This award supports the research in Algebraic Geometry of Professor Yue L. Tong of Purdue University. Dr. Tong's project has two parts. The first is to apply methods that he and D. Toledo developed some years ago in a purely geometric setting, to the problem of verifying the Beilinson-Shechtman conjecture in mathematical physics. This involves getting further refinements of the classical Riemann-Roch Theorem, beyond those developed by Grothendieck over twenty years ago. The second part of Dr. Tong's project is to study harmonic forms and their dual geometric cycles in locally symmetric spaces. The examples and the insights derived will be important tools in attacking the Hodge Con- jecture. This is research 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 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.
