Matrix Factorizations and complete Intersection-rings

Kontsevich在1994年提出的同调镜像对称猜想（Homological Mirror Symmetry conjecture），解释了弦理论中观察到的Calabi-Yau三折之间的对偶性，对数学产生了深远的影响。在研究这个猜想的过程中，康采维奇重新发现了交换代数中的一个构造，即矩阵分解。这个结构最初是由Eisenbud在1980年发现的，他用它来描述超表面环上的自由分辨率。它们已经成为极大Cohen-Macaulay模表示理论的标准工具。这种由矩阵分解提供的数学和物理之间的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的问题，以及更一般的完全相交环，这些问题与矩阵分解和上述连接有关。

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.