My research mainly concerns the following topics, with an eye towards collaborations with computer scientists and physicists:

Automorphic Forms, L-functions

Lattices and Sphere Packing

Cryptography

Representation Theory

Selected Publications:

On the existence and temperedness of cusp forms for SL(3,Z) , Journal für die reine und angewandte Mathematik, 533 (2001), pp. 127-169.

Automorphic Distributions, L-functions, and Voronoi Summation for GL(3), with Wilfried Schmid,
Annals of Mathematics 164 (2006), pp. 423-488.

Do All Elliptic Curves of the Same Order Have the Same Difficulty of Discrete Log?, with David Jao and Ramarathnam Venkatesan, in Advances in Cryptology - ASIACRYPT 2005: 11th International Conference on the Theory and Application of Cryptology and Information Security, Springer Verlag Lecture Notes in Computer Science, 3788 (2005), pp. 21-40.

The Archimedean theory of the exterior square L-functions over Q, with Wilfried Schmid, Journal of the American Mathematical Society 25 (2012), 465-506.

Residual automorphic forms and spherical unitary representations of exceptional groups. Ann. of Math. (2) 177 (2013), no. 3, 1169–1179

Small representations, string instantons, and Fourier modes of Eisenstein series, with Michael Green and Pierre Vanhove. J. Number Theory 146 (2015), 187–309.

The sphere packing problem in dimension 24, with Henry Cohn, Abhinav Kumar, Danylo Radchenko, and Maryna Viazovska, Ann. of Math (2) 185 (2017), no. 3, 1017-1033.

Ph.D. students

Zhuohui Zhang (Ph.D. 2018)

Edmund Karasiewicz (Ph.D. 2017)

Tien Trinh (Ph.D. 2016)

Brandon Bate (Ph.D. 2013)

Current mentees

George Hauser

Edna Jones

Louis Gaudet

Tamar Lichter

Zhi Qi (Hill Instructor, 2015-)

Maksym Radziwill (Hill Instructor, 2015)

Adi Akavia (DiMACS fellow, 2008-2009)

HLL-514

 571 Modular forms