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.
When is Frobenius epic? (joint with J. Carvajal-Rojas, N. Olander and A. Stäbler).
A characteristic p analog of formal lifting properties (joint with N. Olander). Int. Math. Res. Not., vol. 2026 no. 15, 2026.
F-intersection flatness of dagger and Berkovich affinoid algebras (joint with J.J. Garzella and K. Tucker).
Uniform approximation of Abhyankar valuation ideals in mixed characteristic.
Variants on Frobenius intersection flatness and applications to Tate algebras (joint with N. Epstein, K. Schwede and K. Tucker).
Mittag-Leffler modules and intersection flatness (joint with N. Epstein and K. Tucker). Forum Math. Sigma, vol. 14, 2026.
Finite generation of split F-regular monoid algebras (joint with K. Schwede and K. Tucker). J. London Math. Society, vol. 112 no. 1, 2025.
Openness of splinter loci in prime characteristic (joint with K. Tucker). J. Algebra, vol. 629, 2023.
Essential finite generation of extensions of valuation rings. Math. Nachrichten, vol. 296 no. 3, 2023.
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.
Tate algebras and Frobenius non-splitting of excellent regular rings (joint with T. Murayama). J. Eur. Math. Soc. (JEMS), vol. 25 no. 11, 2023.
Valuation rings are derived splinters (joint with B. Antieau). Math. Zeitschrift, vol. 299, 2021.
On some permanence properties of derived splinters (joint with K. Tucker). Michigan Math. J, vol. 73 no. 2, 2023.
Hilbert-Kunz multiplicity of fibers and Bertini's theorems (joint with A. Simpson). J. Algebra, vol. 595 no. 1, 2022.
Annihilators of D-modules in mixed characteristic (joint with N. Switala and W. Zhang). Math. Research Letters, vol. 30 no. 3, 2023.
Permanence properties of F-injectivity (joint with T. Murayama). Math. Research Letters, vol. 31 no. 4, 2024.
Frobenius splitting of valuation rings and F-singularities of centers. Algebra Number Theory, vol. 15 no. 10, 2021.
Uniform approximation of Abhyankar valuations in function fields of prime characteristic. Trans. Amer. Math. Soc., vol. 373 no. 1, 2020.
Excellence in prime characteristic (joint with K.E. Smith). Contemporary Mathematics, Local and Global Methods in Algebraic Geometry, 712 (2018).
Non-vanishing results on local cohomology of valuation rings. J. Algebra, vol. 479 no. 1, 2017.
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.
Free and very free morphisms on a Fermat hyper surface (joint with T. Bridges, J. Eddy, M. Newman and J. Yu). Involve vol. 6 no. 4, 2013.
Polygons in quadratically closed rings and properties of n-adically closed rings.