Mathematical Sciences: Enumerative Geometry of Moduli Spaces

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Professor Bertram will study deformations of the cohomology rings of homogeneous projective varieties as well as smooth hypersurfaces. This deformation ring (usually called the "quantum cohomology ring") encodes a great deal of data about the enumerative geometry of the moduli space of maps of a rational curve to the variety. The primary goal of this project is to obtain a purely algebraic proof of the existence of this ring. This is a 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.