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
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A parametrization of $3$-class groups of quadratic rings over Dedekind domains
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The average number of $3$-torsion elements in class groups of orders in quadratic extensions of number fields
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Second-order terms for orbits of coregular representations, and applications
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Computing the second-order term in the counting function for $2$-Selmer groups of elliptic curves
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Heuristics for class groups of orders in quadratic fields
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$2$-Selmer groups of even-degree hyperelliptic curves, and $2$-class groups of even-degree number fields
Publications
Arithmetic Statistics and Invariant Theory
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The second moment of the size of the $2$-class group of monogenized cubic fieldsSubmitted. Based on Chapter 4 of my PhD thesis. 17 pages.
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The second moment of the size of the $2$-Selmer group of elliptic curvesSubmitted. Based on Chapter 4 of my PhD thesis. 49 pages.
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Counting integral points on symmetric varieties with applications to arithmetic statisticsProceedings of the London Mathematical Society
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Geometry-of-numbers methods in the cuspTo appear in Algebra & Number Theory. Based on Chapter 5 of my PhD thesis.
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The mean number of $2$-torsion elements in the class groups of cubic ordersTo appear in Commentarii Mathematici Helvetici.
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A positive proportion of monic odd-degree hyperelliptic curves have no unexpected quadratic pointsInternational Mathematics Research Notices (2024).
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Most odd-degree binary forms fail to primitively represent a squareCompositio Mathematica. Based on Chapter 3 of my PhD thesis.
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A new parametrization for ideal classes in rings defined by binary forms, and applicationsJournal fur die Reine und Angewandte Mathematik (Crelle's Journal). Based on Chapters 1 and 2 of my PhD thesis.
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Hermite equivalence of polynomialsActa Arithmetica (Special issue for Andrzej Schinzel).
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Hyperelliptic curves with maximal Galois action on the torsion points of their JacobiansIndiana University Mathematics Journal.
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Surjectivity of Galois representations in rational families of abelian varietiesAlgebra & Number Theory.
Algebraic Geometry
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Inflectionary invariants for isolated complete intersection curve singularitiesMemoirs of the American Mathematical Society. Based on my senior thesis.
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On the EKL-degree of a Weyl coverJournal of Algebra.
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Appendix to: An arithmetic count of the lines meeting four lines in $\mathbb{P}^3$Transactions of the American Mathematical Society.
Other Number Theory
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Lifting subgroups of symplectic groups over $\mathbb{Z}/\ell\mathbb{Z}$Research in Number Theory.
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Permutations that destroy arithmetic progressions in elementary $p$-groupsThe Electronic Journal of Combinatorics.
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Elliptic curve variants of the least quadratic nonresidue problem and Linnik's TheoremInternational Journal of Number Theory.
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On logarithmically Benford sequencesProceedings of the American Mathematical Society.
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On arboreal Galois representations of rational functionsJournal of Algebra.
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Linnik’s theorem for Sato-Tate laws on elliptic curves with complex multiplicationResearch in Number Theory.
Analysis and Probability
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Universality theorems for zeros of random real polynomials with fixed coefficientsSubmitted. 23 pages.
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Analysis on surreal numbersJournal of Logic and Analysis.
Theses and Expository Articles
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$2$-Selmer groups, $2$-class groups, and the arithmetic of binary forms (Princeton PhD thesis)
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Inflection points of linear systems on families of curves (Harvard senior thesis)
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Lie algebras and Ado's Theorem
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An introduction to the theory of valued fields (Harvard junior paper)
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On the Selberg-Erdős proof of the Prime Number Theorem
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Arrow's impossibility theorem on social choice systems