Gross 9700761 The principal investigator will continue his study of Calabi-Yau threefolds, which are complex threefolds with vanishing first Chern class. Various aspects of their geometry will be addressed. He will continue developing tools of deformation theory in order to understand how various moduli spaces of Calabi-Yau threefolds interconnect. These tools will also be applied to study the deformation theory of rational Gorenstein threefold singularities, which can give insights into four-dimensional birational geometry. Calabi-Yau threefolds will also be considered in light of the recently proposed construction of Strominger, Yau, and Zaslow for mirror symmetry. Among other issues this will involve studying the nature of special Lagrangian torus fibrations and the role that special Lagrangian submanifolds play in the geometry of Calabi-Yau manifolds. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, 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 physics, theoretical computer science, and robotics.
