- Points on curves. (My PhD thesis at Princeton.)
- The average elliptic curve has few integral points. (My senior thesis at Harvard.)
- Un peu d'effectivité pour les variétés modulaires de Hilbert-Blumenthal.
- Quadrics in arithmetic statistics.
- Note on a theorem of Professor X.
- Modularity and effective Mordell I.
- 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 Manjul Bhargava and Ari Shnidman).
- The second moment of the number of integral points on elliptic curves is bounded (with Wei Ho).
- 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.
- Low-lying zeroes of Maass form L-functions (with Steven J. Miller). Int. Math. Res. Not. (2015), no. 10, 2678-2701.
- Maass waveforms and low-lying zeros (with Nadine Amersi, Geoffrey Iyer, Oleg Lazarev, Steven J. Miller, and Liyang Zhang). In: C. Pomerance, M. Rassias (eds.), Analytic Number Theory: in Honor of Helmut Maier's 60th Birthday, 19-56, Springer, 2015.
- Decidability and shortest strings in formal languages (with Thomas Ang, Luke Schaeffer, and Jeffrey Shallit). Descriptional Complexity of Formal Systems, Lecture Notes in Comput. Sci. 6808 (2011), 55-67.
- A vtk-based, CUDA-optimized non-parametric vessel detection method (with Alark Joshi, Dustin Scheinost, John Onofrey, Xiaoning Qian, and Xenios Papademetris). 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
- Math 123 (= Algebra II) notes (with Dennis Gaitsgory and Gurbir Dhillon).
- Math 122 (= Algebra I) notes (with Dennis Gaitsgory).
some amusing math history links