Castelnuovo-Mumford regularity and the Eisenbud-Goto Conjecture 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 Castelnuovo-Mumford regularity in terms of the number of variables and the degrees of the minimal generators. The bound is nearly sharp since the Mayr-Meyer 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), Bertram-Ein-Lazarsfeld (1991), and Chardin-Ulrich (2002). As discussed in an influential paper by Bayer-Mumford (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 Eisenbud-Goto (1984), predicts the following elegant linear bound in terms of the degree:                 reg(L)  ≤   deg(U/L) -  codim (L) + 1 for every homogeneous non-degenerate prime ideal L in a standard graded polynomial ring U over an algebraically closed field. The Regularity Conjecture holds if U/L is Cohen-Macaulay by a result of Eisenbud-Goto (1984). It is proved for curves by Gruson-Lazarsfeld-Peskine (1983), completing classical work of Castelnuovo. It is also proved for smooth surfaces by Lazarsfeld (1987) and Pinkham (1986), and for most smooth 3-folds 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 non-degenerate homogeneous prime ideals is not bounded by any polynomial function of the degree; this holds over any field k (the case k=C is particularly important). We provide a family of prime ideals Pr, depending on a parameter r, whose multiplicity is singly exponential in r and whose regularity is doubly exponential in r. For this purpose, we introduce an approach, which starting from a homogeneous ideal I, produces a prime ideal P whose projective dimension, regularity, degree, dimension, depth, and codimension are expressed in terms of numerical invariants of I. The construction involves two new concepts:       (1) Rees-like algebras which, unlike the standard Rees algebras, have well-structured defining equations and minimal free resolutions;       (2) a new homogenization technique for prime ideals which, unlike classical homogenization, preserves graded Betti numbers.

	Free Resolutions over Complete Intersections 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 ∑bi(N)ti (where bi(N) are the Betti numbers over R) is rational for every finitely generated R-module N, and that the denominator divides (1-t2)c (where c is the codimension of R). For this purpose, he showed that ExtR(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 bi(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 ExtR(N,k) with a correction term in a natural generalization of the Auslander-Buchsbaum formula. In 1997 Avramov, Gasharov and Peeva showed that the truncated Betti sequence {bi(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 non-zero element f in a regular local ring S is a pair (d,h) of maps of finitely generated S-free modules d: A1→ A0, h: A0→ A1 such that hd = f. IdA1 and dh = f. IdA0. This concept has many other applications: for example, in the study of Cohen-Macaulay modules and Singularity Theory, Cluster Tilting, Hodge Theory, Khovanov-Rozansky 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 non-minimal 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         the research monograph   Minimal Free Resolutions over Complete Intersections         and the paper   Layered resolutions of Cohen-Macaulay modules we believe we have found an answer. 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 f1, ..., fc 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. Matrix factorizations provide a characterization of maximal Cohen-Macaulay modules over a hypersurface. Our new concept of higher matrix factorization provides a characterization of maximal Cohen-Macaulay modules over a complete intersection.