We meet by Zoom this semester.

Zoom ID : 393 193 3864

Password: zero minus the Euler characteristic of a closed oriented surface of genus 4000

Upcoming talks

March 15, 18 (M, Th), 2 - 3:30 pm, Zoom

Junehyuk Jung (Brown)

Equidistribution problems of closed geodesics on hyperbolic surfaces

In this series of lectures, I will introduce the modular intersection kernel and then explain how one can use this to understand intersections of geodesics using it.

(1) In the first lecture I'm going to go over some background topics that are necessary for understanding the main result. This includes Duke's theorem on equidistribution of closed geodesics on the full modular surface $\mathbb{X}=PSL_2(\mathbb{Z})\backslash \mathbb{H}$, discriminant geodesics, period integrals, spectral decomposition of $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})$, Selberg's trace formula, the prime geodesic theorem, etc. This is a comprehensive lecture designated for non-experts, so those who are already familiar with these concepts may skip it.

(2) Two theorems regarding the intersections of closed geodesics on the full modular surface will be introduced in the second lecture. Let $C_d$ be the union of closed primitive geodesics corresponding to indefinite primitive binary quadratic form of discriminant $d$. Then a generalized form of Duke's theorem implies that the lift $\mathscr{C}_d$ of $C_d$ to the unit tangent bundle $S\mathbb{X}$ of $\mathbb{X}$ becomes equidistributed as $d\to \infty$. As an application of generalized Duke's theorem to the modular intersection kernel, for any fixed geodesic segment $\beta$, the intersections between $\beta$ and $C_d$ become equidistributed on $\beta$ as $d \to \infty$. This resolves the main conjecture introduced by Rickards based on numerical experiment. The second theorem I will talk about provides an asymptotic formula on the total number of intersections between $C_{d_1}$ and $C_{d_2}$.

This lecture is based on the collaboration with Naser Talebizadeh Sardari.

April 15, 22 (Th), 10 - 11 am Korea

( = April 14, April 21 (W) 9 - 10 pm EST with daylight saving)

KIAS Distinguished Lecture Series

Danny Calegari (Chicago)

Sausages and Butcher Paper

Abstract TBA

Organizers

Sang-hyun Kim (KIAS)