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

摘要 提案：DMS-9802612 首席研究员：Jim Bryan 这个项目研究拓扑和几何不变量，这些不变量与弦论中物理学中出现的对偶有关，有时也受到其启发。 主要的兴趣是版本的Gromov-Witten和Seiberg-Witten不变量的家庭辛结构。 布赖恩的项目将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。 具体来说，他将使用他的技术来解决大类的枚举问题的K3曲面，阿贝尔品种和其他椭圆曲面。 他将这些技术扩展到计算Gromov-Witten不变量的高维品种，以验证预测所作的镜像对称。 他将使用Seiberg-Witten理论来研究K3表面的扭曲度。 他将延长他的技术开发早期的工作研究循环群行动4流形使用塞伯格-威滕理论。 扩展的技术应适用于其他群体，3-流形，和代数曲面上的真实的结构。 理论物理学家在超对称弦理论方面取得的最新进展，在几何学和拓扑学的不同方面之间产生了许多有趣的、很大程度上是理论性的联系。 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.