Ashvin A. Swaminathan

Current Employment: I'm an NSF Postdoctoral Fellow and Benjamin Peirce Fellow in the Department of Mathematics at Harvard University, working with Melanie Matchett Wood.

For the 2024-25 academic year, I am serving as the Assistant Director of Graduate Studies. To current and prospective graduate students: please do not hesitate to contact me with questions! I am also a member of the Harvard Math Department Community Committee; for more information, feel free to email me or stop by my office (SC236), and check out our website.

In Spring 2024, I am teaching Math 223br: Rational Points on Varieties. Check out the course website.

In Fall 2025, I will be on leave from Harvard, participating as a visiting researcher in the special semester on arithmetic statistics at the new Lodha Mathematical Sciences Institute in Mumbai, India.

I love animals and currently serve as a volunteer at Broken Tail Animal Rescue. If you are interested in adopting an animal, please check out our website.

Education: I received my AB in mathematics from Harvard College, where I did my senior thesis under the supervision of Joe Harris and Anand Patel. I did my PhD in mathematics at Princeton University under the supervision of Manjul Bhargava.

Research interests: number theory and algebraic geometry; more specifically, arithmetic statistics, invariant theory, the Cohen-Lenstra heuristics, rational points on varieties, Diophantine equations, and enumerative geometry.

Email: swaminathan@math.harvard.edu.

Research 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
   Preprint available upon request.
2. The second moment of the size of the $2$-Selmer group of elliptic curves, with Manjul Bhargava and Arul Shankar
   Preprint based on Chapter 4 of my PhD thesis. 49 pages, submitted. [arXiv]
   
3. Counting integral points on symmetric varieties with applications to arithmetic statistics, with Arul Shankar and Artane Siad
   Published in 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
   Preprint based on Chapter 5 of my PhD thesis. To appear in Algebra & Number Theory. [arXiv]
   
5. The mean number of $2$-torsion elements in the class groups of cubic orders.
   Preprint. 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
   Published in International Mathematics Research Notices. [journal]  [arXiv]
   
7. Most odd-degree binary forms fail to primitively represent a square.
   Published in 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.
   Published in 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
   Published in a special issue of Acta Arithmetica dedicated to the memory of 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
    Published in 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
    Published in Algebra & Number Theory. [journal] [arXiv]
    

Algebraic Geometry

1. Inflectionary invariants for isolated complete intersection curve singularities, with Anand Patel
   Published in 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
   Published in 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
   Published in 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
   Published in Research in Number Theory. [journal]   [arXiv]
   
2. Permutations that destroy arithmetic progressions in elementary $p$-groups, with Noam D. Elkies
   Published in 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
   Published in International Journal of Number Theory. [journal]   [arXiv]
   
4. On logarithmically Benford sequences, with Evan Chen and Peter Park
   Published in Proceedings of the American Mathematical Society. [journal]   [arXiv]
   
5. On arboreal Galois representations of rational functions.
   Published in 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
   Published in Research in Number Theory. [journal]   [arXiv]
   

Analysis and Probability

1. Universality theorems for zeros of random real polynomials with fixed coefficients, with Matt King
   Preprint. 23 pages, submitted. [arXiv]
2. Analysis on surreal numbers, with Simon Rubinstein-Salzedo
   Published in 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]

Presentations

1. Toward secondary terms for 2-Selmer groups in families of elliptic curves
   Algebraic and Number Theory Seminar, Dartmouth College, 2025.
   
2. A positive proportion of hyperelliptic curves have no unexpected quadratic points
   Algebraic Geometry Seminar, Harvard University, 2025.
   
3. A positive proportion of hyperelliptic curves have no unexpected quadratic points
   Algebraic Geometry Seminar, Boston College, 2024.
   
4. A positive proportion of hyperelliptic curves have no unexpected quadratic points
   Informal Number Theory Seminar, University of Glasgow, 2024.
   
5. A positive proportion of hyperelliptic curves have no unexpected quadratic points
   Number Theory Seminar, Tufts University, 2024.
   
6. A positive proportion of hyperelliptic curves have no unexpected quadratic points
   Five College Number Theory Seminar, Amherst College, 2024.
   
7. Geometry-of-numbers in the cusp, and class groups of orders in number fields
   Number Theory Seminar, Duke University, 2023.
   
8. Affine symmetric spaces and class groups of cubic fields
   Québec Vermont Number Theory Seminar, McGill University, 2023.
   
9. Geometry-of-numbers in the cusp, and class groups of orders in number fields
   Conference on Arithmetic Statistics, CIRM Luminy, 2023.
   
10. Affine symmetric spaces and class groups of cubic fields
    Conference on Algebra, Algorithms, and Arithmetic, ICMS Edinburgh, 2023.
    
11. Affine symmetric spaces and class groups of cubic fields
    Symposium on Arithmetic Geometry and its Applications, CIRM Luminy, 2023.
    
12. The second moment of the size of the 2-Selmer group of elliptic curves
    Special Session on Arithmetic Statistics, JMM 2023.
    
13. Geometry-of-numbers methods in the cusp, and class groups of orders in number fields
    Number Theory Seminar, MIT, 2022.
    
14. 2-Selmer groups, 2-class groups, and the arithmetic of binary forms
    Online Number Theory Seminar, University of Debrecen, 2022.
    
15. On the distribution of 2-Selmer groups of hyperelliptic Jacobians
    Workshop on Rational Points on Higher-Dimensional Varieties, ICMS Edinburgh, 2022.
    
16. 2-Selmer groups, 2-class groups, and the arithmetic of binary forms
    Number Theory Seminar, Fields Institute, 2022.
    
17. Geometry-of-numbers methods in the cusp, and applications to class groups
    Montreal Online Biweekly Inter-University Seminar on Analytic Number Theory (MOBIUS ANT), 2022.
    
18. The second moment of the size of the $2$-Selmer group of elliptic curves
    Number Theory Seminar, Princeton University/IAS, 2021.
    
19. "$2$-Selmer groups, $2$-class groups, and the arithmetic of binary forms
    Palmetto Joint Arithmetic and Modularity Series (PAJAMAS) III, Invited Graduate Speaker, 2021
    
20. The second moment of the size of the $2$-Selmer group of elliptic curves
    Workshop on Explicit Methods in Number Theory, Mathematisches Forschungsinstitut Oberwolfach, 2021.
    
21. Average $2$-torsion in the class groups of rings associated to binary $n$-ic forms
    Bhargava Seminar, Princeton University, 2020.
    
22. Average $2$-torsion in the class groups of rings associated to binary $n$-ic forms
    Number/Representation Theory Seminar, University of Toronto, 2020.
    
23. Most hyperelliptic curves are pointless
    Bhargava Seminar, Princeton University, 2019.
    
24. Inflectionary invariants for plane curve singularities
    Special Session on Research in Mathematics by Undergraduates, JMM 2018.
    
25. Surjectivity of Galois representations in rational families of abelian varieties
    Algebra, Geometry, and Number Theory Seminar, Tufts University, 2017.
    
26. Inflection points of linear systems on families of curves
    Algebraic Geometry Seminar, Stanford University, 2017.
    
27. Analysis on surreal numbers: functions and integration
    ASL Special Session, JMM 2016.
    
28. Surreal analysis: an analogue of real analysis for surreal numbers
    Siemens Competition Regional Finals, California Institute of Technology, 2012.
    

1. Toward secondary terms for 2-Selmer groups in families of elliptic curves
   Arithmetic Statistics Seminar, Harvard University.
   
2. A positive proportion of monic odd hyperelliptic curves have no unexpected quadratic points
   Lightning Talk at conference on The Mordell conjecture: 100 years later, 2024.
   
3. Most odd-degree binary forms fail to primitively represent a square
   Lightning Talk at Annual Meeting of Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation, 2022.
   
4. Counting lattice points in cusps of fundamental domains
   Number Theory Tea, Princeton University, 2020.
   
5. Three lectures on the average size of $2$-Selmer groups of elliptic curves
   Bhargava Seminar, Stanford University, 2020.
   
6. On the paucity of soluble superelliptic equations
   Bhargava Seminar, Stanford University, 2020.
   
7. On the paucity of soluble superelliptic equations
   Bhargava Seminar, Princeton University, 2019.
   
8. Inflectionary invariants for local complete intersection curve singularities
   Graduate Student Seminar, Princeton University, 2018.
   
9. Inflectionary invariants for plane curve singularities
   LinkedIn Bangalore, 2018.
   
10. Surjectivity of Galois representations in rational families of abelian varieties
    JMM 2017.
    
11. Permutations that destroy arithmetic progressions in elementary $p$-groups
    JMM 2017.
    
12. Elliptic curve variants of the least quadratic nonresidue problem and Linnik's Theorem
    JMM 2016.
    
13. Permutations that destroy arithmetic progressions
    Harvard Math Table, 2016.
    
14. On arboreal Galois representations of rational functions
    JMM 2015.
    
15. How small is that prime?
    Harvard Math Table, 2015.
    
16. Arboreal Galois representations of rational functions
    Harvard Math Table, 2014.
    
17. Surreal analysis: an analogue of real analysis for surreal numbers
    Harvard Math Table, 2013.
    
18. Surreal analysis: an analogue of real analysis for surreal numbers
    CMS Summer Meeting, 2013.
    

Teaching Experience

• [Math 223b] Rational points on varieties, 2025. (expected)

• [Math 295x] Arithmetic statistics, 2024. Q-score: 5.0/5.0

• [Math 202] Linear algebra with applications, 2020. Rating: 4.0/5.0

• [Math 137] Introduction to algebraic geometry (Peter Kronheimer), 2017. Q-score: 5.0/5.0

• [Math 131] Topology I: Topological spaces & the fundamental group (Joe Harris), 2016. Q-score: 4.5/5.0

• [Math 123] Algebra II: Theory of rings & fields (Arul Shankar), 2016. Q-score: 4.9/5.0

• [Math 114] Analysis II: Measure, integration, & Banach spaces (Daniel Cristofaro-Gardiner), 2015. Q-score: 4.9/5.0

• [Math 55b] Honors real & complex analysis (Dennis Gaitsgory), 2015. Q-score: 4.3/5.0

• [Math 55a] Honors abstract algebra (Dennis Gaitsgory), 2014. Q-score: 4.6/5.0

• Matt King (Harvard PhD student), 2022-2024. Coauthored paper: [arXiv]

• Eliot Hodges (Harvard undergrad) 2024-present. Advised senior thesis; winner of the Hoopes Prize; coauthored papers (in preparation).

• Tanav Choudhary (Harvard undergrad) 2025-present.

• Tutor and Classroom Assistant, Cambridge Learning Center, 2022-2023.

• Tutor, MIT Introduction to Technology, Engineering, and Science Program, 2024.

Honors and Awards