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.
For the 202425 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.
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 CohenLenstra heuristics, rational points on varieties, Diophantine equations, and enumerative geometry.
Email: swaminathan@math.harvard.edu.
Research Articles in Preparation

Heuristics for class groups of orders in quadratic fields, with Manjul Bhargava, Arul Shankar, Artane Siad, and Ila Varma

2$2$Selmer groups of evendegree hyperelliptic curves, and 2$2$class groups of evendegree number fields, with Manjul Bhargava and Arul Shankar
Preprints

The second moment of the size of the 2$2$class group of monogenized cubic fields, with Manjul Bhargava and Arul Shankar
Preprint available upon request.

Universality theorems for zeros of random real polynomials with fixed coefficients, with Matt King
Preprint. 23 pages, submitted.
[arXiv]

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]

Counting integral points on symmetric varieties with applications to arithmetic statistics, with Arul Shankar and Artane Siad
Preprint. 39 pages, revisions submitted.
[arXiv]
Publications

Geometryofnumbers 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]

The mean number of 2$2$torsion elements in the class groups of cubic orders.
Preprint. To appear in Commentarii Mathematici Helvetici.
[arXiv]

A positive proportion of monic odddegree hyperelliptic curves have no unexpected quadratic points, with Jef Laga
Preprint. To appear in International Mathematics Research Notices.
[arXiv]

Most odddegree binary forms fail to primitively represent a square.
Published in Compositio Mathematica. Based on Chapter 3 of my PhD thesis.
[journal] [arXiv]

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]

Hermite equivalence of polynomials, with Manjul Bhargava, JanHendrik Evertse, Kálmán Györy, and László Remete
Published in a special issue of Acta Arithmetica dedicated to the memory of Andrzej Schinzel.
[journal] [arXiv]

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]

On the EKLdegree of a Weyl cover, with Joseph Knight and Dennis Tseng
Published in Journal of Algebra. [journal]
[arXiv]

Appendix to: An arithmetic count of the lines meeting four lines in P3${\mathbb{P}}^{3}$, with Borys Kadets, Padmavathi Srinivasan, Libby Taylor, and Dennis Tseng
Published in Transactions of the American Mathematical Society
[journal] [arXiv]

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]

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]

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]

Lifting subgroups of symplectic groups over Z/ℓZ$\mathbb{Z}/\ell \mathbb{Z}$, with Aaron Landesman, James Tao, and Yujie Xu
Published in Research in Number Theory. [journal] [arXiv]

Permuations that destroy arithmetic progressions in elementary p$p$groups, with Noam D. Elkies
Published in The Electronic Journal of Combinatorics. [journal] [arXiv]

On logarithmically Benford sequences, with Evan Chen and Peter Park
Published in Proceedings of the American Mathematical Society. [journal] [arXiv]

On arboreal Galois representations of rational functions.
Published in Journal of Algebra. [journal] [arXiv]

Linnik’s theorem for SatoTate laws on elliptic curves with complex multiplication, with Evan Chen and Peter Park
Published in Research in Number Theory. [journal] [arXiv]

Analysis on surreal numbers, with Simon RubinsteinSalzedo
Published in Journal of Logic and Analysis. [journal] [arXiv]
Theses and Expository Articles

2$2$Selmer groups, 2$2$class groups, and the arithmetic of binary forms (Princeton PhD thesis) [pdf]

Inflection points of linear systems on families of curves (Harvard senior thesis) [pdf]
 Lie algebras and Ado's Theorem [pdf]
 An introduction to the theory of valued fields (Harvard junior paper)[pdf]
 On the SelbergErdős proof of the Prime Number Theorem [pdf]
 Arrow's impossibility theorem on social choice systems
[pdf]
Presentations
Invited Talks

A positive proportion of hyperelliptic curves have no unexpected quadratic points
Algebraic Geometry Seminar, Boston College, 2024.

A positive proportion of hyperelliptic curves have no unexpected quadratic points
Informal Number Theory Seminar, University of Glasgow, 2024.

A positive proportion of hyperelliptic curves have no unexpected quadratic points
Number Theory Seminar, Tufts University, 2024.

A positive proportion of hyperelliptic curves have no unexpected quadratic points
Five College Number Theory Seminar, Amherst College, 2024.

Geometryofnumbers in the cusp, and class groups of orders in number fields
Number Theory Seminar, Duke University, 2023.

Affine symmetric spaces and class groups of cubic fields
Québec Vermont Number Theory Seminar, McGill University, 2023.

Geometryofnumbers in the cusp, and class groups of orders in number fields
Conference on Arithmetic Statistics, CIRM Luminy, 2023.

Affine symmetric spaces and class groups of cubic fields
Conference on Algebra, Algorithms, and Arithmetic, ICMS Edinburgh, 2023.

Affine symmetric spaces and class groups of cubic fields
Symposium on Arithmetic Geometry and its Applications, CIRM Luminy, 2023.

The second moment of the size of the 2Selmer group of elliptic curves
Special Session on Arithmetic Statistics, JMM 2023.

Geometryofnumbers methods in the cusp, and class groups of orders in number fields
Number Theory Seminar, MIT, 2022.

2Selmer groups, 2class groups, and the arithmetic of binary forms
Online Number Theory Seminar, University of Debrecen, 2022.

On the distribution of 2Selmer groups of hyperelliptic Jacobians
Workshop on Rational Points on HigherDimensional Varieties, ICMS Edinburgh, 2022.

2Selmer groups, 2class groups, and the arithmetic of binary forms
Number Theory Seminar, Fields Institute, 2022.

Geometryofnumbers methods in the cusp, and applications to class groups
Montreal Online Biweekly InterUniversity Seminar on Analytic Number Theory (MOBIUS ANT), 2022.
 The second moment of the size of the 2$2$Selmer group of elliptic curves
Number Theory Seminar, Princeton University/IAS, 2021.
 "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
 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.
 Average 2$2$torsion in the class groups of rings associated to binary n$n$ic forms
Bhargava Seminar, Princeton University, 2020.
 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.
 Most hyperelliptic curves are pointless
Bhargava Seminar, Princeton University, 2019.
 Inflectionary invariants for plane curve singularities
Special Session on Research in Mathematics by Undergraduates, JMM 2018.
 Surjectivity of Galois representations in rational families of abelian varieties
Algebra, Geometry, and Number Theory Seminar, Tufts University, 2017.
 Inflection points of linear systems on families of curves
Algebraic Geometry Seminar, Stanford University, 2017.
 Analysis on surreal numbers: functions and integration
ASL Special Session, JMM 2016.
 Surreal analysis: an analogue of real analysis for surreal numbers
Siemens Competition Regional Finals, California Institute of Technology, 2012.
Contributed Talks

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.

Most odddegree binary forms fail to primitively represent a square
Lightning Talk at Annual Meeting of Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation, 2022.
 Counting lattice points in cusps of fundamental domains
Number Theory Tea, Princeton University, 2020.
 Three lectures on the average size of 2$2$Selmer groups of elliptic curves
Bhargava Seminar, Stanford University, 2020.
 On the paucity of soluble superelliptic equations
Bhargava Seminar, Stanford University, 2020.
 On the paucity of soluble superelliptic equations
Bhargava Seminar, Princeton University, 2019.
 Inflectionary invariants for local complete intersection curve singularities
Graduate Student Seminar, Princeton University, 2018.
 Inflectionary invariants for plane curve singularities
LinkedIn Bangalore, 2018.
 Surjectivity of Galois representations in rational families of abelian varieties
JMM 2017.
 Permutations that destroy arithmetic progressions in elementary p$p$groups
JMM 2017.
 Elliptic curve variants of the least quadratic nonresidue problem and Linnik's Theorem
JMM 2016.
 Permutations that destroy arithmetic progressions
Harvard Math Table, 2016.
 On arboreal Galois representations of rational functions
JMM 2015.
 How small is that prime?
Harvard Math Table, 2015.
 Arboreal Galois representations of rational functions
Harvard Math Table, 2014.
 Surreal analysis: an analogue of real analysis for surreal numbers
Harvard Math Table, 2013.
 Surreal analysis: an analogue of real analysis for surreal numbers
CMS Summer Meeting, 2013.
Teaching Experience
Instructor, Department of Mathematics, Harvard University

[Math 223b] Rational points on varieties, 2025. (expected)
[Math 295x] Arithmetic statistics, 2024. Qscore: 5.0/5.0
Assistant Instructor, Department of Mathematics, Princeton University
Course Assistant, Department of Mathematics, Harvard University

[Math 137] Introduction to algebraic geometry (Peter Kronheimer), 2017. Qscore: 5.0/5.0

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

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

[Math 114] Analysis II: Measure, integration, & Banach spaces (Daniel CristofaroGardiner), 2015. Qscore: 4.9/5.0

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

[Math 55a] Honors abstract algebra (Dennis Gaitsgory), 2014. Qscore: 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, 20172019

NSF Graduate Research Fellowship, 20172022

Hertz Fellowship  Finalist, 2017

Centennial Fellowship, Princeton University, 20172021

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

Music Review of Matthew Aucoin's "Its Own Accord," 2016 [pdf]

Letter from Susanna: Handel's Messiah as a universal vehicle of deliverance, 2016 [pdf]

Violinist, Harvard–Radcliffe Orchestra (Conductor: Federico Cortese), 20132014

John Near Grant Recipient, The Harker School, 2013

Violinist, San Francisco Symphony Youth Orchestra (Conductor: Donato Cabrera), 20102013

Violinist, Music@Menlo Summer Chamber Music Festival and Institute (Directors: Wu Han & David Finckel), 2006, 2008, 2009
Last updated: 11 August 2024