Free resolutions give an interesting invariant of a variety embedded in projective space called the ``Betti table''; for example, Green's conjecture relates a basic property of a curve -- essentially the minimal degree of a map from the curve to $P^1$ -- to the Betti table of the curve's canonical model. However, it seems in general extremely difficult to say what values this invariant can take on.

A remarkable conjecture of Boij and Soederberg, now proven (and much generalized), gives remarkable insight into the possibilities, and has led to a new way of thinking about free resolutions. The proof of the conjecture involved relating free resolutions to vector bundles on projective space in a novel way that also yields strong information about the possible cohomology tables of such vector bundles.

I'll describe this connection, which is joint work of mine with Frank-Olaf Schreyer, and some of the related results in this rapidly developing area.

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