## The Thin Obstacle Problem

Yash Jhaveri

Member, School of Mathematics

September 25, 2019

Yash Jhaveri

Member, School of Mathematics

September 25, 2019

Chi Jin

Member, School of Mathematics

September 25, 2019

Clark Butler

Veblen Research Instructor, School of Mathematics

September 24, 2019

Yu Cheng

Member, School of Mathematics

September 24, 2019

Alexander Perry

Member, School of Mathematics

September 24, 2019

Shai Evra

Member, School of Mathematics

September 24, 2019

Will Feldman

Member, School of Mathematics

September 24, 2019

Mikolaj Fraczyk

Member, School of Mathematics

September 24, 2019

Silvia Ghinassi

Member, School of Mathematics

September 24, 2019

Raphael Steiner

Member, School of Mathematics

September 12, 2019

We parametrise elements in the full Hecke algebra in a way such that the parametrisation represents a generic automorphic form. By convolving, we then arrive at pre-trace formulas which are modular in three variables. From here, various identities for higher moments may be derived. We give applications to the sup-norm and fourth-norm of holomorphic Hecke eigenforms as well as Hecke-Maass forms on and furthermore outline future work on higher moments of periods and quantum variance. This is joint work with Ilya Khayutin.