Research
Research Interests:
My primary research interests are prime characteristic methods in commutative algebra and algebraic geometry, and especially Frobenius actions on local cohomology and various degrees of nilpotence they may have. This work has a strong relationship to uniformity properties for Frobenius closure and tight closure of ideals, including the existence of Frobenius test exponents. Furthermore, several surprising properties of certain singularity types (F-nilpotent and its generalizations) in prime characteristic have been discovered throughout my research program.
Publications and Preprints:
"Counting geometric branches via the Frobenius map and F-nilpotent singularities" with Hailong Dao and Vaibhav Pandey; Nagoya Mathematical Journal FirstView (arXiv link)
"Rees algebras and generalized depth-like conditions in prime characteristic" with Alessandra Costantini and Lance Edward Miller; Math. Nach. 00 (2023), 1–20 (arXiv link)
"Homological properties of pinched Veronese rings" with Vaibhav Pandey; J. Algebra 614 (2023), 307-329 (arXiv link)
"Generalized F-depth and graded nilpotent singularities" with Lance Edward Miller; Communications in Algebra (2024) 1-25 (arXiv link)
"F-nilpotent rings and permanence properties" with Jennifer Kenkel, Thomas Polstra, and Austyn Simpson; J. Commut. Algebra 15(4): 559-575 (Winter 2023) (arXiv link)
"A sufficient condition for finiteness of Frobenius test exponents" Proc. Amer. Math. Soc. 147 (2019), 5083-5092 (arXiv link)
Recent Invited Talks and Presentations:
Current:
Georgia State University Algebra Seminar "F-singularities of amalgamated algebras along an ideal" (Fall Spring 2024)
Past:
AMS Spring Southeastern Sectional Meeting "Counting geometric branches via Frobenius" (Spring 2023)
Workshop on Commutative Algebra, Algebraic Geometry, and Related Topics "Homological properties of pinched Veronese rings" (Fall 2022)
ICTP Graduate Course on Tight Closure and its Applications, F-rational rings tutorial for Florian Enescu (Summer 2022)
KUMUNU 2021 "Nilpotent singularity types in prime characteristic" (Spring 2022)