Syzygies and Hilbert Functions (06w5082)

Objectives

For many years, Hilbert functions and Resolutions have been both central objects and fruitful tools in many fields, including Algebraic Geometry, Combinatorics, Commutative Algebra, and Computational Algebra. There has been a surge in interest and research in this direction during the recent years: a variety of new ideas and techniques were introduced, and substantial progress was made. The proposed workshop will give the participants an excellent platform for disseminating their results and will give them a chance to absorb new ideas and techniques. We will invite leading researchers to present a spectrum of ``hot topics".

Notably, many central conjectures and open problems on Resolutions and Hilbert functions are very challenging and remain wide open. We plan to have discussion sessions and one-hour overview talks in the mornings. During the workshop, we would like to organize discussion sessions on the following important and challenging conjectures:
1) the Eisenbud-Green-Harris Conjecture on Hilbert functions;
2) the Eisenbud-Goto Regularity Conjecture;
3) the Herzog-Huneke-Srinivasan Conjecture on bounds for the multiplicity.

The workshop will focus not only on results in Commutative Algebra but also on:
1) relations to Algebraic Geometry, especially bounds on the Castelnuovo-Mumford regularity, Hilbert schemes, and syzygies of toric varieties;
2) relations to Combinatorics, especially monomial resolutions, shellings and Koszul rings;
3) relations to Computational Algebra;
4) relations to Hyperplane Arrangements, especially connections between the topological properties (cohomology and homotopy) of complements of subspace arrangements and certain syzygies.

There are exciting recent developments in the above topics. A workshop in this area would be very useful at the present time since it will provide an opportunity to foster further research and explore in new directions.

The main goals of the workshop are: to spark further interaction, to have expository talks on `` hot topics", to have discussion sessions on challenging conjectures, to spread new problems, and to gather researchers in order to announce the latest developments.