# 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:

Openness of splinter loci in prime characteristic (joint with K. Tucker).

Essential finite generation of extensions of valuation rings.

*To appear in***Math. Nachrichten***.*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).

*To appear in***J. Eur. Math. Soc. (JEMS).**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***, Advance Publication 1-30 (2022)**.*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). To appear in.**Math. Research Letters**Permanence properties of

*F*-injectivity (joint with T. Murayama).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.