9870033 Edidin The focus of this project is intersection theory on quotient moduli spaces that arise in algebraic geometry. This includes using localization methods on moduli spaces of vector bundles to study surface invariants, as well as calculating integral Chow rings of moduli stacks of curves and vector bundles. We also consider the problems of determining which stacks are quotient stacks and of obtaining an explicit Riemann-Roch formula for singular quotient schemes. Much of the work will use the equivariant intersection theory developed jointly by the principal investigator and W. Graham. 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.