The mathematical research supported by this award focuses on the nature of solutions of differential equations in the complex domain. Specifically, work will be done investigating algebraic differential equations. That is, differential equations in which the derivatives of the dependent variable enter only polynomially. A solution to such an equation is called differentially algebraic. The point of view taken in this project is one of determining the complex calculus one can derive with the restricted class of differentially algebraic analytic functions. For example, is it possible to classify all such entire functions? What, if any, are the growth restrictions on such functions? Efforts will also be made to analyze the iterates of such functions and to establish the extent to which interpolation of arbitrary complex sequences is possible at prescribed values by a single function in the class. Differential equations play an essential role in modeling the physical universe. It is often the case that solutions, although known to exist theoretically, may not be expressible by elementary means. The purpose of this research is to develop properties of broad classes of functions that can arise as solutions of differential equations and to use these properties as part of a comprehensive classification scheme.
