
Research Interests:
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, 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 important 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.



Hilbert's Syzygy Theorem provides a nice upper bound on the projective dimension of homogeneous ideals
in a standard graded polynomial ring: projective dimension is smaller than the number of variables.
In contrast, there is a doubly exponential upper bound on the CastelnuovoMumford regularity
in terms of the number of variables and the degrees of the minimal generators.
The bound is nearly sharp since the MayrMeyer construction leads to examples of families of ideals
attaining doubly exponential regularity.
It is
expected that much better bounds hold for the defining ideals of geometrically nice projective varieties.
In the smooth case, important bounds were obtained by Mumford (1993), BertramEinLazarsfeld (1991), and ChardinUlrich (2002).
As discussed in an influential paper by BayerMumford (1993), the biggest missing link between the general case and the smooth case is to
obtain a decent bound on the regularity of all reduced equidimensional ideals.
The longstanding Regularity Conjecture, by EisenbudGoto (1984), predicts the following elegant linear bound in terms of the degree: The Regularity Conjecture holds if U/L is CohenMacaulay by a result of EisenbudGoto (1984). It is proved for curves by GrusonLazarsfeldPeskine (1983), completing classical work of Castelnuovo. It is also proved for smooth surfaces by Lazarsfeld (1987) and Pinkham (1986), and for most smooth 3folds by Ran (1990). In the smooth case, Kwak (1998) gave bounds for regularity in dimensions 3 and 4 that are only slightly worse than the optimal ones. Many other special cases and related bounds have been proved as well.
Jason McCullough and I provide
counterexamples to the Regularity Conjecture . Our main theorem is much stronger and shows that the regularity
of nondegenerate homogeneous prime ideals is not bounded by any polynomial function of the degree; this holds over any field k (the case
k= 

Minimal free resolutions over a local complete intersection R have attracted attention ever since the elegant construction of the minimal free resolution of the residue field by Tate in 1957. The next impressive result was Gulliksen's proof in 1974 that the Poincaré series ∑b_{i}(N)t^{i} (where b_{i}(N) are the Betti numbers over R) is rational for every finitely generated Rmodule N, and that the denominator divides (1t^{2})^{c} (where c is the codimension of R). For this purpose, he showed that Ext_{R}(N,k) can be regarded as a finitely generated graded module over a polynomial ring k[χ _{1},...,χ _{c}]. This also implies that the even Betti numbers b_{i}(N) are eventually given by a polynomial in i, and the odd Betti numbers are given by another polynomial. In 1989 Avramov proved that the two polynomials have the same leading coefficient and the same degree. He also identified the dimension of Ext_{R}(N,k) with a correction term in a natural generalization of the AuslanderBuchsbaum formula. In 1997 Avramov, Gasharov and Peeva showed that the truncated Betti sequence {b_{i}(N)}_{i≥ q} is either strictly increasing or constant for q≫ 0 and proved further properties of the Betti numbers. The theory of matrix factorizations was introduced by Eisenbud in 1980 to describe the asymptotic structure of minimal free resolutions over a hypersurface. A matrix factorization of a nonzero element f in a regular local ring S is a pair (d,h) of maps of finitely generated Sfree modules d: A_{1}→ A_{0}, h: A_{0}→ A_{1} such that hd = f. Id_{A1} and dh = f.Id_{A0}. This concept has many other applications: for example, in the study of CohenMacaulay modules and Singularity Theory, Cluster Tilting, Hodge Theory, KhovanovRozansky Homology, Moduli of Curves, Quiver and Group Representations, Singularity Categories. Starting with Kapustin and Li, who followed an idea of Kontsevich, physicists discovered amazing connections with String Theory. Despite all this work on applications, progress on the structure of minimal free resolutions over complete intersections was scant. The condition of minimality is important. The mere existence of free resolutions suffices for foundational issues such as the definition of Ext and Tor, and there are various methods of producing resolutions uniformly (for example, the Bar resolution). But without minimality, resolutions are not unique, and the very uniformity of constructions like the Bar resolution implies that they give little insight into the structure of the modules resolved. We focus on high syzygies, since there are examples, constructed by Eisenbud, of minimal resolutions over complete intersections that have intricate structure, but exhibit stable patterns when sufficiently truncated. As mentioned above, in 1980 he described the minimal free resolutions of high syzygies over a hypersurface. In 2000, Avramov and Buchweitz analyzed the codimension 2 case. But the general case (of higher codimensions) has remained elusive, even though nonminimal resolutions have been known for over 45 years from the work of Shamash.
Eisenbud and I have wondered, for many years, how to describe the eventual patterns in the minimal resolutions of modules over complete intersections of higher codimension. With the theory developed in For this purpose, we introduce a new concept of higher matrix factorization (d,h) with respect to a regular sequence; this extends the theory of matrix factorizations. We obtain the following results: Let S be a regular local ring with infinite residue field k, and let I⊂ S be an ideal generated by a regular sequence of length c. Let N be a finitely generated module over the complete intersection R := S/I, and M be a sufficiently high syzygy of N over R. We prove that there exists a minimal higher matrix factorization (d,h), with respect to a generic choice of generators f_{1}, ..., f_{c} of I, such that M is its higher matrix factorization module Coker(R⊗ d). We construct the minimal free resolution of M over the complete intersection R. We also construct the minimal free resolution of M over the regular local ring S. 
