Department of Mathematics, Statistics and Computer Science
University of Illinois Chicago
Office: SEO 411
I am a Research Assistant Professor in the Mathematics Department at University of Illinois Chicago. I recently completed my Ph.D. from the University of Michigan, where I was advised by Karen Smith.
I work at the interface of algebraic geometry and commutative algebra, specializing in the use of local prime characteristic methods (in commutative algebra) and their global variants. I often enjoy thinking about non-Noetherian objects such as valuation rings, perfectoid rings, and more recently, rings of differential operators.
Winter 2019: Math 210 (the course website). Here is the website for my sections.
- On some permanence properties of (derived) splinters (joint with Kevin Tucker). In preparation.
- Hilbert-Kunz multiplicity of fibers and Bertini theorems (joint with Austyn Simpson).
- Annihilators of D-modules in mixed characteristic (joint with Nicholas Switala and Wenliang Zhang).
- Permanence properties of F-injectivity (joint with Takumi Murayama).
- Frobenius splitting of valuation rings and F-singularities of centers.
- Uniform approximation of Abhyankar valuation ideals in function fields of prime characteristic. To appear in Transactions of AMS.
- Excellence in Prime Characteristic (joint with Karen Smith). Contemporary Mathematics, Local and Global Methods in Algebraic Geometry, 712
- (Non)Vanishing results on local cohomology of valuation rings. Journal of Algebra 479-1(2017), 413--436. DOI 10.1016/j.jalgebra.2016.12.03.
- Frobenius and valuation rings (joint with Karen Smith). Algebra & Number Theory 10-5 (2016), 1057-1090. DOI 10.2140/ant.2016.10.1057; corrigendum: Correction to the article Frobenius and valuation rings. Algebra & Number Theory 11-4 (2017), 1003-1007. DOI 10.2140/ant.2017.11.1003.
- Free and very free morphisms on a Fermat hypersurface (joint with T. Bridges, J. Eddy, M. Newman, J. Yu). Involve 6 (2013), No. 4, 437-445.
- Polygons in quadratically closed rings and properties of n-adically closed rings (undergraduate thesis supervised by Aise Johan de Jong).
Here is a copy of my dissertation. Corrections are welcome!
Notes are subject to change without notice.
- Very rough notes (prepared for a Grad seminar at UMich) on a proof of a theorem of Kunz, characterizing regularity of Noetherian rings in terms of flatness of Frobenius, using the surprising homological properties of perfect rings. Based loosely on a talk given by Bhatt at Gennady Lyubeznik's 60th birthday conference, the notes are a more verbose version of the original proof appearing in a paper by Bhatt-Scholze.
- Notes on Huber rings for a learning seminar at the University of Michigan (Winter 2017). Last updated Feb 18, 2017. A new section was added on uniform Huber rings (not discussed in the lectures), following Bhargav’s discussion of uniform K-Banach algebras in his course. In particular, we prove equivalence of categories results generalizing [Bha17, Thm 9.7 and Cor 9.9].
- On a vanishing result in sheaf cohomology. An example is given of a non-quasicompact scheme that violates a vanishing result in sheaf cohomology that holds for certain quasicompact spaces [Stacks Project, Tag02UX]. This example can be interpreted purely topologically (without mentioning schemes), and is incorporated in the latter form in Tag0BX0.
- Notes from a summer mini-course I taught at Michigan on notions of singularities in prime characteristic in 2016. The notes have not been proof-read and do not cover a lot of material.