Thomas Polstra

Thomas Polstra

JWB 130
polstra@math.utah.edu
Curriculum Vitae


About me

I am a National Science Foundation postdoctoral fellow at the University of Utah. Before joining the math department at the University of Utah I was a PhD student at the University of Missouri under the advisement of Ian Aberbach. Beginning August 2020 I will be a research associate and lecturer at the University of Virginia.


Research

Commutative algebra is the universal language connecting the subjects of algebraic geometry, homological algebra, and number theory. I am a commutative algebraist. However, I often find myself motivated by the considerations and techniques of algebraic geometry. Algebraic geometry concerns itself with the theory of varieties and schemes and commutative algebra is the local theory of these objects. In other words, commutative algebra is to algebraic geometry as analysis is to differential geometry. A primary goal of my research is to understand, measure, and relate singularities appearing in prime characteristic algebraic geometry and commutative algebra. Motivation to do so stems from algebraic techniques surrounding the theory of tight closure, geometric considerations of birational geometry, and classical results concerning traditional measurements of singularities.


Publications and preprints

  1. Compatible ideals in Gorenstein rings, with Karl Schwede, submitted.
  2. Local cohomology bounds and test ideals, with Ian Aberbach, submitted.
  3. Covers of rational double points in mixed characteristic, with Javier Carvajal-Rojas, Linquan Ma, Kevin Tucker, and Karl Schwede, submitted.
  4. Global F-splitting ratio, with Alessandro De Stefani and Yongwei Yao, submitted.
  5. A theorem about maximal Cohen-Macaulay modules, to appear in Int. Math. Res. Not.
  6. Global Frobenius Betti numbers and Euler characteristic, with Alessandro De Stefani and Yongwei Yao, to appear in Mich. M. Journal.
  7. F-nilpotent rings and permanence properties, with Jennifer Kenkel, Kyle Maddox, and Austyn Simpson, to appear in J. Commutative Algebra.
  8. Equimultiplicity theory of strongly F-regular rings, with Ilya Smirnov, to appear in Mich. M. Journal.
  9. Nilpotence of Frobenius actions on local cohomology and Frobenius closure of ideals, with Pham Hung Quy, J. Algebra 529 (2019), 196--225.
  10. F-signature under birational morphisms, with Linquan Ma, Karl Schwede, and Kevin Tucker, Forum Math. Sigma 7 (2019), e11, 20 pp.
  11. Continuity of Hilbert-Kunz multiplicity and F-signature, with Ilya Smirnov, Nagoya Mathematical Journal, 1-24. doi:10.1017/nmj.2018.43.
  12. Globalizing F-invariants, with Alessandro De Stefani and Yongwei Yao, Adv. Math. 350 (2019), 359--395.
  13. F-signature and Hilbert-Kunz Multipicity: a combined approach and comparison, with Kevin Tucker, Algebra & Number Theory 12-1 (2018), 61--97.
  14. Generalizing Serre's Splitting Theorem and Bass's Cancellation Theorem via free-basic elements, with Alessandro De Stefani and Yongwei Yao, Proc. Amer. Math. Soc. 146 (2018), no. 4, 1417--1430.
  15. Uniform bounds in F-finite rings and lower semi-continuity of the F-signature, Trans. Amer. Math. Soc. 370 (2018), no. 5, 3147--3169.
  16. Depths and Stanley depths of path ideals of spines,, with Daniel Campos, Susan Morey, and Chelsey Paulsen, Involve, 9(1): 155-170, 2016.
  17. Depths and Cohen-Macaulay Properties of Path Ideals, with Daniel Campos, Susan Morey, and Chelsey Paulsen, Journal of Pure and Applied Algebra, 218: 1537-1543, 2014.

Teaching

At the University of Virginia

To be determined.

At the University of Utah

MATH 4800, Undergraduate Research Mathematics, Fall 2019.
MATH 1260, Honors multivariable calculus, Fall 2018.

At the University of Missouri

MATH 8411 (Algebra II), Teaching Assistant, Spring 2016, Spring 2017.
MATH 8410 (Algebra I), Teaching Assistant and Reciatation Instructor, Fall 2016.
MATH 4720 (Introduction to Abstract Algebra) Recitation Instructor, Spring 2015, Fall 2015, Spring 2017.
MATH 4140 (Matrix Theory) Instructor, Summer 2015, Summer 2017.
MATH 3000 (Introduction to Advanced Mathematics) Recitation Instructor, Fall 2016.
MATH 2320 (Discrete Mathematical Structures) Instructor, Spring 2017.
MATH 2300 (Calculus III) Instructor, Summer 2014, Summer 2016.
MATH 1700 (Calculus II) Recitation Instructor, Fall 2015.
MATH 1100 (College Algebra) Instructor, Fall 2013, Fall 2014.
MATH 0110 (Intermediate Algebra) Instructor, Spring 2014.