Combinatorial Model for Geometry and Analysis Based on the Algebraic Topology of Closed Curves

基于闭曲线代数拓扑的几何与分析组合模型

• 批准号: 9975527
• 负责人: Dennis Sullivan
• 金额: $ 19.43万
• 依托单位: CUNY Graduate School University Center
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-15 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9975527&HistoricalAwards=false
• 关键词: Combinatorial; Model; Geometry; Analysis; Based

Proposal: DMS-9975527Principal Investigator: Dennis SullivanABSTRACT: This project concerns the construction of a combinatorial model of geometry to be used in theoretical studies where differential forms and tensors appear inadequate or in practical work, where scientific computation is required. We have discovered an operator "delta" in the cell structure of the space of closed curves in a manifold (string space), which records how finite collections of strings interact with themselves and with each other. This operator induces a Lie bialgebra structure on the homology of the space of the locally embedded strings. After developing the abstract theory behind these string interactions, we will build an explicit combinatorial model using any cell decomposition of the manifold. The concrete model will be designed to implement nonlinear computations that arise in the motion of fluids, gases, or plasmas. The motion of these media is closely related to the way curves in space twist and curl under the influence of physical forces. Also, the interactions of the basic structure "delta" alluded to above, can be readily seen in such phenomena as the recoupling of vertex trails behind the wing tips of airplanes. We have some optimism that some of our goals can be realized because the mathematical structure of closed curves seems very well adapted geometrically to many of the problems we care about on the one hand, and on the other hand once revealed this structure seems to flow forth rather willingly and with some pageantry and richness.

摘要：本课题旨在建立一个几何组合模型，用于微分形式和张量不足的理论研究或需要科学计算的实际工作中。我们在流形（弦空间）的闭曲线空间的单元结构中发现了一个算子“delta”，它记录了有限的弦集合如何相互作用以及彼此之间的相互作用。该算子在局部嵌入字符串空间的同调上推导出李双代数结构。在发展了这些弦相互作用背后的抽象理论之后，我们将使用流形的任何单元分解建立一个显式组合模型。具体模型将被设计用于实现流体、气体或等离子体运动中出现的非线性计算。这些介质的运动与空间曲线在物理力的作用下扭曲和旋度的方式密切相关。此外，上面提到的基本结构“三角洲”的相互作用，可以很容易地在飞机机翼尖端后面的顶点尾迹的重新耦合等现象中看到。我们乐观地认为，我们的一些目标是可以实现的，因为一方面，封闭曲线的数学结构似乎在几何上很好地适应了我们关心的许多问题，另一方面，一旦揭示出来，这种结构似乎就会自然而然地出现，而且带有一些华丽和丰富。