Title "Betti numbers of Graded Moduls and Cohomology tables of Coherent Sheaves" Abstract: It can be very difficult to analyze for a given system of polynomials equations qualitative properties, such as the geometry of the corresponding variety. The theory of syzygies offers a tool for looking at systems of equations, which might help to make their subtle properties visible. In a recent paper Boij and S\"oderberg introduced a series of conjectures, which characterize all possible syzygy numbers of graded modules over the polynomial ring up to rational multiples. In the talk, I report on joined work with David Eisenbud, which proves these conjectures and an analogous statements for cohomology tables of coherent sheaves.