851 S Morgan St.
Department of Mathematics, Statistics and Computer Science
University of Illinois at Chicago
Science and Engineering Offices
Chicago, IL 60607
Office: SEO 411
I am a Research Assistant Professor in the Mathematics Department at the University of Illinois at Chicago, mentored by
Kevin Tucker. I was previously at the University of Michigan, where I completed my PhD under 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.
Fall 2020: Graduate commutative algebra (Math 520).
New online seminar
There will be a first of its kind joint NU/UIC/UofC online seminar on algebraic geometry and commutative algebra beginning the second week of May. Please check its webpage for further details.
- Excellence, F-singularities, and solidity (with T. Murayama). In preparation.
- Tate algebras and Frobenius non-splitting of excellent regular rings (with T. Murayama).
- Valuation rings are derived splinters (with B. Antieau).
- On some permanence properties of (derived) splinters (with K. Tucker).
- Hilbert-Kunz multiplicity of fibers and Bertini theorems (with A. Simpson).
- Annihilators of D-modules in mixed characteristic (with N. Switala and W. Zhang).
- Permanence properties of F-injectivity (with T. Murayama).
- Frobenius splitting of valuation rings and F-singularities of centers.
- Uniform approximation of Abhyankar valuation ideals in function fields of prime characteristic. Transactions of American Mathematical Society 373-1 (2020), 319--341. DOI 10.1090/tran/7917.
- Excellence in Prime Characteristic (with K. E. 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 (with K. E. 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 (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 A.J. de Jong).
Here is a copy of my dissertation. Corrections are welcome!
Notes are subject to change without notice.
- A note, written with Takumi Murayama, gives an example of a noetherian domain whose localizations at prime ideals are Japanese (aka N-2), but the domain itself is not. In a sense, our example is as nice as possible. This answers an old question on MathOverflow. Our motivation was a somewhat related question for prime characteristic rings (mentioned at the end of the note) that we still don't know the answer to. Let us know if you do!
- This note, written with Remy van Dobben de Bruyn, gives two examples which illustrate that formally unramified and flat morphisms need not be formally étale. The note exists so that both of us don't forget that such examples exist. It also provides a reinterpretation of the excellence condition for regular rings of prime characteristic in terms of a certain map being formally étale.
- 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, Tag 02UX]. This example can be interpreted purely topologically (without mentioning schemes), and is incorporated in the latter form in Tag 0BX0.
- 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.