This mathematical work will center on the algebraic structure of solutions of differential equations. By this one means the analysis of how solutions of differential equations can be expressed in terms of solutions of certain special differential equations. Particular instances of these problems include solving differential equations implicitly or explicitly in terms of elementary or liouvillian functions, or more generally, in terms of solutions of differential equations of lower order. Related investigations into the calculation of associated Galois groups will also be carried out. The work divides into three parts. In the first, work will be done on the algebraic structure of elementary and other special functions, with the aim of improving and extending existing algorithms for integration in finite terms. The second thrust involves a continuation of efforts to develop a theory of liouvillian first integrals - first integrals expressible in terms of algebraic combinations of integrals and exponentials, and developing algorithms for finding such integrals. By a first integral, one understands a function of several variables which vanishes on solutions of systems of differential equations. A third goal of this project is to understand the algebraic structure of solutions of linear differential equations. Here one is interested in using algebraic arguments, Lie theory and Galois theory to determine whether solutions of given differential equations can be expressed as combinations of solutions of equations of lower order. Coupled with this are fundamental questions of providing algorithms for accomplishing the desired representations. In addition to its applications to the general theory of differential equations, this research bears direct relationship with important work at the frontiers of computer science, particularly with regard to symbolic manipulation algorithms.
