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. I led Axiom's bounded gaps between primes project, which produced a complete Lean 4 formalization of the theorem that $p_{n+1}-p_n \leq 246$ for infinitely many $n$.

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. New preprint: “A new bound for small gaps between primes”, proving $\liminf_{n \to \infty}(p_{n+1}-p_n) \leq 212$. With François Charton, Letong Hong, Kenny Lau, Ken Ono, Guillaume Remy, Ho Chung Siu, Jesse Thorner and Yunzhou Xie
  2. I begin as a Member of Technical Staff at Axiom Math.
  3. I spoke with MTS about the bounded gaps formalization and AxiomProver. Video interview
  4. IEEE Spectrum covers the formalization: “AI Used to Verify Toughest Mathematics Proof Yet”. Benjamin Skuse
  5. Bounded gaps between primes, the project I led at Axiom, is public: a complete Lean 4 formalization of the theorem that $p_{n+1}-p_n \leq 246$ for infinitely many $n$. primegaps.axiommath.ai
  6. New version of “Parity of the partition function in quadratic progressions” (with Ken Ono), with the algebraic identities verified in Lean.
  7. New preprint: “Beyond mock modularity: elliptic corrections for higher Dyson ranks” (with Claudia Alfes and Ken Ono).
  8. I joined Axiom Math as a Resident Research Scientist.
  9. Attended the ICARM Workshop on Milestones of Autonomous Mathematics. San Francisco, CA
  10. New preprint: “On the quartic invariant of odd-degree binary forms”, accompanied by a complete Lean 4 formalization.
  11. New preprint: “Positive proportion results for elliptic curves over $\mathbb{Q}$ with small $2$-Selmer rank” (with Manjul Bhargava, Wei Ho, Ari Shnidman and Alexander Smith). Available upon request
  12. New preprint: “The average number of $3$-torsion elements in class groups of orders in quadratic extensions of number fields” (with Eliot Hodges). Available upon request