Seminar| Institute of Mathematical Sciences
Time: Wednesday, December 31th, 2025,16:00-17:00
Location: RS408, IMS
Speaker: Jialun Li, Fudan University
Abstract: I will begin with the classical setting of the modular surface: the unit tangent bundle of the modular curve $ T^1(SL(2,Z)\backslash H)=SL(2,Z)\backslash SL(2,R) $, where periodic geodesics correspond to periodic orbits of the diagonal subgroup. Margulis and Bowen established dynamical frameworks for studying these orbits, while Selberg and Sarnak developed spectral and number-theoretic approaches. We then turn to the higher-rank generalization, periodic diagonal orbits on $SL(3,Z)\backslash SL(3,R)$. In this case, each periodic diagonal orbit is a two-dimensional flat torus embedded in the space. Different counting problems emerge, depending on how one orders these tori--whether by dynamical ordering, geometric ordering or arithmetic ordering. The dynamical and geometric ordering are closely connected through the shapes of the flat tori. Finally, I will discuss my recent joint work with Thi Dang and Nihar Gargava on the density of shapes of periodic tori.