Hilbert Functions and Betti Numbers

Ever since Hilbert introduced Hilbert functions in 1890, they have been intensely studied. Numerous problems in various areas of mathematics can be reinterpreted as questions about Hilbert functions. Closely tied to Hilbert functions, but conveying much more information, is the notion of minimal free resolutions and graded Betti numbers. The main objects of study in this proposal are Gorenstein algebras and monomial ideals. These are in the mainstream of current research in Commutative Algebra and its ties with Algebraic Geometry and Combinatorics - they have been used as important tools, and they are of great interest in and of themselves. This proposal builds on recent results of the PI's and on their track record of collaboration. It contains new approaches to very old and difficult problems in this area, as well as new problems and perspectives, such as the connection with syzygy bundles. Although these topics are related, the questions and applications presented here are broad in nature, and we present a variety of approaches to their solution. Elements of algebra, geometry and combinatorics play important roles. The first part of the proposal concerns the possible Hilbert functions of Gorenstein algebras. This 1S an intractable problem with current methods, so the PI's propose to study important special cases. Notable among these is when the algebra is presented by quadrics. A special case of this was studied by Eisenbud, Green and Harris. and another is a famous conjecture of Charney and Davis. Several well-known conjectures indicate that the Hilbert functions ofreduced Gorenstein algebras admit a nice characterization. Stanley conjectured that the h-vector of a Gorenstein domain is an SI-sequence. Similarly, the so-called g-Conjecture predicts that the Stanley-Reisner rings of simplicial spheres have SI-sequences as h-vectors. Settling this conjecture is a central problem in Convex Geometry. An affirmative answer would greatly extend the famous g-Theorem of Billera, Lee, and Stanley that characterizes the f-vectors of simplicial polytopes. The PI's propose an approach via the Weak Lefschetz Property. The second part of the proposal studies this latter property more deeply, and in particular it investigates the connection with the semistability of syzygy bundles. These bundles are obtained from the minimal free resolution, and have already been shown to be important for the study of the Weak Lefschetz Property. The PI's propose independent ways of studying the Weak Lefschetz Property, and consequently to investigate whether these will have implications for the semistability of these bundles. The last part of the proposal concerns the free resolutions themselves, and in particular asks whether there are reasonable lower bounds for the graded Betti numbers. It is proposed to study a very recent conjecture that states sharp lower bounds and identifies a candidate ideal that simultaneously attains the bounds. The methods proposed are combinatorial in nature.