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
   with Eliot Hodges
2. The average number of $3$-torsion elements in class groups of orders in quadratic extensions of number fields
   with Eliot Hodges
3. Second-order terms for orbits of coregular representations, and applications
   with Jef Laga and Artane Siad
4. Computing the second-order term in the counting function for $2$-Selmer groups of elliptic curves
   with Jef Laga and Artane Siad
5. Heuristics for class groups of orders in quadratic fields
   with Manjul Bhargava, Arul Shankar, Artane Siad, and Ila Varma
6. $2$-Selmer groups of even-degree hyperelliptic curves, and $2$-class groups of even-degree number fields
   with Manjul Bhargava and Arul Shankar

Publications

Arithmetic Statistics and Invariant Theory

1. The second moment of the size of the $2$-class group of monogenized cubic fields
   with Manjul Bhargava and Arul Shankar
   Submitted. Based on Chapter 4 of my PhD thesis. 17 pages.
   [arXiv]
2. The second moment of the size of the $2$-Selmer group of elliptic curves
   with Manjul Bhargava and Arul Shankar
   Submitted. Based on Chapter 4 of my PhD thesis. 49 pages.
   [arXiv]
3. Counting integral points on symmetric varieties with applications to arithmetic statistics
   with Arul Shankar and Artane Siad
   Proceedings of the London Mathematical Society
   [journal] [arXiv] [github]
4. Geometry-of-numbers methods in the cusp
   with Arul Shankar, Artane Siad, and Ila Varma
   To appear in Algebra & Number Theory. Based on Chapter 5 of my PhD thesis.
   [arXiv]
5. The mean number of $2$-torsion elements in the class groups of cubic orders
   To appear in Commentarii Mathematici Helvetici.
   [arXiv]
6. A positive proportion of monic odd-degree hyperelliptic curves have no unexpected quadratic points
   with Jef Laga
   International Mathematics Research Notices (2024).
   [journal] [arXiv]
7. Most odd-degree binary forms fail to primitively represent a square
   Compositio Mathematica. Based on Chapter 3 of my PhD thesis.
   [journal] [arXiv]
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.
   [journal] [arXiv]
9. Hermite equivalence of polynomials
   with Manjul Bhargava, Jan-Hendrik Evertse, Kálmán Györy, and László Remete
   Acta Arithmetica (Special issue for Andrzej Schinzel).
   [journal] [arXiv]
10. Hyperelliptic curves with maximal Galois action on the torsion points of their Jacobians
    with Aaron Landesman, James Tao, and Yujie Xu
    Indiana University Mathematics Journal.
    [journal] [arXiv]
11. Surjectivity of Galois representations in rational families of abelian varieties
    with Aaron Landesman, James Tao, and Yujie Xu
    Algebra & Number Theory.
    [journal] [arXiv]

Algebraic Geometry

1. Inflectionary invariants for isolated complete intersection curve singularities
   with Anand Patel
   Memoirs of the American Mathematical Society. Based on my senior thesis.
   [journal] [arXiv]
2. On the EKL-degree of a Weyl cover
   with Joseph Knight and Dennis Tseng
   Journal of Algebra.
   [journal] [arXiv]
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.
   [journal] [arXiv]

Other Number Theory

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

Analysis and Probability

1. Universality theorems for zeros of random real polynomials with fixed coefficients
   with Matt King
   Submitted. 23 pages.
   [arXiv]
2. Analysis on surreal numbers
   with Simon Rubinstein-Salzedo
   Journal of Logic and Analysis.
   [journal] [arXiv]

Theses and Expository Articles

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