Methods of deRham Topology Applied to Nonlinear Problems

deRham 拓扑方法应用于非线性问题

• 批准号: 1309228
• 负责人: Dennis Sullivan
• 金额: $ 17.7万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-15 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1309228&HistoricalAwards=false
• 关键词: Methods; deRham; Topology; Applied; Nonlinear

AbstractAward: DMS 1309228, Principal Investigator: Dennis SullivanOne knows the algebra of differential forms up to an equivalence relation from algebraic topology contains much more information than the usual cohomology. Define the deRham coHomotopy to be the cohomology of the linearized deRham complex defined as follows: first construct a resolution of the deRham algebra (A,d). This is a differential algebra map of a free differential graded commutative algebra (T,d) to the deRham algebra (A,d) inducing a bijection on cohomology.The linearized complex is that induced on the generators [or indecomposables] of T by its differential. This deRham Cohomotopy can be related to ordinary homotopy for simply connected spaces and to algebraic K theory of associative algebras using spaces with contractible universal covers. The concrete proof that this recipe is meaningful is the key non-trivial point of deRham Topology and to the applications of this proposal. It is based on developing explicitly the nonlinear notion of homotopy between maps of differential algebras. One obtains an illuminated picture of algebraic structures and maps between them that closely resembles that of maps between cell complexes and fibrations in usual topology. Thus one may analyze the theory of algebraic objects defined by any number of multilinear operations with j inputs and k outputs for j and k positive.In one envisaged set of applications mathematical models in geometry and analysis that involve infinitely many degrees of freedom and nonlinear structures, finite dimensional approximating models are constructed with coherent mappings between the different levels of approximations. The Hodge star operator [which associates to a linear subspace its orthogonal complement] is not immediately amenable to this method. Thus the proposal also focuses on several algebraic structures like string theories and geometric structures like singularities in open and closed strings particular to manifolds in order to finesse the Hodge Star difficulty.Some questions are linear problems. The quantitative answer depends linearly on the data of the problem. There are good techniques for these problems. Nonlinear problems like the mathematical models of ocean currents, flows of oil in a reservoir and the weather are much more difficult to get a grip on mathematically and computationally. The proposal claims that a technique of nonlinear topology holds some promise to give a new technique for treating quite general nonlinear mathematical models of physical processes. The technique will be easy to apply by offering finite models approximating nonlinear problems. The likelihood these models will fit with reality and have predictive value is enhanced because their derivation is based on the underlying mathematical structure of the nonlinearities.

AbstractAward：DMS 1309228，首席研究员：Dennis SullivanOne知道微分形式的代数直到代数拓扑的等价关系包含比通常的上同调更多的信息。定义deRham余同伦为线性化deRham复形的上同调，定义如下：首先构造deRham代数（A，d）的一个分解。 这是一个自由微分分次 交换代数（T，d）到deRham代数（A，d） 在上同调上导出一个双射。线性化复形是由T的微分在T的生成元上导出的复形。这个deRham同伦可以与普通同伦的简单连通空间和代数K理论的结合代数使用的空间与可收缩的普遍覆盖。这个配方是有意义的具体证明是deRham拓扑的关键非平凡点和该建议的应用。它是基于发展明确的非线性概念同伦映射之间的微分代数。人们得到一个照明图片的代数结构和它们之间的地图，非常类似于细胞复合体和纤维化之间的地图在通常的拓扑结构。因此，人们可以分析理论的代数对象定义的任何数量的多线性操作与j输入和k输出j和k positive.In一个设想的一套应用数学模型的几何和分析，涉及无限多个自由度和非线性结构，有限维近似模型构造与相干映射之间的不同层次的近似。 霍奇星星算子[它与线性子空间及其正交补相关联]不能立即适用于这种方法。因此，该建议还侧重于几个代数结构，如弦理论和几何结构，如奇点在开放和封闭的字符串特别是流形，以巧妙的霍奇星星的困难。定量的答案线性地依赖于问题的数据。对于这些问题有很好的技术。非线性问题，如洋流的数学模型，油藏中石油的流动和天气， 在数学和计算上更难掌握。该提案声称，非线性拓扑技术有一定的希望，给一个新的技术处理相当普遍的非线性数学模型的物理过程。通过提供近似非线性问题的有限模型，该技术将易于应用。这些模型符合现实并具有预测价值的可能性得到了增强，因为它们的推导是基于非线性的基本数学结构。