Mathematical Sciences: "Fundamental Groups Of Quasiprojective Varieties"

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This award supports work on the fundamental group of a smooth quasiprojective variety defined over the field of complex numbers. If such a variety maps properly onto a hyperbolic curve, its fundamental group surjects onto a nonabelian free group. The principal investigator will determine to what extent the converse holds. The problem will be attacked by a variety of methods. These include a generalization of the Castenuovo-De Franchis lemma, the construction of L2 harmonic 1-forms on the universal cover, and a study of certain homologically defined subsets of the groups of 1-dimensional characters of the fundamental group. This is reasearch in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in theoretical computer science and robotics.