- Points on curves. (My PhD thesis at Princeton.)
- The average elliptic curve has few integral points. (My senior thesis at Harvard.)
- (with Manjul Bhargava and Ari Shnidman) A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so.
- (with Manjul Bhargava and Ari Shnidman) A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so.
- (with Wei Ho) The second moment of the number of integral points on elliptic curves is bounded.
- The average number of rational points on odd genus two curves is bounded.
- Square-root cancellation for the signs of Latin squares. Combinatorica.
- The average number of integral points on elliptic curves is bounded.
- van der Waerden and the primes. Amer. Math. Monthly 122 (2015), no. 8, 784-785.
- Proof of a conjecture of Stanley-Zanello. J. Combin. Theory Ser. A. 125 (2014), 166-176.
- Self-conjugate core partitions and modular forms. J. Number Theory 140 (2014), 60-92.
- (with Steven J. Miller) Low-lying zeroes of Maass form L-functions. Int. Math. Res. Not. (2015), no. 10, 2678-2701.
- (with Nadine Amersi, Geoffrey Iyer, Oleg Lazarev, Steven J. Miller, and Liyang Zhang) Maass waveforms and low-lying zeros. In: C. Pomerance, M. Rassias (eds.), Analytic Number Theory: in Honor of Helmut Maier's 60th Birthday, 19-56, Springer, 2015.
- (with Thomas Ang, Luke Schaeffer, and Jeffrey Shallit) Decidability and shortest strings in formal languages. Descriptional Complexity of Formal Systems, Lecture Notes in Comput. Sci. 6808 (2011), 55-67.
- (with Alark Joshi, Dustin Scheinost, John Onofrey, Xiaoning Qian, and Xenios Papademetris) A vtk-based, CUDA-optimized non-parametric vessel detection method. VTK Journal. (My Intel STS project from high school.)
- Analytic number theory and quadratic reciprocity. Harvard College Mathematics Review.
(I once thought I'd discovered a new and "purely analytic" proof of quadratic reciprocity. In fact the argument was known to Dirichlet! So I turned it into a little expository article.
See the fun!! section for a non-expository version, which amounts to a few lines.)
recent recorded talks
- (with Dennis Gaitsgory and Gurbir Dhillon) Math 123 (= Algebra II) notes.
- (with Dennis Gaitsgory) Math 122 (= Algebra I) notes.
some amusing math history links