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

秦振波教授将从事代数几何、顶点代数、Gromov-Witten不变量及其与物理的相互作用等方面的研究。 主要的工具是代数曲面上Gieseker半稳定丛的模空间;顶点算子代数特征的模不变性，以及来自交理论的局部化公式。 这些研究将有助于揭示稳定丛、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上某些枚举不变量与Gromov-Witten不变量之间的关系。代数几何是世纪数学的重大进展之一。在过去的20年里，数学的所有领域和物理学的某些分支都受益于我们对这门学科抽象而有力的思想的日益广泛的应用。在代数几何中，几何对象被代数对象所取代，代数对象保留了几何的基本属性。 这就为代数的抽象方法的研究打开了大门。 其结果是，复杂的几何连接更容易研究和理解。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.