Gauge Theoretic Invariance and Applications to an Enumerative Geometry and Low-dimensional Topology

摘要建议：dms-9802612首席研究员：Jim Bryan这个项目研究与弦理论中的物理学中出现的对偶性有关的拓扑和几何不变量，有时还受到这些对偶的启发。主要感兴趣的是辛结构族的Gromov-Witten和Seiberg-Witten不变量的版本。布莱恩的项目将把这些不变量应用于枚举代数几何、辛几何和低维拓扑中的各种问题。具体地说，他将使用他的技巧来解决K3曲面、阿贝尔簇和其他椭圆曲面的大类计数问题。他将把这些技术扩展到计算高维变种的Gromov-Witten不变量，以验证镜像对称性做出的预测。他将使用Seiberg-Witten家族理论来研究K3表面具有旋风家族特征的程度。他将利用Seiberg-Witten理论扩展他在早期工作中开发的用于研究4-流形上的循环群作用的技术。推广的技巧应该适用于其他群、3-流形和代数曲面上的真实结构。理论物理学家在超对称弦理论方面取得的最新进展导致了几何学和拓扑学中非常不同的方面之间有许多有趣的、主要是猜测的相互联系。布莱恩的项目将阐明这些物理理论背后的一些几何学；物理学家希望超对称弦理论最终能给出一个完整的宇宙理论。

Abstract Proposal: DMS-9802612 Principal Investigator: Jim Bryan This project investigates topological and geometric invariants that are related to, and sometimes inspired by, dualities that arise in physics in string theory. Of primary interest are versions of the Gromov-Witten and Seiberg-Witten invariants for families of symplectic structures. Bryan's project will apply these invariants to various problems in enumerative algebraic geometry, symplectic geometry, and low-dimensional topology. Specifically, he will use his techniques to solve large classes of enumerative problems for K3 surfaces, Abelian varieties and other elliptic surfaces. He will extend these techniques to compute Gromov-Witten invariants for higher dimensional varieties in order to verify predictions made by mirror symmetry. He will use Seiberg-Witten theory for families to investigate the extent to which the K3 surface is characterized by its twistor family. He will extend his techniques developed in earlier work for studying cyclic group actions on 4-manifolds using Seiberg-Witten theory. The extended techniques should apply to other groups, to 3-manifolds, and to real structures on algebraic surfaces. Recent progress made by theoretical physicists in super-symmetric string theories have led to many intriguing and largely conjectural interconnections between very different aspects of geometry and topology. Bryan's project will elucidate some of the geometry that underlies these physical theories; physicists hope super-symmetric string theory will eventually give a complete theory of the universe.