# 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). To appear in J. Algebra.

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.