Publications and Preprints

• Moduli interpretations for noncongruence modular curves (survey article for conference proceedings) (2025) [submitted]

• Noncongruence presentations of finite simple groups, with Alex Lubotzky and Pham Huu Tiep, submitted (2024). [arXiv]

• Families of elliptic curves over the four-pointed configuration space and exceptional sequences for the braid group on four strands, with Nick Salter (2024). To appear in Proceedings of the AMS [arxiv]

• Finite simple characteristic quotients of the free group of rank 2, with Alex Lubotzky and Pham Huu Tiep (2023). To appear in Commentarii Mathematici Helvetici. [arXiv]

• Remarks on the Z/p Dijkgraaf-Witten invariants of 3D mapping tori, with Alex Kontorovich and Shehryar Sikander, published in Communications in Mathematical Physics. (2023) [arXiv] [published]

• Nonabelian level structures, Nielsen equivalence, and Markoff triples. Published in Annals of Mathematics (2024). [arXiv] [published]

  Video for in person talk at the Arithmetic Groups seminar at the IAS
  Video for online talk at the Joint Columbia-CUNY-NYU number theory seminar
  Video for online talk at the Joint IAS-Princeton number theory seminar

• Tamely ramified covers of the projective line with alternating and symmetric monodromy, with Renee Bell, Jeremy Booher, and Yuan Liu. Published in Algebra and Number Theory (2021). [arXiv] [published]

• Arithmetic monodromy actions on pro-metabelian fundamental groups of once-punctured elliptic curves, with Pierre Deligne (2017) [Latest revision] [arXiv]

• Moduli Interpretations for noncongruence modular curves. Published in Mathematische Annalen (2018) [arXiv] [published]
  Video for online talk during a special trimester at Bhaskaracharya Pratishthana, Pune

Expository Notes

• Katz modular forms - Expository notes I wrote to explain the relationship between classical and geometric notions of modular forms (the latter in the sense of Katz) for finite index subgroups of SL(2,Z), including algebraic and integral properties of q-expansions for their modular forms.

• Two base change results for rings of invariants - A short note explaining two base change results for rings of invariants. The first is a classical theorem in invariant theory, the second is a base change result for character varieties of representations of a free group in a split reductive group.

Computing

• Given a finite group G, the geometry of the connected components of the moduli stacks of elliptic curves with G-structures can be readily computed from its monodromy as a finite etale cover of the moduli stack of elliptic curves. The results of these computations have led to observations that inspired the papers on Markoff triples and on metabelian groups (joint with Deligne). To facilitate these computations, I have written some code which you can find here (includes documentation and examples. Requires GAP to run). Data from some of these computations can be found below in csv and pdf formats. However, the best way to examine the data is to generate them for yourself in GAP and query the database via the SGD function. The data below took 4 days to generate using a 2021 Macbook Pro.

• Data for M(G) as G ranges over all finite simple groups of order at most the order of the Janko group |J1| = 175560 can be found here: [csv] [pdf] [README]

• Data for M(G)/Out(G) can be found here: [csv] [pdf] [README] (readme is the same)