Voronoi formulas and its generalizations
Voronoi formulas are Poisson-style summation formulas for automorphic forms (on GL(n)). They have been powerful tools in analytic number theory for a long time (with applications to the divisor problem, the circle problem, subconvexity of L-functions, etc). Firstly, we present a Voronoi formula for coefficients of a large class of L-functions, in the style of the classical converse theorem of Andre Weil. Our formula applies to full-level cusp forms, Rankin-Selberg convolutions, and isobaric sums. Secondly, we present the balanced Voronoi formula, with its both sides twisted by hyper-Kloosterman sums. They are joint work with Eren Kiral and Stephen D. Miller, respectively.