My primary work is in Commutative Algebra, and my primary research is focused on Free Resolutions and Hilbert Functions. I have also done work on the many connections
of Commutative Algebra with Algebraic Geometry, Combinatorics, Computational Algebra, Noncommutative Algebra, and Subspace Arrangements.
The study of free resolutions and Hilbert functions is a beautiful and core area in Commutative Algebra. It contains a number of challenging conjectures and open problems. The idea to associate a free resolution to a module was introduced by Hilbert in his famous paper "Über the Theorie von algebraischen Formen." Resolutions provide a method for describing the structure of modules.
- Counterexamples to the Eisenbud-Goto regularity conjecture, (with J. McCullough), Journal of the AMS 31 (2018), 473–496.
- Tor as a module over an exterior algebra, (with D. Eisenbud and F.-O. Schreyer), Journal of the EMS, 21 (2019), 873–896.
- Minimal free resolutions over complete intersections, (with D. Eisenbud), research monograph, Lecture Notes in Mathematics 2152, Springer, 2016.
- Hilbert schemes and Betti numbers over Clements-Lindström rings, (with S. Murai), Compositio Mathematica 148 (2012), 1337–1364.
- Graded Syzygies, 312 pages, Springer, London, 2011.
- Flips and Hilbert schemes over exterior algebras, (with M. Stillman), Mathematische Annalen 339 (2007), 545-557.
- Connectedness of Hilbert schemes, (with M. Stillman), Journal of Algebraic Geometry 14 (2005), 193–211.
- Finite regularity and Koszul algebras, (with L. Avramov), American Journal of Mathematics 123 (2001), 275–281.
- The lcm-lattice in monomial resolutions, (with V. Gasharov and V. Welker), Mathematical Research Letters 6 (1999), 521–532.
- Generic lattice ideals, (with B. Sturmfels), Journal of the AMS 11 (1998), 363–373.
- Complete intersection dimension, (with L. Avramov and V. Gasharov), Publications Mathématiques IHÉS 86 (1997), 67–114.
In the news
- Cornell mathematicians featured at International Congress of Mathematicians
- Two mathematics professors honored with 2019 Simons Fellowships