Title: A two term Kuznecov sum formula

Abstract: A period integral is the average of a Laplace eigenfunction over a compact, embedded submanifold. Bounds on period integrals tell us something about where eigenfunctions tend to oscillate. Much like for the Weyl law, one can obtain improved estimates on period integrals given geometric or dynamical assumptions on the geodesic flow. These improvements have been enthusiastically pursued over the last few years under gradually weaker assumptions, the gold standard now being that the set of recurrent directions of geodesics normal to the submanifold has measure zero.

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 joint work with Yakun Xi, 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, and as a corollary, we obtain improvements under yet weaker dynamical hypotheses.

About the Speaker: Emmett Wyman is a Visiting Assistant Professor at the University of Rochester.  He received his Ph.D. in 2018 from Johns Hopkins University under the direction of Christopher Sogge.  While there, he was awarded the Professor Joel Dean Award for extraordinary performance in undergraduate teaching. Prior to Rochester, Dr. Wyman was an RTG Postdoctoral Research Fellow at Northwestern University.  His research interests span several facets of harmonic and microlocal analysis as well as geometric combinatorics.  For the past 3 years, Dr. Wyman has been a teacher, mentor, and co-organizer of the NSF Tripods REU/STEM for All program at the University of Rochester.