Mats Boij: Decompositions of Betti diagrams and parameter spaces of modules

Two years ago, Eisenbud and Schreyer proved a set of conjectures on the set of Betti diagrams of graded Cohen-Macaulay modules up to scaling stated in 2006 by me and Jonas Söderberg. The main idea is that Betti diagrams can be decomposed into sums of pure diagrams, with possibly rational entries, in a very precise manner. Since then, there has been further progress in the area and recently Eisenbud, Erman and Schreyer came with the first results indicating what the numerical decompositions of Betti diagrams correspond to in terms of structure of the minimal free resolutions. I have myself been studying parameter spaces of modules using the known structure of the cone of Betti diagrams. I will give an introduction to the area and report on some of
the new results.