Homological Mirror Symmetry for Singularities

在奇点理论中，迄今观察到了一些相当神秘的现象：McKay对应，Arnold奇异对偶，Saito对偶，以及先验无关的代数和拓扑不变量之间的令人惊讶的关系，Poincar_e级数和单列的关系。这些现象中的大多数还没有被很好地理解。有一些证据表明，对这些现象的可能解释可能来自一种令人兴奋的数学猜想，它起源于理论物理(弦理论)。这就是同源镜像对称猜想，这就是表象理论发挥作用的地方。该建议的主要目的是调查奇点的同调镜像对称性的含义，该猜想在多大程度上成立，以及人们是否可以通过研究该猜想来更好地理解上述现象。

In singularity theory some up to now rather mysterious phenomena were observed: the McKay correspondence, Arnold's strange duality, Saito's duality, and a surprising relation between a priori unrelated algebraic and topological invariants, Poincar_e series on one hand and monodromy on the other hand. Most of these phenomena are not yet well understood. There is some evidence that a possible explanation for these phenomena might come from an exciting mathematical conjecture which has its origin in theoretical physics (string theory). This is the homological mirror symmetry conjecture and this is where representation theory comes into play. The main goal of the proposal is to investigate what the meaning of homological mirror symmetry for singularities is, to what extent the conjecture holds, and whether one can gain a better understanding of the above mentioned phenomena from the study of this conjecture.