**Zu Chongzhi Center Mathematics Research Seminar**

**Date and Time (China standard time): **Thursday, November 24, 9:00-10:00 am

**Zoom ID**: 910 1234 3548

**Passcode: **dkumath

**Title: **A two term Kuznecov sum formula

**Speaker:** Yakun Xi

**Bio**: Yakun Xi is a harmonic analyst at School of Mathematical Sciences, Zhejiang University. In 2017, he received his Ph.D. from Johns Hopkins University, advised by Prof. Christopher D. Sogge. He worked as a visiting assistant professor at University of Rochester from 2017 to 2020, and joined Zhejiang University in 2020.

**Abstract**: A period integral is the average of a Laplace eigenfunction over a compact submanifold. Much like for the Weyl law, one can obtain improved estimates on period integrals given geometric or dynamical assumptions on the geodesic flow. While there are many results improving bounds on period integrals, there have been none which improve the remainder of the corresponding sum formula. In this talk, we discuss a recent joint work with Emmett Wyman. We show that an improvement to the remainder term of this sum formula reveals a lower-order oscillating term whose behavior can be described by the dynamics of the geodesic flow. Moreover, this oscillating second term illuminates bounds on period integrals.