I am an assistant professor at the University of Illinois at Urbana-Champaign. Before that I have been a member at the Institute for Advanced Study, as well as an NSF Postdoctoral Fellow at Columbia and McGill. I completed my PhD in 2016 at Penn State, under the supervision of Winnie Li.
Email: oxeimon[at]gmail[dot]com
My work concerns arithmetic geometry, broadly construed. I am especially interested in studying the relation between group theory, algebraic geometry, and number theory, as mediated by monodromy actions of etale fundamental groups.
In my thesis I showed that noncongruence modular curves can be viewed as moduli spaces of elliptic curves with nonabelian level structures. From this I proved certain integrality properties about the Fourier coefficients of noncongruence modular forms. More recently I showed that these moduli spaces can be used to construct new tamely ramified covers of the projective line in characteristic p. By studying the degree of the Hodge bundle on these spaces, one can deduce a divisibility theorem on the cardinalities of Nielsen equivalence classes of generating pairs of finite groups. Up to a finite computation, this resolves a conjecture of Bourgain, Gamburd, and Sarnak on the Markoff equation.
This semester I’m supervising a reading seminar on Katz-Mazur’s book Arithmetic moduli of elliptic curves.
Publications and Preprints
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 a sequence of Dyer-Formanek-Grossman, with Nick Salter (2024). [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, submitted. (2023) [arXiv]
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 seminarTamely 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)
Teaching at University of Illinois Urbana-Champaign
Fall 2024 - Math 415 Applied Linear Algebra
Teaching at Rutgers
Spring 2023 - Math 356 Theory of Numbers
Spring 2023 - Math 135 Calculus I for the life and social sciences
Fall 2022 - Math 350 Linear Algebra [course website]
Teaching at Columbia
Summer 2021 - Math 3020 Number Theory and Cryptography [course materials]
Spring 2020 - Math 1101 Calculus I [course materials]
Teaching at Penn State
Spring 2016 - Math 251 Ordinary and partial differential equations [course materials]
Fall 2014 - Math 41 Trigonometry and analytic geometry
Fall 2013 - Math 21 College Algebra I
Fall 2012 - Math 21 College Algebra I
Fall 2011 - Math 220 Matrices [course materials]
Summer 2011 - Math 220, Matrices [course materials]
Spring 2011 - Math 17 Finite Mathematics
Other
I enjoy taking pictures. You can find some of my photography here.