Deformations of Representations; and Prounipotent Groups in Differential Galois Theory

University Of Oklahoma Norman Campus, Norman OK

Investigators

Abstract

This project will investigate (1) liftings and deformations of representations of finitely generated groups; and (2) the structure of the antiderivative closure of the field of rational complex functions, and its extensions and their groups of differential automorphisms. Regarding (1), a simple complex matrix representation of a finitely generated group is deformable if it extends to a representation in matrices over complex formal power series. Such an extension exists if there is a compatible family of liftings of the representation to the finite dimensional quotients of formal power series. Thus the extension problem can be viewed as a sequence of (relative) lifting problems. Each such lifting problem is obstructed by an element in a certain cohomology group of a module constructed from the original simple representation. The proposed research will investigate the lifting problem by studying these cohomological elements. Regarding (2), the antiderivative closure of the field of rational functions is the maximal extension obtained by adjoining full sets of solutions of linear homogeneous ordinary differential equations whose solutions can be obtained by repeated antiderivatives. The group of differential automorphisms of such an extension is free prounipotent. The research will exploit this structure to describe the antiderivative closure directly in terms of the functions in the coordinate ring of the free prounipotent group. Antiderivative closures may not be antiderivative closed, so a tower of such closures may result. The research will analyze the differential Galois groups of the steps in this tower over the base field of rational functions. These groups have a normal series with free prounipotent sections. However, they are not known to be free prounipotent themselves. Groups are objects that mathematically encode the concept of symmetry, and have been a mainstay in mathematics, chemistry, and elsewhere for over a century. Representations of groups are realizations of groups as transformations of space. A given group usually has infinitely many representations. These can be organized into a geometrical set. Representations which are not just isolated points of the set are called deformable. This research is aimed at understanding deformable representations and their applications to group representations in mathematics, chemistry, and elsewhere. Integral calculus deals with reconstructing a function, such as one describing the position of an object, from the instantaneous rates of change of the function; the reconstructed function is called an antiderivative. The planned research can be viewed as a method to solve, in principle, and all at once, all the repeated integral calculus problems arising from polynomial functions. This would have many applications in calculus.