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A brief review of some work on BPS states, wall-crossing, hyperkahler geometry, Hitchin systems, etc.

Greg Moore
Rutgers University
September 18, 2019

This talk will give a brief overview focused on six papers written by D. Gaiotto, G. Moore, and A. Neitzke
loosely centered around the topics mentioned in the title. The papers are on the arXiv. 
Background reading:  Some short reviews are: 
https://arxiv.org/pdf/1211.2331.pdf
https://arxiv.org/pdf/1308.2198.pdf
https://arxiv.org/pdf/1412.7120.pdf

Taking the Hecke algebra to its limits

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.