Cohomological methods in symplectic topology

该项目涉及辛拓扑、镜像对称及其关系。在拓扑方面，我们研究了拉格朗日交点的辛上同调及其类比（包花上同调）。Bourgeois-Ekholm-Eliashberg和作者的最新进展提供了强大的计算工具，我们打算利用这些工具来研究开流形上辛结构的非唯一性问题。在镜像对称方面，我们考虑将辛流形分解成Mikhalkin意义上的裤子。正在考虑的一个问题是，是否可以通过将每条裤子对应的碎片粘合在一起来重建深谷类别。一般来说，这是不正确的，必须通过即时贡献来修改。然而，在某些情况下，这种贡献应该是不存在的，这将给Kontsevich的同调镜像对称猜想提供一个新的观点。从更广泛的角度来看，当前许多物理和数学研究中的一个关键问题是从量子描述中出现的经典空间概念。在本文所研究的情况下，其主要特征是经典空间的构建涉及一系列渐进的扭曲，这些扭曲可以通过复杂但明确的公式来确定。这是一般物理理论的简化数学模型，并不现实。然而，研究这些模型向我们展示了我们面临的概念困难，这对于进一步发展我们的理解是很重要的。

The project concerns symplectic topology, mirror symmetry, and their relationship. On the topological side, we study symplectic cohomology and its analogue for Lagrangian intersections (wrapped Floer cohomology). Recent advances of Bourgeois-Ekholm-Eliashberg and the author yield powerful computational tools, which we intend to exploit in order to study non-uniqueness questions for symplectic structures on open manifolds. On the mirror symmetry side, we consider decompositions of symplectic manifolds into pairs-of-pants, in the sense of Mikhalkin. One question under consideration is whether the Fukaya category can be reconstructed by gluing together pieces corresponding to each pair-of-pants. In general, this is not true and has to be modified by instanton contributions. However, there are cases where such contributions should be absent, and this would give a new viewpoint on Kontsevich's homological mirror symmetry conjecture.From a broader perspective, a crucial issue in many current investigations in physics and mathematics is the emergence of the classical notion of space from a quantum description. In the situation under study here, the main feature is that the construction of the classical space involves a series of gradual distortions, which can be determined by complicated but explicit formulae. This is a simplified mathematical model of a general class of physical theories, and not realistic as such. However, studying such models shows us where we are facing conceptual difficulties, which is important in order to develop our understanding further.