Office: Math 740

Office Phone: 765 49-41956

I am an Associate Professor of mathematics at Purdue University. I obtained my Ph.D in May 2014 at the University of Michigan under the direction of Melvin Hochster. My research interests are commutative algebra and algebraic geometry. Here is my CV.

- Test ideals in mixed characteristic: a unified theory up to perturbation (with Bhargav Bhatt, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron and Jakub Witaszek).

- Colength, multiplicity, and ideal closure operations II (with Pham Hung Quy and Ilya Smirnov).

- Vanishing and non-negativity of the first normal Hilbert coefficient (with Pham Hung Quy).

- Perfectoid signature, perfectoid Hilbert-Kunz multiplicity, and an application to local fundamental groups (with Hanlin Cai, Seungsu Lee, Karl Schwede and Kevin Tucker). (Perfectoid signature and applications, Oberwolfach Report 6/2023, 338-339.)

- Regularity, singularities and h-vector of graded algebras (with Hailong Dao and Matteo Varbaro),
*Trans. Amer. Math. Soc.*, to appear.

- F-singularities: a commutative algebra approach (with Thomas Polstra), last updated 2023.

- Lim Ulrich sequences and Boij-Soderberg cones (with Srikanth Iyengar and Mark Walker),
*Forum Math. Sigma*11 (2023), Paper No. e115.

- Uniform Lech's inequality (with Ilya Smirnov),
*Proc. Amer. Math. Soc.*151 (2023), no. 6, 2387-2397.

- Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic (with Bhargav Bhatt, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron and Jakub Witaszek),
*Publ. Math. Inst. Hautes Etudes Sci.*, 138 (2023), 69-227.

- Lim Ulrich sequences and Lech's conjecture,
*Invent. Math.*231 (2023), no. 1, 407-429.

- Symbolic power containments in singular rings in positive characteristic (with Eloisa Grifo and Karl Schwede),
*Manuscripta Math.*170 (2023), no. 3-4, 471-496.

- Lim Cohen-Macaulay sequences of modules (with Bhargav Bhatt and Melvin Hochster), last updated 2022.

- A Buchsbaum theory for tight closure (with Pham Hung Quy),
*Trans. Amer. Math. Soc.*375 (2022), no. 11, 8257-8276.

- Multiplicities and Betti numbers in local algebra via lim Ulrich points (with Srikanth Iyengar and Mark Walker),
*Algebra Number Theory*16 (2022), no. 5, 1213-1257.

- F-stable secondary representations and deformation of F-injectivity (with Alessandro De Stefani),
*Acta Math. Vietnam.*47 (2022), no. 1, 141-150 (special issue dedicated to Nguyen Tu Cuong).

- An analog of adjoint ideals and PLT singularities in mixed characteristic (with Karl Schwede, Kevin Tucker, Joe Waldron and Jakub Witaszek),
*J. Algebraic Geom.*31 (2022), no. 3, 497-559.

- A short proof of the direct summand theorem via the flatness lemma, unpublished note 2021.

- Maximal Cohen-Macaulay complexes and their uses: a partial survey (with Srikanth Iyengar, Karl Schwede and Mark Walker),
*Commutative Algebra*, 475-500, Springer, Cham, 2021 (dedicated to David Eisenbud).

- Koszul and local cohomology, and a question of Dutta (with Anurag Singh and Uli Walther),
*Math. Z.*298 (2021), no. 1-2, 697-711.

- Covers of rational double points in mixed characteristic (with Javier Carvajal-Rojas, Thomas Polstra, Karl Schwede and Kevin Tucker),
*J. Singul.*23 (2021), 127-150.

- Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras (with Karl Schwede),
*Duke Math. J.*170 (2021), no. 13, 2815-2890.

- Cohomologically full rings (with Hailong Dao and Alessandro De Stefani),
*Int. Math. Res. Not. IMRN*2021, no. 17, 13508-13545.

- Extended plus closure in complete local rings (with Raymond Heitmann),
*J. Algebra*571 (2021), 134-150 (special issue dedicated to Craig Huneke).

- Asymptotic Lech's inequality (with Craig Huneke, Pham Hung Quy and Ilya Smirnov),
*Adv. Math.*372 (2020), 107296, 33 pp.

- Filter regular sequence under small perturbations (with Pham Hung Quy and Ilya Smirnov),
*Math. Ann.*378 (2020), no. 1-2, 243-254.

- Colength, multiplicity, and ideal closure operations (with Pham Hung Quy and Ilya Smirnov),
*Comm. Algebra*48 (2020), no. 4, 1601-1607.

- A Kunz-type characterization of regular rings via alterations (with Karl Schwede),
*J. Pure Appl. Algebra*224 (2020), no. 3, 1124-1131.

- Recent applications of p-adic methods to commutative algebra (with Karl Schwede),
*Notices Amer. Math. Soc.*66 (2019), no. 6, 820-831.

- F-signature under birational morphisms (with Thomas Polstra, Karl Schwede and Kevin Tucker),
*Forum Math. Sigma*7 (2019), e11, 20 pp.

- Lech's inequality, the Stuckrad-Vogel conjecture, and uniform behavior of Koszul homology (with Patricia Klein, Pham Hung Quy, Ilya Smirnov and Yongwei Yao),
*Adv. Math.*, 347 (2019), 442-472. (Lech's inequality and the Stuckrad-Vogel conjecture, Oberwolfach Report 6/2019, 428-429.)

- Regular rings and perfect(oid) algebras (with Bhargav Bhatt and Srikanth Iyengar),
*Comm. Algebra*47 (2019), no. 6, 2367-2383 (special issue dedicated to Gennady Lyubeznik).

- Maximal Cohen-Macaulay modules over certain Segre products,
*Comm. Algebra*47 (2019), no. 6, 2488-2493 (special issue dedicated to Gennady Lyubeznik).

- Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers (with Karl Schwede),
*Invent. Math.*214 (2018), no. 2, 913-955.

- Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic (with Raymond Heitmann),
*Algebra Number Theory*12 (2018), no. 7, 1659-1674.

- D-module and F-module length of local cohomology modules (with Mordechai Katzman, Ilya Smirnov and Wenliang Zhang),
*Trans. Amer. Math. Soc.*370 (2018), no. 12, 8551-8580.

- F-singularities under generic linkage (with Janet Page, Rebecca R.G., William Taylor and Wenliang Zhang),
*J. Algebra*505 (2018), 194-210.

- Frobenius actions on local cohomology modules and deformation (with Pham Hung Quy),
*Nagoya Math. J.*232 (2018), 55-75.

- The dualizing complex of F-injective and Du Bois singularities (with Bhargav Bhatt and Karl Schwede),
*Math. Z.*288 (2018), no. 3-4, 1143-1155.

- The vanishing conjecture for maps of Tor and derived splinters,
*J. Eur. Math. Soc. (JEMS)*20 (2018), no. 2, 315-338.

- Lech's conjecture in dimension three,
*Adv. Math.*322 (2017), 940-970. (Lech's conjecture, Oberwolfach Report 57/2016, 3214-3216.)

- Local cohomology of Du Bois singularities and applications to families (with Karl Schwede and Kazuma Shimomoto),
*Compos. Math.*153 (2017), no. 10, 2147-2170.

- F-injectivity and Buchsbaum singularities,
*Math. Ann.*362 (2015), no. 1-2, 25-42.

- Eulerian graded D-modules (with Wenliang Zhang),
*Math. Res. Lett.*21 (2014), no. 1, 149-167.

- A sufficient condition for F-purity,
*J. Pure Appl. Algebra*218 (2014), 1179-1183.

- The category of F-modules has finite global dimension,
*J. Algebra*402 (2014), 1-20.

- Finiteness properties of local cohomology for F-pure local rings,
*Int. Math. Res. Not. IMRN*2014, no. 20, 5489-5509.

- The Frobenius endomorphism and multiplicities, Thesis, University of Michigan (2014).

- Math 37500 Introduction to Discrete Mathematics (Spring 2023).
- Math 69000 Topics in Commutative Algebra: Complete intersections (Fall 2022).
- Math 35100 Elementary Linear Algebra (Spring 2022).
- Math 45000 Algebra Honors (Fall 2021, Fall 2020, Fall 2019).
- Math 69000 Topics in Commutative Algebra: Positive characteristic methods (Spring 2021).
- Math 69000 Topics in Commutative Algebra: Homological conjectures/theorems and perfectoid spaces (Spring 2020).
- Math 26500 Linear Algebra (Fall 2018, Spring 2015, Fall 2014).

- Math 7830 Topics in Commutative Algebra: Perfectoid spaces and the homological conjectures (Spring 2018).
- Math 2210 Calculus 3 (Fall 2017, Spring 2016).
- Math 2200 Introduction to Discrete Mathematics (Spring 2017, Fall 2016).
- Math 2250 Differential Equations and Linear Algebra (Fall 2015).

- TA for the course on Fibonacci numbers for Michigan Math and Science Scholars (Summer 2014).
- TA for Math 215 Calculus 3 (Spring 2014, Fall 2013, Fall 2012, Spring 2012).

- Math 116 Calculus 2 (Spring 2011).

- Math 115 Calculus 1 (Spring 2010, Fall 2009).

- Math 105 Data, Functions & Graphs (Spring 2009, Fall 2008).

- Ph.D's Life, with Jingchen Wu, English Crosstalk.

- Go Back to The Past, with Qi Gong, Chinese Crosstalk.