Matrix Factorizations and complete Intersection-rings

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The Homological Mirror Symmetry conjecture, proposed by Kontsevich in 1994 as an explanation for a duality between Calabi-Yau three-folds observed in string theory, has had a profound impact on mathematics. In the course of his work on this conjecture Kontsevich rediscovered a construction in commutative algebra known as a matrix factorization. This construction was originally discovered by Eisenbud in 1980 and he used it to describe free resolutions over hypersurface rings. They have since been a standard tool in the representation theory of maximal Cohen-Macaulay modules. This connection between mathematics and physics afforded by matrix factorizations is just beginning to be explored. In this project we would work on several different problems over hypersurface rings, and more generally complete intersection rings, that are related to matrix factorizations and the above connection.