Research

Research summary:

I specialize in studying solution sets of polynomial equations whose coefficients are in finite fields.  Finite fields have a special function associated with them called the Frobenius (p-th power) map, and I exploit  algebraic properties of the Frobenius map to study geometric properties of polynomial solutions. My work has a significant overlap with valuation theory, with an eye toward understanding non-noetherian rings that arise naturally in geometry and arithmetic, and using this understanding to attack problems about noetherian rings.

Research publications:

• Uniform approximation of Abhyankar valuation ideals in mixed characteristic, In Preparation.

• Finite generation of split F-regular monoid algebras (joint with K. Schwede and K. Tucker).

• Mittag-Leffler modules and Frobenius (joint with N. Epstein and K. Tucker). 

• 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.