Mathematical Sciences: Algebraic Geometry and Gauge Theory

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This awards supports the research of Professor R. Friedman to work in algebraic geometry. He will continue to study the connections between the algebraic and smooth properties of 4-manifolds. In particular, he will try to decide if every complex surface diffeomorphic to a rational surface is itself rational and whether one can characterize smoothly embedded 2-spheres of certain types lying on a complex surface. 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.