (n is at least 3) then the lattice should be commensurable with SLnZ. This is the work in progress with Aaron Brown.

**Date:**2023-03-31 16**:**00 ~ 17:00-
**Place:**27-325 (SNU) **Speaker**: Homin Lee (Northwestern University)**Title**: Height gap theorem and almost law**Abstract**:

E. Breuillard showed that finite subsets $F$ of matrices in $GL_{d}(\overline{Q})$ generating non-virtually solvable groups have normalized height $\widehat{h}(F) \ge \epsilon$, for some positive $\epsilon>0$. This can be thought of as a non-abelian analog of Lehmer’s Mahler measure problem and has a nice application such as uniform Tits alternative. Recently, it also leads to the arithmetic Margulis lemma by M.Fraczyk, S.Hurtado, and J. Raimbault. In this talk, we will discuss a relatively elementary proof of E.Breuillard's height gap theorem which avloids Bruhat-Tits geometry, and deep results on algebraic tori that are used in the original E.Breuillard's proof. The key idea is a usage of a mysterious word map so-called "almost law". This is joint work with Lvzhou Chen (Joe Chen) and Sebastian Hurtado.