Floer Theory in examples of interest to Mirror Symmetry

AbstractAward：DMS 1108397，首席研究员：加勒特阿尔斯通镜像对称是数学的一个令人兴奋的分支，它与卡-丘流形的辛几何和代数几何之间的对偶性有关。有两个互补的猜想来解释这种对偶性--Kontsevich的同调镜像对称猜想（HMS）和Strominger-Yau-Zaslow猜想（SYZ）。主要研究者的目的是通过研究明确的例子来研究镜像对称性。首先，将对五次三重流形的镜像进行研究。PI将搜索镜像流形上的对象，这些对象表现出类似于五次曲线上某类拉格朗日子流形的行为。这将为HMS提供直接和具体的说明。结合这项工作，研究的Floer上同调的拉格朗日K3曲面将进行。这项工作将利用显式拉格朗日环面纤维化结构的格罗斯，Castano-Bernard，Matessi和其他人，这反过来又受到了SYZ猜想的启发。代数几何和辛几何是深刻而重要的领域，其起源可以追溯到数百年前。从这些领域产生的问题和想法启发了许多数学家和物理学家，并导致了伟大的科学进步。镜像对称有望增加这一遗产。镜像对称起源于弦理论，如今已成为物理学的一个重要分支。对数学家来说，镜像对称是一个令人兴奋和诱人的主题，因为它暗示了代数几何和辛几何之间的联系。往往正是这些类型的联系--不同学科之间的联系--导致突破。这个项目的目标是通过应用新开发的理论和技术来发现代数和辛几何之间的联系，这些理论和技术对镜像对称具有重要意义。

AbstractAward: DMS 1108397, Principal Investigator: Garrett AlstonMirror symmetry is an exciting branch of mathematics that is concerned with duality between the symplectic and algebraic geometry of Calabi-Yau manifolds. There are two complementary conjectures that purport to explain this duality-Kontsevich's Homological Mirror Symmetry conjecture (HMS) and the Strominger-Yau-Zaslow conjecture (SYZ). The principal investigator aims to study mirror symmetry in the context of these conjectures by studying explicit examples. First, an investigation of the manifold mirror to the quintic threefold will be undertaken. The PI will search for objects on the mirror manifold that exhibit behavior similar to a certain class of Lagrangian submanifolds on the quintic. This will provide a direct and concrete illustration of HMS. In conjunction with this work, a study of Floer cohomology of Lagrangians in K3 surfaces will be undertaken. This work will exploit explicit Lagrangian torus fibration constructions of Gross, Castano-Bernard, Matessi and others, which in turn are inspired by the SYZ conjecture.Algebraic geometry and symplectic geometry are deep and important fields whose origins go back hundreds of years. The questions and ideas arising from these fields have inspired many mathematicians and physicists and led to great scientific advances. Mirror symmetry promises to add to this legacy. Mirror symmetry has its roots in string theory and today is an important branch of physics. To mathematicians, mirror symmetry is an exciting and tantalizing subject because it hints at a link between algebraic and symplectic geometry. It is often these types of links-links between different subjects-that lead to breakthroughs. The goal of this project is to discover links between algebraic and sympletic geometry by applying newly developed theory and techniques to certain examples that are of central importance to mirror symmetry.