Teaching (Fall 2023)
Math 417: Group Theory, TR 11:00-12:15 in AvH 118
Research
My research interests lie in the field of commutative algebra, a branch of mathematics focused on the study of algebraic equations and their solutions. The field has its origins in number theory, algebraic geometry, and invariant theory.
Below are some of my publications:
- Hilbert coefficients and the depths of associated graded rings, joint with Sam Huckaba, Journal of the London Math. Soc (2) 56 (1997) 64-76.
- Cofinite modules and local cohomology, joint with Donatella Delfino, Journal of Pure and Applied Algebra 121 (1997) 45-52.
- On associated graded rings of normal ideals, joint with Sam Huckaba, Journal of Algebra 222 (1999), 146-163.
- The associated primes of local cohomology modules over rings of small dimension, Manuscripta Mathematica 104 (2001), 519-525.
- Cofiniteness and associated primes of local cohomology modules, joint with Janet Vassilev, Journal of Algebra 256 (2002), 180-193.
- Local cohomology modules with infinite dimensional socles, joint with Janet Vassilev, Proc. of the American Math. Soc 132 (2004), 3485-3490.
- Non-Noetherian Cohen-Macaulay rings, joint with Tracy Hamilton, Journal of Algebra 307 (2007), 343-360.
- On rings for which finitely generated ideals have only finitely many minimal components, Communications in Algebra 35 (2007), no. 5, 1757-1760.
- Gorenstein rings and irreducible parameter ideals, joint with Mark Rogers and Hideto Sakurai, Proc. Amer. Math. Soc. 136, no. 1 (2008), 49-53.
- On the support of local cohomology, joint with Craig Huneke and Daniel Katz, Journal of Algebra 322 (2009), no. 9, 3194-3211.
- The Auslander-Bridger formula and the Gorenstein property for coherent rings, joint with Livia (Miller) Hummel, Journal of Commutative Algebra 1 (2009), no. 2, 283-314.
- The Frobenius functor and injective modules, Proceedings of the American Math. Soc. 142 (2014), no. 6, 1911-1923.
- The acyclicity of the Frobenius functor for modules of finite flat dimension, joint with Marcus Webb, J. Pure Appl. Algebra 220 (2016), 2886-2896.
- A change of rings result for Matlis reflexivity, joint with Douglas Dailey, Proc. American Math. Soc. 145 (2017), no. 5, 1837-1841.
- Rigidity of Ext and Tor with coefficients in residue fields of a commutative noetherian ring, joint with Lars Christensen and Sri Iyengar, Proceedings of the Edinburgh Math. Soc. 62 (2019), 305-321.
- Detecting finite flat dimension of modules via iterates of the Frobenius endomorphism, joint with Douglas Dailey and Sri Iyengar, Journal of Commutative Algebra 12 (2020), no. 1, 71-76..
- Characterizing Gorenstein rings using contracting endomorphisms, joint with Brittney Falahola, Journal of Algebra 571 (2021), 168-178.
- Frobenius and homological dimensions of complexes, joint with Taran Funk, Collectanea Mathematica 71 (2020), 287-297.
- Level and Gorenstein dimension, joint with Laila Awadalla, Journal of Algebra 609 (2022), 606--618.
Slides from recent talks:
- Frobenius and injective modules, Riverside AMS meeting, November 2013.
- Cohen-Macaulay dimension for coherent rings, Albuquerque AMS meeting, April 2014.
- Frobenius and modules of finite flat dimension, San Antonio JMM meeting, January 2015.
- The beauty, mystery, and utility of prime numbers, talk given to high school students at the University of New Mexico, February 2015.
- The Binomial theorem without middle terms: putting prime numbers to work in algebra, Missouri MAA meeting in St. Joseph, MO, April 2016.
- Matlis reflexivity and change of rings, Charleston AMS meeting, March 2017.
- Characterizing Gorenstein rings using the Frobenius endomorphism, Denton AMS meeting, September 2017.
- Frobenius and homological dimensions, Honolulu AMS meeting, March 2019.
- Thick subcategories and Gorenstein projective modules, Cincinnati AMS meeting, April 2023.
Miscellaneous:
- Hilbert functions of ideals in Cohen-Macaulay rings, Ph.D. Thesis, Purdue University, 1989.
- Graded rings and modules. These are some notes based on a five-week course I taught in the summer of 1993.
- Introduction to local cohomology. These are notes from a 1999 summer course. The notes were recently put into LaTeX by Laura Lynch.
- Math 901/902. These are notes created by Robert Huben on the Math 901-902 course I taught in 2015-16.
- The Homological Conjectures. These are notes on a course given in the spring semester, 2009. Laura put these notes into LaTeX as well.
- A theorem of Hochster and Huneke concerning tight closure and Hilbert-Kunz multiplicity, by Lori McDonnell. This gives a mostly self-contained proof of Hochster and Huneke's theorem characterizing Hilbert-Kunz multiplicity in terms of tight closure.
- Gruson's Theorem, by Brian Johnson. This gives a self-contained proof of Gruson's Theorem, following an argument of Vasconcelos.
Ph.D. Students
- Victoria Sapko, Ph.D. 2001
- Ed Loeb, Ph.D. 2006
- Livia Hummel, Ph.D. 2008
- Laura Lynch, Ph.D. 2011
- Lori McCune, Ph.D. 2011
- Brian Johnson, Ph.D. 2012
- Marcus Webb, Ph.D. 2015
- Doug Dailey, Ph.D. 2016
- Becky Egg, Ph.D. 2016
- Brittney Falahola, Ph.D. 2017
- Taran Funk, Ph.D., 2021
- Laila Awadalla, Ph.D. 2022
- Zach Nason, current student
- Shalom Echalaz, current student