9622564 In this project, seven problems in the general context of algebraic geometry, quantum cohomology, and their interplay with physics are proposed to be investigated. The main tools are Grothendieck-Knudsen-Mumford determinant line bundles over moduli space of Gieseker semistable bundles on algebraic surfaces and Gromov-Witten symplectic invariants. The investigation would increase the knowledge about the geometry including the quantum cohomology ring structure of the moduli space of Gieseker semistable bundles, the S-duality conjecture from physics in the form of Vafa and Witten, and the deformation type of certain minimal surfaces of general type (a conjecture due to Catanese and Reid). 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 the 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.
