This is a series of talks on my work with Bernhard Leeb and Joan Porti, aiming to extend the theory of geometrically finite discrete groups from rank 1 symmetric spaces to higher rank. I will start by reviewing various equivalent definitions of convex-cocompact subgroups of rank 1 Lie groups. Then I review basic geometry of higher rank symmetric spaces and their ideal boundaries. After that, I will define several notions of geometric finiteness in higher rank which generalize (some of) the rank one definitions and prove at least some of the implications between them.  My lectures will be continued by the ones by Bernhard Leeb, who will prove further equivalences of different definitions. 

We describe several uses of the SL(2, R) action on moduli spaces of translation surfaces, in particular how to use ergodic theorems and recurrence results for the SL(2, R) action to understand counting and ergodicity properties for linear flows. Most of this will be based on work of Veech, Masur, Eskin-Masur, Forni, and Zorich. The talk will be elementary and assume very little background.

This is a series of talks on my work with Bernhard Leeb and Joan Porti, aiming to extend the theory of geometrically finite discrete groups from rank 1 symmetric spaces to higher rank. I will start by reviewing various equivalent definitions of convex-cocompact subgroups of rank 1 Lie groups. Then I review basic geometry of higher rank symmetric spaces and their ideal boundaries. After that, I will define several notions of geometric finiteness in higher rank which generalize (some of) the rank one definitions and prove at least some of the implications between them.  My lectures will be continued by the ones by Bernhard Leeb, who will prove further equivalences of different definitions. 

A complex projective structure is a geometric structure on a surface modeled on the Riemann sphere. Then a complex projective structure has a holonomy representation from the fundamental group of the surface into PSL(2, C), which is not necessarily discrete.

We discuss about ``neck-pinching'’ type degeneration of complex projective structures when their holonomy representations converge in the character variety.

We consider the orbits {pu(n^{1+γ})|n ∈ N} in Γ\PSL(2,R), where Γ is a non-uniform lattice in PSL(2,R) and u(t) is the standard unipotent group in PSL(2,R). Under a Diophantine condition on the intial point p, we show that {pu(n^{1+γ})|n ∈ N} is equidistributed in Γ\PSL(2,R) for small γ>0, which generalizes a result of Venkatesh.

This is a series of talks on my work with Bernhard Leeb and Joan Porti, aiming to extend the theory of geometrically finite discrete groups from rank 1 symmetric spaces to higher rank. I will start by reviewing various equivalent definitions of convex-cocompact subgroups of rank 1 Lie groups. Then I review basic geometry of higher rank symmetric spaces and their ideal boundaries. After that, I will define several notions of geometric finiteness in higher rank which generalize (some of) the rank one definitions and prove at least some of the implications between them.  My lectures will be continued by the ones by Bernhard Leeb, who will prove further equivalences of different definitions.