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

秦教授将在代数几何、顶点代数、Gromov-Witten不变量及其与物理学的相互作用等方面进行研究。所使用的主要工具是代数曲面上Gieseker半稳定束的模空间；顶点算子代数的模不变性，以及由交理论导出的局部化公式。研究应阐明稳定束、顶点代数和物理学中的s -对偶猜想之间的关系。秦教授还希望进一步了解复射影平面的某些枚举不变量与Gromov-Witten不变量之间的关系。代数几何是20世纪数学的一大进步。在过去的二十年里，数学的所有领域和物理学的一些分支都受益于我们越来越多地应用这一学科的抽象而有力的思想。在代数几何中，几何对象被保留几何基本性质的代数对象所取代。这开启了对代数抽象方法的研究。结果是复杂的几何关系更容易研究和理解。秦教授的研究有许多数学物理的根源，应该有助于我们理解我们的世界的一些基本组成部分。

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.