(D. Buchsbaum)

The idea is to note the various ways it has helped to look for or at complexes that may or may not be resolutions of significant modules. Another way to say this is that we're looking at resolutions or complexes with appropriate grade-sensitivity or other useful properties. Among the complexes looked at will be the generalized Koszul complex, resolutions of determinantal ideals and resolutions of Weyl modules.

(R. Hartshorne)

The title describes a fruitful interaction between algebra and geometry, which has been useful recently in studying the notion of Gorenstein liaison.

In this talk I will report on joint work with M.V. Catalisano and A.Gimigliano concerning the dimensions of the Higher Secant Varieties of the Segre Varieties. This area of research is filled with open problems and interesting theorems and I would like to explain some of the work that has been done to try and "clear the debris".

(E. Gover)

We review some results based on Poincare' series formulas that give upper bounds for Betti numbers of finitely generated modules over local rings.

(R. Miró-Roig)

(M. Manaresi)

(S. Nollet)

It is not yet known whether the Hilbert scheme H_d,g of locally Cohen-Macaulay space curves of degree d and genus g is connected. I will discuss the current state of this problem. I'll then describe the irreducible components of H_4,g and explain some of the techniques used to show that these Hilbert schemes are connected, including deformations on the doubling of a smooth surface.

(A. P. Rao)

Generalized quadrics are hypersurfaces whose equations resemble that of a usual quadric hypersurface, except that the regular sequence of linear forms is replaced by a regular sequence of forms of other degrees. Such hypersurfaces have been used in characteristic p constructions of low rank bundles on projective four and five space. We study such hypersurfaces in characteristic zero and show that they must always be reduced. Hence analogous constructions of vector bundles in characteristic zero cannot be done. This is joint work with Mohan Kumar and Ravindra.

(G. Valla)

The notion of Koszulness is a remarkable refinement of the property of a graded algebra to be non degenerate and defined by quadratic relations. The homogeneuos coordinate algebra of a non-degenerate projectively normal variety X is Koszul iff the same is true for a hyperplane section of X. Because of this one can often reduce the question of Koszulness of a homogeneous coordinate algebra of X to that of a finite set of points. Thus, it is important to know which configurations of points are Koszul. More generally one can ask when the defining ideal of X has a Groebnerbases of quadrics in some coordinate system and with respect to some term order.

It is well known that if such a Groebner bases exists, then the algebra is Koszul and this implication is generally strict. I will discuss some results and many problems related to this circle of ideas.