Please note that the papers below may differ from their published versions.
My current research program
Most of my current work focuses on arithmetic and geometric aspects of fundamental groups — that is, “anabelian geometry” — very broadly construed. I’m interested in connections to arithmetic, geometry, and topology.
papers on Arithmetic and topology of algebraic varieties
• Density of integral points in the Betti moduli of quasi-projective varieties (arXiv version, 2025, joint with Simone Coccia)
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• Algebraicity and integrality of solutions to differential equations (arXiv version, 2025, joint with Josh Lam)
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• Motives, mapping class groups, and monodromy (arXiv version, 2024)
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• Big monodromy for higher Prym representations (accepted at Geometry and Topology, arXiv version, 2024, joint with Aaron Landesman and Will Sawin)
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• Finite braid group orbits on SL2-character varieties (arXiv version, 2023, joint with Yeuk Hay Joshua Lam and Aaron Landesman)
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• Geometric local systems on the projective line minus four points (accepted at Compositio Mathematica, arXiv version, 2023, joint with Yeuk Hay Joshua Lam)
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• Prill’s problem (accepted to Compositio, Algebraic Geometry, arXiv version, 2022, joint with Aaron Landesman)
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• An introduction to the algebraic geometry of the Putman-Wieland conjecture (European Journal of Mathematics, 2023, arXiv version, 2022, joint with Aaron Landesman)
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• Applications of the algebraic geometry of the Putman-Wieland conjecture (Proceedings of the LMS, 2023, arXiv version, 2022, joint with Aaron Landesman)
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• Canonical representations of surface groups (accepted at Annals of Mathematics, arXiv version, 2022, joint with Aaron Landesman)
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• p-adic iterated integration on semi-stable curves (arXiv version, 2022, joint with Eric Katz)
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• Geometric local systems on very general curves and isomonodromy (accepted at Journal of the AMS, arXiv version, 2022, joint with Aaron Landesman — see here for a shorter version that only proves the main results with the assumption of unipotent monodromy at infinity)
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• Surface bundles and the section conjecture (arXiv version, 2020, joint with Wanlin Li, Nick Salter, and Padma Srinivasan, published in Mathematische Annalen)
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• Level structures, arithmetic representations, and non-commutative Siegel linearization (arXiv version, 2021, joint with Borys Kadets, published in Journal für die reine und angewandte Mathematik - Crelle's Journal)
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• Representations of surface groups with universally finite mapping class group orbit (arXiv version, joint with Brian Lawrence, Mathematical Research Letters, 2023)
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• Arithmetic representations of fundamental groups II: finiteness (arXiv version, 2021, Duke Mathematics Journal version here)
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• Arithmetic representations of fundamental groups I (Inventiones Mathematicae (or here), 2018, arXiv version)
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• Arithmetic restrictions on geometric monodromy (not intended for publication, arXiv version)
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• Integral points on algebraic subvarieties of period domains (manuscripta mathematica, arXiv version here, 2019, joint with Ariyan Javanpeykar)
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• Tamely ramified morphisms of curves and Belyi’s theorem in positive characteristic (arXiv version, 2020, joint with Kiran Kedlaya and Jakub Witaszek, published in IMRN)
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• Group-theoretic Johnson classes and a non-hyperelliptic curve with torsion Ceresa class (EPIGA, arXiv version, 2020, joint with Dean Bisogno, Wanlin Li, and Padma Srinivasan)
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• Semisimplicity and weight-monodromy for fundamental groups (arXiv version, joint with Alexander Betts, 2019, appeared in Proceedings of the Simons Symposium on p-adic Hodge Theory)
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Classical Algebraic Geometry
I have pretty broad algebro-geometric interests: Lefschetz theorems, positivity and vanishing theorems for vector bundles in positive characteristic, the fine classification of algebraic varieties, algebraic dynamics, and birational geometry.
Miscellaneous
• The Manin-Mumford conjecture in genus 2 and rational curves on K3 surfaces (arXiv version, 2022, joint with Philip Engel and Raju Krishnamoorthy)
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Vanishing and positivity
• Vanishing for Frobenius twists of ample vector bundles (Tohoku Math Journal, 2018, arXiv version)
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• Non-abelian Lefschetz hyperplane theorems (Journal of Algebraic Geometry, 2018, arXiv version)
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• Manifolds containing an ample P1-bundle (Manuscripta Mathematica, 2017, arXiv version)
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Dynamics
• Dynamical Mordell-Lang and automorphisms of blowups (joint with John Lesieutre, Algebraic Geometry, Foundation Compositio Mathematica, 2016, arXiv version)
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The Grothendieck ring of varieties
• Zeta functions of curves with no rational points (Michigan Math Journal, 2015, arXiv version)
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• Symmetric powers do not stabilize (Proceedings of the AMS, 2014, arXiv version)