Summer Graduate School

Let S be a polynomial ring over any field k, and let P in S be a non-degenerate homogeneous prime ideal of height h. When k is algebraically closed, a classical result attributed to Castelnuovo establishes an upper bound on the number of linearly independent quadric contained in P which only depends on h. We significantly extend this result by proving that the number of minimal generators of P in any degree j can be bounded above by an explicit function that only depends on j and h. In addition to providing a bound for generators in any degree j, not just for quadrics, our techniques allow us to drop the assumption that k is algebraically closed. By means of standard techniques, we also obtain analogous upper bounds on higher graded Betti numbers of any radical ideal.

This is a joint work with Alessandro De Stefani.

Multiplicities of algebras have their roots in intersection theory in the first half of the 20th century with the work of several authors including Hilbert, Noether, and Van der Waerden. In the 1950s developments by Samuel, Serre, and Rees brought multiplicities to the interest of commutative algebraists. Since then multiplicities has become a very active research theme in commutative algebra with connections with other fields such as algebraic combinatorics, convex geometry, and singularity theory. In this talk I will survey the history and some results in this topic, and will include some open problems along the way.

This will be a survey about the theory of (complete intersection) linkage and residual intersection. After reviewing results about linkage, we will focus on the more general notion of residual intersection. Residual intersections arise naturally in intersection theory, enumerative geometry, the study of Rees rings and integral closures, for instance. Of interest are the dimension and depth, the canonical module, and the defining equations of residual intersections.

Differential graded (dg) algebra structures on complexes, and more particularly on free resolutions, have played a key role in commutative algebra. Some of the history will be mentioned, from its origins in algebraic topology to the motivations for having such structures on minimal free resolutions. In addition, we will discuss some general constructions producing (mostly non-minimal) resolutions with dg algebra structures. Time permitting, we may very briefly mention the related roles of A-infinity structures and Massey operations.

As discussed in Anurag's lectures, the ring of differential operators over a polynomial ring has many nice homological and representation-theoretic properties; these yield striking applications to finiteness properties of local cohomology modules. More generally, one can associate to any K-algebra a ring of differential operators. Outside of the polynomial case, rings of differential operators are much less well understood, and most of the results from the polynomial case fail in general. In this talk we will survey some results, both positive and negative, on the extent to which differential operators over singular rings behave as in the regular case.

Suppose R=KK[x_1,...,x_n] is a polynomial ring over a field KK of characteristic zero. If I is an ideal in R, then the local cohomology modules H^j_I(R) can be computed (at least in principle) in Macaulay2. In this talk we discuss the basic ideas that go into this (Groebner basis driven) algorithm. This involves a tour through the land of D-modules, with specific focus on the concept of Bernstein—Sato polynomials attached to elements f of R (which we will define and discuss).