The purpose of this project is to bring techniques from Hodge theory to bear on the properties of solutions of differential equations found throughout mathematics and physics. To elaborate, algebraic geometry is the study of systems of polynomial equations. Hodge theory seeks to understand the shape of the algebraic spaces defined by their solution sets. In so doing, it produces powerful connections between algebraic geometry and other parts of mathematics and physics, such as number theory, differential equations, and string theory. Normal functions are solutions to differential equations arising from algebraic cycles (or subspaces) on an algebraic space. By recognizing a solution as a normal function, which is a nontrivial task, one gets access to its behavior at numerous "special" points. In the scenarios to be considered in this project, these "special values" have important applications to identities and conjectures in number theory, the classification of algebraic spaces such as Fano varieties and algebraic curves, and the spectra of operators in topological string theory. This project will also help to train the next generation of researchers in pure mathematics, by integrating the PI's graduate students in work on specific problems in Hodge theory and bringing outside consultants to Washington University. Results will be disseminated through conferences, summer schools, journal articles, and websites. Normal functions originated in the work of Poincare and