Low Dimensional Geometry and Monopoles

低维几何和单极子

• 批准号: 0305130
• 负责人: Gang Tian
• 金额: $ 10.61万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0305130&HistoricalAwards=false
• 关键词: Low; Dimensional; Geometry; Monopoles

PROPOSAL DMS-0305130LOW-DIMENSIONAL GEOMETRY AND MONOPOLESP.Is: Yann Rollin, Gang TianABSTRACTThe aim of the proposed project is to study a manifold with boundary, andsome additional structure, in relation with the boundary values of thestructure. This generic framework is a very natural approach to manyimportant questions of differential geometry. Some typical cases are thoseof contact structures fillable by symplectic forms, complex geometry andCauchy-Riemann boundary, Einstein geometry and conformal infinity. It isinteresting to study how deformation theory of the boundary inducesdeformations of a filling. This problem is often related to some positivefrequency condition, which is of particular interest for physicists. Thiswill be used to study singular Yang-Mills connections of (a conjectural)Donaldson theory on Calabi-Yau 3-manifolds. More generally, we look at thepicture of a moduli space on the manifold which projects on acorresponding moduli space defined on the boundary. Whenever it can beshown that the projection has a finite nonzero degree in some sense, weobtain a powerfull tool to prove the existence of fillings. Another way totackle the questions of existence is to develop gluing theorems betweenmoduli spaces. Some applications of gluing for Seiberg-Witten equationsare already being developped thus giving new perspectives on contactgeometry.A physical theory is defined by a configuration space, together withobjects (for example a metric) verifying particular equations.The state of a physical system is constrained by its configuration onthe boundary of the space, or, at infinity. Therefore, it is anatural question to ask wether a theory is rich or empty: is itpossible to find a physical system with a given behavior on theboundary, and if so, do we have many solutions? In other words is thesystem soft or rigid? Beyond the physical flavor, this approach, knownas a field theory, has deep implications in mathematics. First, it isrequired to elaborate original tools in analysis, called moduli spacetheories, that have their own beauty. Secondly, the method relatesvery different problems. To illustrate that, we mention the followingsituation: we consider a 3-dimensional space (it could be the3-dimensional sphere) bounding a 4-dimensional space (like the ballinside the sphere). Then, we look at knots on the boundary (they couldbe strands of DNA). These knots can be thought of as the boundary ofa surface lying into the 4-dimensional space. Then, it can be shownthat there is a subtle relation between the knot theory in dimension 3and the theory of surfaces lying in a 4-dimensional space.

提案DMS-0305130低维几何和单极子是：杨罗林，田刚摘要该项目的目的是研究一个流形的边界，和一些额外的结构，与边界值的结构。这个通用的框架是一个非常自然的方法来解决微分几何的许多重要问题。一些典型的情况是那些接触结构可填充辛形式，复杂的几何和Cauchy-Riemann边界，爱因斯坦几何和共形无穷大。边界变形理论如何引起充填体的变形是一个值得研究的问题。这个问题经常与一些正的自由度条件有关，这对物理学家来说是特别感兴趣的。这将被用来研究Calabi-Yau三维流形上的（一个代数）唐纳森理论的奇异Yang-Mills联络.更一般地说，我们来看看流形上的模空间投影在边界上定义的相应模空间上的图像。只要在某种意义下投影有有限个非零度，我们就得到了证明填充存在性的有力工具。另一种解决存在性问题的方法是发展模空间之间的胶合定理。Seiberg-Witten方程的胶合的一些应用已经被开发，从而给接触几何带来了新的视角。物理理论由位形空间以及验证特定方程的对象（例如度量）定义。物理系统的状态由其在空间边界或无穷远处的位形约束。因此，问一个理论是丰富的还是空洞的是一个自然的问题：是否有可能找到一个在边界上具有给定行为的物理系统，如果有，我们有很多解吗？换句话说，这东西是软的还是硬的？除了物理学的味道，这种方法，被称为场论，在数学中有着深刻的含义。首先，它需要精心制作原始的分析工具，称为模空间理论，有自己的美丽。第二，这种方法涉及非常不同的问题。为了说明这一点，我们提到以下情况：我们考虑一个3维空间（它可以是3维球体）包围一个4维空间（就像球体旁边的球）。然后，我们看看边界上的结（它们可能是DNA链）。这些节点可以被认为是一个表面的边界，位于四维空间。由此可见，三维纽结理论与四维曲面理论之间存在着微妙的联系。