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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, and I remain very interested in these fields as well.
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, to appear.
- 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.