Date & Time: Wednesday, 2nd November, 2011 at 4 p.m.
Venue: Ramanujan Hall
Title: Vector Bundles on Projective Spaces and Modules over Polynomial Rings.
Speaker: Prof. Manoj Kummini Chennai Mathematical Institute
Abstract:For a finitely generated graded module over a polynomial ring its Betti table consists of the ranks of the free modules in a minimal graded free resolution. For a vector bundle on the projective space, its cohomology table consists of the dimensions of its cohomology groups, under all twists.
Instead of considering a single Betti table, we look at the positive rational cone generated by all the Betti tables of finite-length graded modules over a polynomial ring. Similarly, one considers the positive rational cone generated by all the cohomology tables of vector bundles on the corresponding projective space.
D. Eisenbud and F.-O. Schreyer showed that the facets of any of these cones can be obtained from the extremal rays of the other. This result and the conjectures that led to them have given rise to more work exploring the structure of resolutions and cohomology tables.