The algebraic structures behind higher homotopies in symplectic topology.

摘要：dms -1105837项目负责人：Sikimeti Mau最近在辛拓扑中出现了a -∞代数结构，并将通过这些项目进行研究和扩展。主要研究者打算基于复杂曲线或复杂曲面上点的希尔伯特格式在具体实例中研究这些代数结构。我们的部分目标是利用与物理学家的弦图密切相关的辛结构，将这些结构扩展到更高的范畴结构。扩展的代数结构是由相同希尔伯特格式的代数几何构造所激发的，这些希尔伯特格式应该具有镜像对称的辛类似物。有两个例子已经在辛方面得到了很好的研究，并且可以作为指导，它们是由低维拓扑学者提出的Heegaard flower理论，以及Seidel-Smith的辛Khovanov同调，一个结点和连杆的辛构造不变量。短期目标是找到一个新理论的具体例证和潜在应用，这个新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“quilts”的代数语言尽可能多地描述拉格朗日花理论的代数结构，quilts是Wehrheim和Woodward在辛拓扑中提出的一种新技术。辛结构是哈密顿力学的几何面，在辛结构中，运动粒子系统的位置和动量坐标被跟踪，并用于写出符合牛顿定律的运动方程。承载这种结构的空间总是偶数维的，它们的基础几何是关于二维面积和高维体积的，而不是关于长度和角度的，这是许多熟悉的几何的根源。辛几何的新方法来自于其他学科，如低维拓扑学，似乎可以设计出一种代数形式体系来承载这些新结构，并揭示它们的有用性质。, # 711;

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