Cohomological methods in symplectic topology

该项目涉及辛拓扑、镜像对称以及它们之间的关系。在拓扑学方面，我们研究了拉格朗日交的辛上同调及其类似(包裹Floer上同调)。Bourgeis-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，我们考虑了在Mikhalkin意义下辛流形到裤子对的分解。一个正在考虑的问题是，是否可以通过将每条裤子对应的碎片粘合在一起来重建Fukaya类别。一般来说，这不是真的，必须通过瞬子贡献来修正。然而，在某些情况下，这种贡献应该是不存在的，这将为康采维奇的同源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了经典的空间概念。在这里研究的情况下，主要特征是古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。这是一类普通物理理论的简化数学模型，本身并不现实。然而，研究这些模型向我们展示了我们在概念上面临的困难，这对于进一步发展我们的理解是重要的。

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.