Topics in commutative algebra

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BETTI NUMBERS STANLEY REISNER IDEALSAny graded, finitely generated module affords a minimal graded resolution, and these resolutions provide a wealth of information about the original module. One such piece of information are the Betti numbers of the module; these are the degrees of the free generators in a minimal free resolution.One of the most exciting recent developments in commutative algebra is the emergence of the Boij-Soderberg Theory which describes the convex cone generated by all the Betti numbers of Cohen-Macaulay modules over a polynomial ring, in particular it describes the extremal rays of this cone and ways to express any given betti diagram as a convex combination of points on these rays.David Carey's project studies the subcone generated by Betti diagrams of square free monomial ideals in general, and edge ideals in particular. The aim of the project is to exploit the combinatorial structure of these ideals to gain an understanding of the betti cones, e.g., the classification of extremal rays in the Betti cones of edge ideals in terms of the properties of the underlying graphs.