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

建议：DMS-9975527首席研究员：Dennis Sullivan摘要：本项目涉及几何组合模型的构建，用于微分形式和张量不充分的理论研究或需要科学计算的实际工作。 我们已经发现了一个运营商“三角洲”的细胞结构中的封闭曲线的空间在一个流形（字符串空间），它记录了有限的字符串集合如何相互作用，并与对方。 该算子在局部嵌入弦空间的同调上诱导出一个李双代数结构。 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细胞分解来建立一个显式的组合模型。具体的模型将被设计来实现在流体、气体或等离子体的运动中出现的非线性计算。 这些介质的运动与在物理力影响下的空间扭曲和卷曲的方式曲线密切相关。 此外，上面提到的基本结构“三角形”的相互作用，可以很容易地在飞机翼尖后的顶点尾迹的重新耦合等现象中看到。 我们对我们的某些目标能够实现持乐观态度，因为封闭曲线的数学结构一方面在几何上似乎非常适合我们所关心的许多问题，另一方面，一旦揭示了这种结构，它似乎就心甘情愿地向前流动，并带有某种华丽和丰富。

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.