Ashvin A. Swaminathan

Member of Technical Staff
Axiom Math
Formerly NSF Postdoctoral Fellow & Benjamin Peirce Fellow, Harvard University

Download CV ashvins@alumni.princeton.edu

Research

My research lies in number theory and algebraic geometry, with a focus on arithmetic statistics, invariant theory, the Cohen-Lenstra heuristics, rational points on varieties, Diophantine equations, and enumerative geometry.

At Axiom I build workflows that combine large language models, symbolic computation, and formal verification to attack research problems in number theory. See my projects for the formalization work.

Background

I received my AB in mathematics from Harvard College, with a senior thesis supervised by Joe Harris and Anand Patel, as well as an AM in Physics. I completed my MA and PhD at Princeton under Manjul Bhargava.

Before joining Axiom I was an NSF Postdoctoral Fellow and Benjamin Peirce Fellow at Harvard, mentored by Melanie Matchett Wood, and in Fall 2025 I was a visiting researcher in the special semester on arithmetic statistics at the Lodha Mathematical Sciences Institute in Mumbai, India.

Outside of mathematics, I love animals and volunteer at Broken Tail Animal Rescue. If you are interested in adopting an animal, please check out their website!

Ashvin A. Swaminathan

News

  1. Axiom's Lean formalization of the Stacks Project is public. Project lead; mathematical and engineering contributor
  2. Axiom's Lean formalization of the classification of finite simple groups is public. Mathematical and engineering contributor
  3. New preprint: “Local factors in the BSD conjecture: a unified statistical view”. With David Kurniadi Angdinata, Kenny Lau, Ken Ono and Sameera Vemulapalli
  4. New preprint: “On a conjecture of Browning and Sawin on random hypersurfaces”. With Ken Ono
  5. New preprint: “A new bound for small gaps between primes”. $\liminf (p_{n+1}-p_n) \leq 212$
  6. Started as a Member of Technical Staff at Axiom Math.
  7. New version of “Parity of the partition function in quadratic progressions”. Adds $\#\{n \leq X : p(n) \text{ odd}\} \gg \sqrt{X}$. With Ken Ono
  8. MTS interview on the bounded gaps formalization and AxiomProver. Video
  9. IEEE Spectrum: “AI Used to Verify Toughest Mathematics Proof Yet”. Benjamin Skuse
  10. Axiom's Lean formalization of bounded gaps between primes is public. Project lead; mathematical and engineering contributor. $p_{n+1}-p_n \leq 246$
  11. New preprint: “Beyond mock modularity: elliptic corrections for higher Dyson ranks”. With Claudia Alfes and Ken Ono
  12. Joined Axiom Math as a Resident Research Scientist.
  13. Attended the ICARM Workshop on Milestones of Autonomous Mathematics. San Francisco, CA
  14. New preprint: “On the quartic invariant of odd-degree binary forms”. With a complete Lean formalization
  15. New preprint: “Positive proportion results for elliptic curves with small $2$-Selmer rank”. With Manjul Bhargava, Wei Ho, Ari Shnidman and Alexander Smith. Available on request
  16. New preprint: “The average number of $3$-torsion elements in class groups of quadratic orders”. With Eliot Hodges. Available on request