Mathematical Sciences: Moduli Problems in Algebraic Geometry

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9622912 Li The goal of this project is to study enumerative geometry of moduli spaces in algebraic geometry. As the first step, this project will investigate how excessive intersection theory based on normal cone construction can be applied to study enumerative problems on moduli spaces whose dimensions are bigger than their expected dimensions. Afterward, Li will apply this technique to study Gromov-Witten invariants of smooth projective varieties, to study enumerative problems of rational curves in Calabi-Yau manifolds and Fano varieties and to derive recursion formulas for Donaldson polynomial invariants of algebraic surfaces. Some of them are known using analytic method. Li believes that progress along this line will provide new insight to these problems, new technique to attack similar problems and will be a source of good mathematical problems in the future. 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.