Methods of deRham Topology Applied to Nonlinear Problems

摘要奖：DMS 1309228，首席研究员：Dennis Sullivan从代数拓扑到等价关系，人们知道微分形式的代数包含的信息比通常的上同调包含的信息多得多。定义Derham上同伦为线性化的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。这是自由可微分次交换代数(T，d)到诱导上同调上的双射的Derham代数(A，d)的微分代数映射。线性化复形是由T的微分在T的生成元[或不可分解的]上诱导的。这种Derham上同伦可以与单连通空间的普通同伦和具有可压缩泛覆盖空间的结合代数的代数K理论相联系。这个配方有意义的具体证明是德勒姆拓扑学和这一提议的应用的关键的非平凡点。它的基础是显式地发展了微分代数映射之间的非线性同伦概念。人们得到了代数结构和它们之间的映射的照明图，这与通常拓扑学中的细胞复合体和纤丝之间的映射非常相似。在一个设想的几何和分析数学模型的应用集合中，涉及无限多个自由度和非线性结构，有限维近似模型用不同的近似级别之间的相干映射来构造。Hodge星运算符[与其正交补空间相关联的线性子空间]不能立即服从于这种方法。因此，为了巧妙地解决Hodge Star困难，该建议还侧重于几个代数结构，如弦理论和几何结构，如开弦和闭弦中的奇点，特别是流形。定量的答案取决于问题的数据。有一些很好的技术可以解决这些问题。像洋流、水库中的石油流动和天气的数学模型这样的非线性问题，在数学和计算上都要困难得多。该建议声称，非线性拓扑技术有望为处理物理过程的非常一般的非线性数学模型提供一种新的技术。通过提供逼近非线性问题的有限模型，该技术将易于应用。这些模型符合实际并具有预测价值的可能性增加了，因为它们的推导是基于非线性的基本数学结构。

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.