Vertex Algebras, S-duality Conjecture, and Counting Plane Curves

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AbstractZhenbo QinDMS-98 77103Professor Qin will work in the general area of algebraic geometry, vertex algebra, Gromov-Witten invariants, and their interplay with physics. The main tools to be employed are the moduli space of Gieseker semistable bundles on algebraic surfaces; the modular invariance of the characters of vertex operator algebras, and the localization formula from intersection theory. The investigation should shed light on the relations between stable bundles, vertex algebras, and the S-duality conjecture from physics. Professor Qin also expects to learn more about the relation between certain enumerative invariants and Gromov-Witten invariants of the complex projective plane.Algebraic geometry is one of the great advances in mathematics of the 20-th century. In the last twenty years all areas of mathematics and some branches of physics have benefited from our increased application of the abstract but powerful ideas of this subject. In algebraic geometry, geometric objects are replaced by algebraic objects which retain the basic properties of the geometry. This opens the study to the abstract methods of algebra. The result is that complicated geometric connections are easier to study and understand. Professor Qin's research has many of its roots in mathematical physics and should help us understand some of the basic building blocks of our world.