Matrix Factorizations and complete Intersection-rings

同调镜像对称猜想，由孔采维奇在1994年提出，作为弦理论中观察到的卡-丘三重对偶的解释，对数学产生了深远的影响。在他的工作过程中，这一猜想Kontsevich重新发现了建设交换代数称为矩阵分解。这个结构最初是由Eisenbud在1980年发现的，他用它来描述超曲面环的自由分辨率。它们从此成为极大Cohen-Macaulay模的表示论中的标准工具。矩阵分解所提供的数学和物理之间的这种联系才刚刚开始被探索。在这个项目中，我们将研究超曲面环上的几个不同问题，更一般地说，是与矩阵分解和上述连接有关的完全相交环。

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.