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.
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 variants on intersection flatness (joint with N. Epstein and K. Tucker).
Finite generation of split F-regular monoid algebras (joint with K. Schwede and K. Tucker). To appear in J. London Math. Society.
Openness of splinter loci in prime characteristic (joint with K. Tucker). J. Algebra, 629 (2023).
Essential finite generation of extensions of valuation rings. Math. Nachrichten, 296(3) (2023).
Excellence, F-singularities and solidity (joint with T. Murayama with an appendix by K.E. Smith).
Tate algebras and Frobenius non-splitting of excellent regular rings (joint with T. Murayama). J. Eur. Math. Soc. (JEMS), 25(11) (2023).
Valuation rings are derived splinters (joint with B. Antieau). Math. Zeitschrift, 299 (2021).
On some permanence properties of derived splinters (joint with K. Tucker). Michigan Math. J, 73(2) (2023).
Hilbert-Kunz multiplicity of fibers and Bertini's theorems (joint with A. Simpson). J. Algebra, 595(1) (2022).
Annihilators of D-modules in mixed characteristic (joint with N. Switala and W. Zhang). Math. Research Letters, 30(3) (2023).
Permanence properties of F-injectivity (joint with T. Murayama). To appear in Math. Research Letters.
Frobenius splitting of valuation rings and F-singularities of centers. Algebra Number Theory, 15(10) (2021).
Uniform approximation of Abhyankar valuations in function fields of prime characteristic. Trans. Amer. Math. Soc. 373-1 (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 479-1 (2017).
Frobenius and valuation rings (with K. E. Smith). Algebra Number Theory 10-5 (2016); corrigendum: Correction to the article Frobenius and valuation rings. Algebra Number Theory 11-4 (2017).
Free and very free morphisms on a Fermat hypersurface (joint with T. Bridges, J. Eddy, M. Newman and J. Yu). Involve 6-4 (2013).
Polygons in quadratically closed rings and properties of n-adically closed rings.