- Points on curves. (My PhD thesis at Princeton.)
- The average elliptic curve has few integral points. (My senior thesis at Harvard.)
- Integers expressible as the sum of two rational cubes (with Manjul Bhargava and Ari Shnidman).
- Local systems and Suzuki groups (with Nick Katz, Gabriel Navarro, Eamonn O'Brien, and Pham Huu Tiep).
- 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.
- Nagell-Lutz, quickly.
- The average number of integral points on elliptic curves is bounded.
- van der Waerden and the primes.
- Analytic number theory and quadratic reciprocity.*
- Proof of a conjecture of Stanley-Zanello.
- Self-conjugate core partitions and modular forms.
- Low-lying zeroes of Maass form L-functions (with Steven J. Miller).
- Maass waveforms and low-lying zeros (with Nadine Amersi, Geoffrey Iyer, Oleg Lazarev, Steven J. Miller, and Liyang Zhang).
- Decidability and shortest strings in formal languages (with Thomas Ang, Luke Schaeffer, and Jeffrey Shallit).
- A vtk-based, CUDA-optimized non-parametric vessel detection method (with Alark Joshi, Dustin Scheinost, John Onofrey, Xiaoning Qian, and Xenios Papademetris). (My Intel STS project from high school.)
*: (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