Professor Xu will work on the geometry and topology of Calabi-Yau manifolds as well as higher dimensional projective varieties. In particular, he wants to study rational curves on Calabi-Yau threefolds. He also intends to work on adjoint linear series and pluricanonical series on projective varieties of higher dimension. Finally, he intends to work on higher dimensional sphere-packing problems. This is research in the field of algebraic geometry, yet it directly connects to two of the great advances in theoretical physics in this century--quantum mechanics and general relativity. Algebraic geometry itself 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.
