Georgia Institute of Technology College of Sciences

I will describe two new local limit theorems on the Heisenberg group, and on an arbitrary connected, simply connected nilpotent Lie group. The limit theorems admit general driving measures and permit testing against test functions with an arbitrary translation on the left and the right. The techniques introduced include a rearrangement group action, the Gowers-Cauchy-Schwarz inequality, and a Lindeberg replacement scheme which approximates the driving measure with the corresponding heat kernel. These results generalize earlier local limit theorems of Alexopoulos and Breuillard, answering several open questions. The work on the Heisenberg group is joint with Persi Diaconis.