Research

Research Interests

My research interests lie in number theory and algebraic geometry. I focus specifically on areas including:

  • Arithmetic statistics
  • Invariant theory
  • The Cohen-Lenstra heuristics
  • Rational points on varieties
  • Diophantine equations
  • Enumerative geometry

Articles in Preparation

  1. A parametrization of $3$-class groups of quadratic rings over Dedekind domains
  2. The average number of $3$-torsion elements in class groups of orders in quadratic extensions of number fields
  3. Second-order terms for orbits of coregular representations, and applications
  4. Computing the second-order term in the counting function for $2$-Selmer groups of elliptic curves
  5. Heuristics for class groups of orders in quadratic fields
  6. $2$-Selmer groups of even-degree hyperelliptic curves, and $2$-class groups of even-degree number fields

Publications

Arithmetic Statistics and Invariant Theory

  1. The second moment of the size of the $2$-class group of monogenized cubic fields
    Submitted. Based on Chapter 4 of my PhD thesis. 17 pages.
  2. The second moment of the size of the $2$-Selmer group of elliptic curves
    Submitted. Based on Chapter 4 of my PhD thesis. 49 pages.
  3. Counting integral points on symmetric varieties with applications to arithmetic statistics
    Proceedings of the London Mathematical Society
  4. Geometry-of-numbers methods in the cusp
    To appear in Algebra & Number Theory. Based on Chapter 5 of my PhD thesis.
  5. The mean number of $2$-torsion elements in the class groups of cubic orders
    To appear in Commentarii Mathematici Helvetici.
  6. A positive proportion of monic odd-degree hyperelliptic curves have no unexpected quadratic points
    International Mathematics Research Notices (2024).
  7. Most odd-degree binary forms fail to primitively represent a square
    Compositio Mathematica. Based on Chapter 3 of my PhD thesis.
  8. A new parametrization for ideal classes in rings defined by binary forms, and applications
    Journal fur die Reine und Angewandte Mathematik (Crelle's Journal). Based on Chapters 1 and 2 of my PhD thesis.
  9. Hermite equivalence of polynomials
    Acta Arithmetica (Special issue for Andrzej Schinzel).
  10. Hyperelliptic curves with maximal Galois action on the torsion points of their Jacobians
    Indiana University Mathematics Journal.
  11. Surjectivity of Galois representations in rational families of abelian varieties
    Algebra & Number Theory.

Algebraic Geometry

  1. Inflectionary invariants for isolated complete intersection curve singularities
    Memoirs of the American Mathematical Society. Based on my senior thesis.
  2. On the EKL-degree of a Weyl cover
    Journal of Algebra.
  3. Appendix to: An arithmetic count of the lines meeting four lines in $\mathbb{P}^3$
    with Borys Kadets, Padmavathi Srinivasan, Libby Taylor, and Dennis Tseng
    Transactions of the American Mathematical Society.

Other Number Theory

  1. Lifting subgroups of symplectic groups over $\mathbb{Z}/\ell\mathbb{Z}$
    Research in Number Theory.
  2. Permutations that destroy arithmetic progressions in elementary $p$-groups
    with Noam D. Elkies
    The Electronic Journal of Combinatorics.
  3. Elliptic curve variants of the least quadratic nonresidue problem and Linnik's Theorem
    International Journal of Number Theory.
  4. On logarithmically Benford sequences
    Proceedings of the American Mathematical Society.
  5. On arboreal Galois representations of rational functions
    Journal of Algebra.
  6. Linnik’s theorem for Sato-Tate laws on elliptic curves with complex multiplication
    Research in Number Theory.

Analysis and Probability

  1. Universality theorems for zeros of random real polynomials with fixed coefficients
    Submitted. 23 pages.
  2. Analysis on surreal numbers
    Journal of Logic and Analysis.

Theses and Expository Articles

  1. $2$-Selmer groups, $2$-class groups, and the arithmetic of binary forms (Princeton PhD thesis)
  2. Inflection points of linear systems on families of curves (Harvard senior thesis)
  3. Lie algebras and Ado's Theorem
  4. An introduction to the theory of valued fields (Harvard junior paper)
  5. On the Selberg-Erdős proof of the Prime Number Theorem
  6. Arrow's impossibility theorem on social choice systems