I am a number theorist working on automorphic representations and arithmetic geometry. Graduate students
that I advise typically work on topics that involve some
combination of the following:
- Generalizations of the Poisson summation formula conjectured by Braverman and Kazhdan, L. Lafforgue,
Ngo and Sakellaridis
- Integral representations of L-functions
- Comparisons of trace formulae, both globally and locally
- Variants of the Hitchin fibration following Ngo
- Cycles on Shimura varieties
- Analytic number theory arising out of limits of trace formulae
- Representation theory of reductive groups over local fields
PhD Mathematics, University of Wisconsin at Madison, 2007
AB Mathematics, Harvard University, 2004
Current and Past PhD students
- Chung-Ru Lee, Duke University, Fall 2016-present.
- Huong Tran, Duke University, Fall 2015-present
- Jason Polak, McGill Univesity, Fall 2011-Spring 2016: Currently postdoc with Vilonen and Xue at the University of Melbourne.
Springer GTM project
Heekyoung Hahn and I are writing a book on automorphic representations.
The chapters that are in reasonable shape will be posted
here. The old course notes are here.
Research papers and preprints
These may differ slightly from the published versions. If you have any
comments or corrections, please email me at the address above.
- A summation formula for triples of quadratic spaces, (with B. Liu), submitted.
- A refined Poisson summation formula for certain Braverman-Kazhdan spaces, (with B. Liu), submitted.
- Secondary terms in asymptotics for the number of zeros of quadratic forms over
number fields, J. London Math. Soc., Vol. 98, No. 2 (2018) 275-305. arXiv.
- Nonabelian Fourier transforms for spherical representations,
Pacific J. Math., Vol. 294, No. 2.
- A summation formula for the
Rankin-Selberg monoid and a nonabelian trace formula American J. of Math., accepted.
- A four variable automorphic kernel function,
Research in the Mathematical Sciences (2016), 3:20
- A general simple relative trace formula (with H. Hahn), Pacific J. Math., Vol. 277, No. 1
- A nonabelian trace formula (with P. E. Herman),
Research in the Mathematical Sciences (2015) 2:14.
- Isolating Rankin-Selberg lifts (with J. Klassen), Proc. Amer. Math. Soc.,
Vol. 143 (2015) 3319-3329.
- Algebraic cycles and Tate classes on Hilbert modular varieties
(with H. Hahn), Int. J. Number Theory, Vol. 10, No. 1 (2014) 161--176.
- An approach to nonsolvable base change and descent, J.
Ramanujan Math. Soc., Vol. 27, No. 2 (2012) 143-211.
Twisted relative trace formulae with a view towards unitary groups
(with E. Wambach),
American J. Math., Vol. 136 (2014) 1-57.
Note: in section 10.4 of the published version the Jacobson density theorem was erroneously invoked where the Dixmier-Malliavin lemma should have been invoked.
- Hilbert Modular Forms with Coefficients in
Intersection Homology and Quadratic Base Change (with M. Goresky), winner of the 2011 Ferran Sunyer i Balaguer Prize, Progress in Math.
- Intersection numbers of
Hecke cycles on Hilbert modular varieties, American J. Math., Vol. 129, No. 6 (2007), 1623-1658.
- Systems of orthogonal
polynomials arising from the modular j-function (with S. Basha, H. Nover, E. Smith), J.
Math. Anal. and Appl., Vol. 289, No. 1 (2004), 336-354.
generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular
forms, Proc. Amer. Math. Soc., 132 (2004), 2221-2231.
- Partition identities
and a theorem of Zagier (with K. Mahlburg), J. Comb. Theory (A),
100, (2002), 27-43.
- Extension of a theorem of Kiming and Olsson for the partition
function, Ramanujan J., Vol. 5, No. 1 (2001), 47-51.
- On congruence properties of the partition function, Int. J. Math. and Math. Sci.,
Vol. 23, No. 7 (2000), 493-496.
- Sixth Abel Conference: Celebrating the Mathematical Impact of Robert Langlands, November 14-16, 2018.
- BC-MIT Number Theory Seminar, December 11, 2018.
- On the Langlands Program: Endoscopy and Beyond, IMS-NUS, Singapore, December 17, 2018-Jan 18, 2019.
- AMS Special Session on Counting Methods in Number Theory, AMS-MAA Joint Meeting, Janurary 2019.
- Recent Developments in Automorphic Forms, AMS meeting, University of Hawaii Manoa, March 22-24, 2019.
- Relative Aspects of the Langlands Program, L-Functions and Beyond Endoscopy, CIRM, France, May 25-29, 2020.