Award: DMS 1309228, Principal Investigator: Dennis Sullivan
One 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.
