Symplectic Structures on Closed Manifolds

闭流形上的辛结构

• 批准号: 0604748
• 负责人: Tian-Jun Li
• 金额: $ 12.18万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604748&HistoricalAwards=false
• 关键词: Symplectic; Structures; Closed; Manifolds

DMS-0604748Tian-Jun LiThe PI proposes to develop new techniques and apply existing methods in gauge theory, pseudo-holomorphic curve theory and equivariant stable homotopy theory to gain some fundamental understanding of the generalshape of closed symplectic manifolds. Four avenues of investigationare addressed. 1) The PI, joint with M. Furuta, is developing acomprehensive treatment of a refined invariant associated with aproper non-linear Fredholm map, which involves a twistedPontrjagin-Thom construction. 2) Such a construction applied to theSeiberg-Witten map turns out to be particular useful in theclassification of symplectic 4--manifolds with torsion symplecticcanonical classes. The PI has made progress bounding their Bettinumbers and is hoping to fully determine their homoeomorphism types. 3)The PI searches for a simple characterization of the symplectic coneof a K\"ahler surface. The case of geometric genus 0 has been completely settled by the PI and A. Liu. Joint with M. Usher, promising progress in the case of positive geometric genus has been made, and more is expected by exploring thesomewhat surprising interplay between symplectic forms and embeddedsymplectic surfaces with negative self-intersections. 4) Jointmainly with Y. Ruan, the PI is investigating the properties ofuniruled symplectic manifolds in dimension 6 and above, usingrelative invariants, their gluing formula, and localization techniques. We also search for the criterion such that symplecticblow-downs can be performed in dimension 6.This project belongs to the relatively new and increasingly important subject the PI has pursued a successful line of research. The proposedactivity raises several fundamental questions in this field and laysout plans to answer them. It also creates original concepts in thisfield and reveals connections with other fields. This proposed activity will advance knowledgein symplectic topology as well as other areas including differentialtopology, mathematical physics and algebraic geometry.An n manifold is a space that locally looks like the Euclidean space of dimension n. For example, the space-time universe we live in is a 4--manifold. A symplectic structure is a very basic structure that underlies almost all the equations of classical and quantum physics. A manifold equipped with a symplectic structure is called a symplecticmanifold. Studies of symplectic manifolds, such as proposed here,will thus enhance our understandings of mathematics, physics, andscience in general.

李天军：本课题提出在规范理论、伪全纯曲线理论和等变稳定同伦理论中发展新技术和应用现有方法，对闭辛流形的一般形状有一些基本的认识。提出了四种调查途径。1) PI与M. Furuta合作，正在开发一种与适当的非线性Fredholm映射相关的精炼不变量的综合处理，该映射涉及扭曲的pontrjagin - thom构造。2)应用于theseberg - witten映射的这种构造在具有扭转辛正则类的辛4—流形的分类中被证明是特别有用的。PI在确定它们的bettinnumber方面取得了进展，并希望完全确定它们的同态类型。3) PI寻找K\ ahler曲面辛锥的一个简单表征。几何属0的情况已经被PI和A. Liu完全解决了。与M. Usher一起，在正几何属的情况下取得了有希望的进展，并且通过探索辛形式和具有负自交的嵌入辛曲面之间令人惊讶的相互作用，人们期待更多。4) PI主要与阮Y.合作，利用相对不变量、胶合公式和局部化技术，研究了6维及以上的无规辛流形的性质。我们也寻找能在第6维中进行辛吹落的判据。这个项目属于相对较新的和日益重要的主题，PI已经追求成功的研究路线。拟议的活动提出了该领域的几个基本问题，并制定了回答这些问题的计划。它还创造了本领域的原创概念，并揭示了与其他领域的联系。这项活动将促进辛拓扑以及其他领域的知识，包括微分拓扑、数学物理和代数几何。一个n流形是一个局部看起来像n维欧几里得空间的空间。例如，我们生活的时空宇宙是一个4-流形。辛结构是一种非常基本的结构，它是几乎所有经典和量子物理方程的基础。具有辛结构的流形称为辛流形。对辛流形的研究，比如这里提出的，将增强我们对数学、物理和一般科学的理解。