In studying a vector bundle E on projective space, one often reasons about the dimensions of the cohomology groups of various twists of E, information that is often collectively called the "cohomology table" of E. Similarly, in reasoning
about a graded module over a polynomial ring, one is often interested in the number of generators of each degree of the syzygy modules -- the "Betti table" of the module.
Mats Boij and Jonas Soederberg recently proposed a conjectural description of the rational cone of Betti tables of Cohen-Macaulay modules. Frank Schreyer and I have proven their conjecture, and along the way discovered a description of the cone of cohomology tables of vector bundles. A remarkable feature of the picture that emerges is that these two cones are "almost" dual to one another. I will explain this circle of ideas and some applications.
I'm especially glad to be able to speak about this at Mark Green's Birthday celebration in view of the enormouos impetus his work has give to the study of syzygies in algebraic geometry.
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