Mathematical Sciences: Fundamental Groups of Complex Manifolds

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This awards supports the research of Professors J. Carlson and D. Toledo to work in algebraic geometry. They will work on fundamental groups of complex manifolds and their representations. They intend to study the representation theory of the recently constructed examples of manifolds with non-residually finite fundamental groups. They also intend to apply the theory of harmonic maps to enlarge the class of groups for which the Shafarevic conjecture is known to be true. The research is in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which blossomed to the point where it has, in the past 10 years, solved problems that have stood for centuries. Originally, it treated figures defined in the plane by the simplest of equations, namely polynomials. Today, the field uses methods not only from algebra, but also from analysis and topology, and conversely it is extensively used in those fields. Moreover, it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics.