This supports the work of Dr. Tony Pantev a post-doctoral associate to Professor Kleiman. Three problems will be studied. The first is to analyze the monodromy group of generalized theta functions. The second problem asks for a geometric description of the locus of motivic local systems on curves. The third problem to be studied is a conjectural construction of Hyperkahler manifolds. 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. ***
