The algebraic structures behind higher homotopies in symplectic topology.

AbstractAward：DMS-1105837首席研究员：Sikimeti Mau A-无穷代数结构最近出现在辛拓扑中，并将通过这些项目进行研究和扩展。主要研究人员打算研究这些代数结构的具体例子的基础上希尔伯特计划的点复杂曲线或复杂曲面。部分目标是将结构扩展到更高的范畴结构，使用与物理学家的弦图密切相关的辛结构。扩展的代数结构的动机是在代数几何结构相同的希尔伯特计划，这应该有辛类似物的镜像对称。两个例子，特别是已经很好地研究了辛方面，并可以作为指导，是Heegaard弗洛尔理论开发的低维拓扑学家，和塞德尔-史密斯的辛Khovanov同调，一个辛构造的不变量的结和链接。短期目标是找到一个新理论的具体例证和潜在应用，这个理论在很大程度上是抽象的，但有可能解释这些领域的代数现象。更广泛的目标是用一种来自“被子”的代数语言尽可能多地描述拉格朗日弗洛尔理论的代数结构，“被子”是Wehrheim和Woodward在辛拓扑学中的一种最新技术。辛结构是哈密顿力学的几何面，其中跟踪运动粒子系统的位置和动量坐标，并用于写出相应的运动方程，牛顿定律。承载这种结构的空间总是偶数维的，它们的基本几何是关于二维面积和更高维的体积，而不是关于长度和角度，这是许多熟悉几何的根源。新的方法正在进入辛几何从其他学科，如低维拓扑，它似乎是一个代数形式主义可以设计进行一些这些新的建设，并揭示有用的性质。ˇ

AbstractAward: DMS-1105837Principal Investigator: Sikimeti Mau A-infinity algebra structures have recently emerged in symplectic topology and will be investigated and extended by these projects. The principal investigator intends to study these algebraic structures in concrete examples based on Hilbert schemes of points on complex curves or complex surfaces. Part of the goal will be to extend the structures to higher categorical structures, using symplectic constructions closely related to the string diagrams of physicists. The extended algebraic structures are motivated by constructions in algebraic geometry for the same Hilbert schemes, which should have symplectic analogues by mirror symmetry. Two examples in particular that have been well studied on the symplectic side, and can function as guides, are the Heegaard Floer theory developed by low-dimensional topologists, and Seidel-Smith's symplectic Khovanov homology, a symplectically constructed invariant of knots and links. The short-term objective is to find concrete illustrations, and potential applications, of a new theory that is largely abstract, but has the potential to explain algebraic phenomena in these fields. The broader goal is to describe as much of the algebraic structure of Lagrangian Floer theory as possible in a single algebraic language coming from "quilts", a recent technique in symplectic topology due to Wehrheim and Woodward.A symplectic structure is the geometric face of Hamiltonian mechanics, in which the position and momentum coordinates of a system of moving particles are tracked and used to write out equations of motion that correspond to Newton's laws. Spaces that carry such structures are always even-dimensional, and their underlying geometry is about two-dimensional area and higher-dimensional volume rather than about length and angle, which are at the root of much of familiar geometry. New methods are coming into symplectic geometry from other subjects such as low--dimensional topology, and it appears that an algebraic formalism can be devised to carry a number of these new constructions and to reveal useful properties of them. ˇ