I specialize in studying rings over finite fields. Finite fields have a distinguished function associated with them called the Frobenius (p-th power) map, and I exploit algebraic properties of the Frobenius map to study singularities of Noetherian rings. My work has a significant overlap with valuation theory. More recently I have been thinking a lot about rings arising in rigid analytic geometry.
Frobenius splitting of valuation rings revisited (joint with K.E. Smith). In preparation.
When is Frobenius epic? (joint with J. Carvajal-Rojas, N. Olander and A. Stäbler). arXiv link.
A characteristic p analog of formal lifting properties (joint with N. Olander). Int. Math. Res. Not., vol. 2026 no. 15, 2026. arXiv link.
F-intersection flatness of dagger and Berkovich affinoid algebras (joint with J.J. Garzella and K. Tucker). To appear in Proceedings of AMS. arXiv link.
Uniform approximation of Abhyankar valuation ideals in mixed characteristic. arXiv link.
Variants on Frobenius intersection flatness and applications to Tate algebras (joint with N. Epstein, K. Schwede and K. Tucker). arXiv link.
Mittag-Leffler modules and intersection flatness (joint with N. Epstein and K. Tucker). Forum Math. Sigma, vol. 14, 2026. arXiv link.
Finite generation of split F-regular monoid algebras (joint with K. Schwede and K. Tucker). J. London Math. Society, vol. 112 no. 1, 2025. arXiv link.
Openness of splinter loci in prime characteristic (joint with K. Tucker). J. Algebra, vol. 629, 2023. arXiv link.
Essential finite generation of extensions of valuation rings. Math. Nachrichten, vol. 296 no. 3, 2023. arXiv link.
Excellence, F-singularities and solidity (joint with T. Murayama with an appendix by K.E. Smith). To appear in London Mathematical Society Lecture Note Series. arXiv link.
Tate algebras and Frobenius non-splitting of excellent regular rings (joint with T. Murayama). J. Eur. Math. Soc. (JEMS), vol. 25 no. 11, 2023. arXiv link.
Valuation rings are derived splinters (joint with B. Antieau). Math. Zeitschrift, vol. 299, 2021. arXiv link.
On some permanence properties of derived splinters (joint with K. Tucker). Michigan Math. J, vol. 73 no. 2, 2023. arXiv link.
Hilbert-Kunz multiplicity of fibers and Bertini's theorems (joint with A. Simpson). J. Algebra, vol. 595 no. 1, 2022. arXiv link.
Annihilators of D-modules in mixed characteristic (joint with N. Switala and W. Zhang). Math. Research Letters, vol. 30 no. 3, 2023. arXiv link.
Permanence properties of F-injectivity (joint with T. Murayama). Math. Research Letters, vol. 31 no. 4, 2024. arXiv link.
Frobenius splitting of valuation rings and F-singularities of centers. Algebra Number Theory, vol. 15 no. 10, 2021. arXiv link.
Uniform approximation of Abhyankar valuations in function fields of prime characteristic. Trans. Amer. Math. Soc., vol. 373 no. 1, 2020. arXiv link.
Excellence in prime characteristic (joint with K.E. Smith). Contemporary Mathematics, Local and Global Methods in Algebraic Geometry, 712 (2018). arXiv link.
Non-vanishing results on local cohomology of valuation rings. J. Algebra, vol. 479 no. 1, 2017. arXiv link.
Frobenius and valuation rings (with K. E. Smith). Algebra Number Theory, vol. 10 no. 5, 2016; corrigendum: Correction to the article Frobenius and valuation rings. Algebra Number Theory vol. 11 no. 4, 2017. arXiv link.
Free and very free morphisms on a Fermat hypersurface (joint with T. Bridges, J. Eddy, M. Newman and J. Yu). Involve vol. 6 no. 4, 2013. arXiv link.
Polygons in quadratically closed rings and properties of n-adically closed rings. arXiv link.