This talk is divided into two parts. In the first part, after some background on manifolds with negative curvature, I’ll state Margulis’s celebrated counting result proved in his thesis [Mar]. After that, I’ll briefly mention later works inspired by/generalizing his argument. Finally, I’ll sketch an argument proving a counting result due to Paulin and Parkkonen [PP]. This part should be accessible to general audience!
In the second part, which is the more technical part, I’m going to explain how to adapt Paulin and Parkkonen’s argument to prove a similar result for Teichmuller space. My intention is to mainly focus on deeper results that have made this adaptation possible (e.g. [ABEM], [Fr]).

References:
[ABEM] J. Athreya, A. Bufetov, A. Eskin and M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space. Duke Math. J. 161, No. 6, 1055-1111 (2012).
[Fr] I. Frankel, CAT(-1)-Type Properties for Teichmuller Space. arXiv:1808.10022, (2018).
[PP] J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature. Ergodic Theory Dyn. Syst. 37, No. 3, 900-938 (2017)
[Mar] G. A. Margulis, On some aspects of the theory of Anosov systems. With a survey by Richard Sharp: Periodic orbits of hyperbolic flows. Transl. from the Russian by S. V. Vladimirovna. Berlin: Springer (2004).

[IPM Youth Seminars on Topology and Dynamics]
Venue: meet.google.com/qix-iqiy-txt