Conference on 100 Years of Noetherian Rings

Abstract:

Motivated by applications in Invariant Theory, Hilbert introduced an approach to describe the structure of modules by free resolutions. Hilbert's Syzygy Theorem shows that minimal free resolutions over a polynomial ring are finite. Most minimal free resolutions over quotient rings are infinite. We will discuss the properties of such resolutions. The Serre-Kaplansky Problem was a central question in Commutative Algebra for many years; it asked whether the generating function of the minimal free resolution of the residue field of a commutative noetherian local ring is always rational. Anick constructed the first example  providing a negative answer and a strong indication that the structure of these resolutions can be quite complex.

The concept of matrix factorization was introduced by Eisenbud 35 years ago, and it describes completely the asymptotic structure of minimal free resolutions over a hypersurface. Matrix factorizations have applications in many fields of mathematics: for the study of cluster algebras, Cohen-Macaulay modules, knot theory, moduli of curves,  quiver and group representations, and singularity theory.  Starting with Kapustin and Li, physicists discovered  amazing connections with string theory. The talk will focus on the concept of matrix factorization for complete intersection rings, which we introduced and showed that it suffices to describe the asymptotic structure of minimal free resolutions over complete intersections.