Ashvin Swaminathan's homepage

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.

I am co-organizing the Harvard Number Theory Seminar for the academic year 2022-2023.

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. Heuristics for class groups of orders in quadratic fields, with Manjul Bhargava, Arul Shankar, Artane Siad, and Ila Varma
2. Failure of monogenicity in sextic number fields
3. Average $2$$2$-torsion in the class groups of cubic orders, and related problems
4. Affine symmetric spaces and class groups of cubic fields, with Iman Setayesh, Arul Shankar, and Artane Siad
5. The second moment of the size of the $2$$2$-class group of monogenized cubic fields, with Manjul Bhargava and Arul Shankar
6. $2$$2$-Selmer groups of even-degree hyperelliptic curves, and $2$$2$-class groups of even-degree number fields, with Manjul Bhargava and Arul Shankar

Preprints and Publications

1. Geometry-of-numbers methods in the cusp, with Arul Shankar, Artane Siad, and Ila Varma
Preprint based on Chapter 5 of my PhD thesis. 29 pages, submitted. [arXiv]
2. Hermite equivalence of polynomials, with Manjul Bhargava, Jan-Hendrik Evertse, Kálmán Györy, and László Remete
Preprint. 30 pages, submitted. [arXiv]
3. The second moment of the size of the $2$$2$-Selmer group of elliptic curves, with Manjul Bhargava and Arul Shankar
Preprint based on Chapter 4 of my PhD thesis. 49 pages, submitted. [arXiv]
4. A new parametrization for ideal classes in rings defined by binary forms, and applications.
Preprint based on Chapters 1 and 2 of my PhD thesis. 37 pages, to appear in Journal fur die Reine und Angewandte Mathematik (Crelle's Journal) [pdf]   [arXiv]
5. Most odd-degree binary forms fail to primitively represent a square.
Preprint based on Chapter 3 of my PhD thesis. 32 pages, submitted. [pdf]   [arXiv]
6. Appendix to: An arithmetic count of the lines meeting four lines in ${\mathbb{P}}^{3}$$\mathbb{P}^3$, with Borys Kadets, Padmavathi Srinivasan, Libby Taylor, and Dennis Tseng
Published in Transactions of the American Mathematical Society [journal]   [arXiv]
7. On the EKL-degree of a Weyl cover, with Joseph Knight and Dennis Tseng
Published in Journal of Algebra. [journal] [arXiv]
8. Inflectionary invariants for isolated complete intersection curve singularities, with Anand Patel
Preprint based on my senior thesis. 100 pages, to appear in Memoirs of the American Mathematical Society [arXiv]
9. 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]
10. 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]
11. 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]
12. Lifting subgroups of symplectic groups over $\mathbb{Z}/\ell \mathbb{Z}$$\mathbb{Z}/\ell\mathbb{Z}$, with Aaron Landesman, James Tao, and Yujie Xu
Published in Research in Number Theory. [journal] [arXiv]
13. Permuations that destroy arithmetic progressions in elementary $p$$p$-groups, with Noam D. Elkies
Published in The Electronic Journal of Combinatorics. [journal] [arXiv]
14. On logarithmically Benford sequences, with Evan Chen and Peter Park
Published in Proceedings of the American Mathematical Society. [journal] [arXiv]
15. On arboreal Galois representations of rational functions.
Published in Journal of Algebra. [journal] [arXiv]
16. 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]
17. Analysis on surreal numbers, with Simon Rubinstein-Salzedo
Published in Journal of Logic and Analysis. [journal] [arXiv]

Theses and Expository Articles

1. $2$$2$-Selmer groups, $2$$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

Invited Talks

1. TBA
Conference on Algebra, Algorithms, and Arithmetic, ICMS Edinburgh, 2023.
2. The second moment of the size of the 2-Selmer group of elliptic curves
Special Session on Arithmetic Statistics, JMM 2023.
3. Geometry-of-numbers methods in the cusp, and class groups of orders in number fields
Number Theory Seminar, MIT, 2022.
4. 2-Selmer groups, 2-class groups, and the arithmetic of binary forms
Online Number Theory Seminar, University of Debrecen, 2022.
5. On the distribution of 2-Selmer groups of hyperelliptic Jacobians
Workshop on Rational Points on Higher-Dimensional Varieties, ICMS Edinburgh, 2022.
6. 2-Selmer groups, 2-class groups, and the arithmetic of binary forms
Number Theory Seminar, Fields Institute, 2022.
7. 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.
8. 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.
9. The second moment of the size of the $2$$2$-Selmer group of elliptic curves
Number Theory Seminar, Princeton University/IAS, 2021.
10. "$2$$2$-Selmer groups, $2$$2$-class groups, and the arithmetic of binary forms
Palmetto Joint Arithmetic and Modularity Series (PAJAMAS) III, Invited Graduate Speaker, 2021
11. The second moment of the size of the $2$$2$-Selmer group of elliptic curves
Workshop on Explicit Methods in Number Theory, Mathematisches Forschungsinstitut Oberwolfach, 2021.
12. Average $2$$2$-torsion in the class groups of rings associated to binary $n$$n$-ic forms
Bhargava Seminar, Princeton University, 2020.
13. Average $2$$2$-torsion in the class groups of rings associated to binary $n$$n$-ic forms
Number/Representation Theory Seminar, University of Toronto, 2020.
14. Most hyperelliptic curves are pointless
Bhargava Seminar, Princeton University, 2019.
15. Inflectionary invariants for plane curve singularities
Special Session on Research in Mathematics by Undergraduates, JMM 2018.
16. Surjectivity of Galois representations in rational families of abelian varieties
Algebra, Geometry, and Number Theory Seminar, Tufts University, 2017.
17. Inflection points of linear systems on families of curves
Algebraic Geometry Seminar, Stanford University, 2017.
18. Analysis on surreal numbers: functions and integration
ASL Special Session, JMM 2016.
19. Surreal analysis: an analogue of real analysis for surreal numbers
Siemens Competition Regional Finals, California Institute of Technology, 2012.

Contributed Talks

1. Counting lattice points in cusps of fundamental domains
Number Theory Tea, Princeton University, 2020.
2. Three lectures on the average size of $2$$2$-Selmer groups of elliptic curves
Bhargava Seminar, Stanford University, 2020.
3. On the paucity of soluble superelliptic equations
Bhargava Seminar, Stanford University, 2020.
4. On the paucity of soluble superelliptic equations
Bhargava Seminar, Princeton University, 2019.
5. Inflectionary invariants for local complete intersection curve singularities
Graduate Student Seminar, Princeton University, 2018.
6. Inflectionary invariants for plane curve singularities
7. Surjectivity of Galois representations in rational families of abelian varieties
JMM 2017.
8. Permutations that destroy arithmetic progressions in elementary $p$$p$-groups
JMM 2017.
9. Elliptic curve variants of the least quadratic nonresidue problem and Linnik's Theorem
JMM 2016.
10. Permutations that destroy arithmetic progressions
Harvard Math Table, 2016.
11. On arboreal Galois representations of rational functions
JMM 2015.
12. How small is that prime?
Harvard Math Table, 2015.
13. Arboreal Galois representations of rational functions
Harvard Math Table, 2014.
14. Surreal analysis: an analogue of real analysis for surreal numbers
Harvard Math Table, 2013.
15. Surreal analysis: an analogue of real analysis for surreal numbers
CMS Summer Meeting, 2013.

Teaching Experience

Assistant Instructor, Department of Mathematics, Princeton University

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

Course Assistant, Department of Mathematics, Harvard University

• [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

Honors and Awards

• Princeton Engineering Council Teaching Award (Math 202), 2021

• Frank and Brennie Morgan Prize, 2017

• David B. Mumford Prize, 2017

• Paul and Daisy Soros Fellowship for New Americans, 2017-2019

• NSF Graduate Research Fellowship, 2017-2022

• Hertz Fellowship -- Finalist, 2017

• Centennial Fellowship, Princeton University, 2017-2021

• Harvard University Certificate of Distinction in Teaching (Math 114, 123, 131)

• John Harvard Scholar for Academic Achievement (2016, 2015, 2014)

• Robert Fletcher Rogers Prize, 2016

• Phi Beta Kappa -- Junior 24, 2016

• Barry M. Goldwater Scholarship (Fellow -- 2016, Honorable Mention -- 2015)

• Detur Book Prize for Academic Excellence, 2014

• Siemens Competition in Math, Science, and Technology -- Regional Finalist, 2013

• Siemens AP Science Award -- National Winner, 2013

Journal Refereeing

• International Mathematics Research Notices

• Research in Number Theory

• Research in the Mathematical Sciences

• Proceedings of the American Mathematical Society

• Mathematical Proceedings of the Cambridge Philosophical Society

• Involve, a Journal of Mathematics

Classical Music

Last updated: 1 September 2022