Floer Theory in examples of interest to Mirror Symmetry

摘要奖：DMS 1108397，首席研究员：Garrett AlstonMirror对称性是一个令人兴奋的数学分支，它涉及到Calabi-Yau流形的辛几何和代数几何之间的对偶性。有两个互补的猜想可以解释这种二元性--康采维奇的同源镜像对称猜想(HMS)和Strominger-Yau-Zaslow猜想(SYZ)。主要研究人员旨在通过研究明确的例子来研究这些猜想背景下的镜像对称性。首先，将对五次曲面的三重流形镜进行调查。PI将在镜像流形上搜索表现出类似于五次曲面上的某类拉格朗日子流形的行为的对象。这将为HMS提供一个直接和具体的例证。结合这项工作，我们将研究K3曲面上拉格朗日函数的Floer上同调。这项工作将利用Gross，Castano-Bernard，Matessi等人的显式拉格朗日环面纤维结构，而这些结构又受到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，并导致了巨大的科学进步。镜面对称有望为这一遗产增添色彩。镜像对称起源于弦理论，今天是物理学的一个重要分支。对于数学家来说，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。通常正是这种类型的联系--不同学科之间的联系--导致了突破。这个项目的目标是通过将新发展的理论和技术应用于对镜像对称至关重要的某些例子来发现代数几何和辛几何之间的联系。

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.