Low Dimensional Geometry and Monopoles

建议dms -0305130低维几何和垄断。摘要本课题的目的是研究具有边界和附加结构的流形与结构的边界值之间的关系。这个通用框架是解决微分几何许多重要问题的一种非常自然的方法。一些典型的例子是可由辛形式、复几何和柯西-黎曼边界、爱因斯坦几何和保形无穷填充的接触结构。研究边界变形理论如何引起充填体变形是一个有趣的问题。这个问题通常与某些正频率条件有关，这是物理学家特别感兴趣的。这将用于研究Calabi-Yau 3-流形上（一个推测的）Donaldson理论的奇异Yang-Mills连接。更一般地说，我们观察流形上模空间的图像，流形投影在边界上定义的相应模空间上。只要在某种意义上能证明投影具有有限的非零度，我们就得到了证明填充存在的有力工具。另一种解决存在性问题的方法是发展模空间间的粘接定理。粘接法在Seiberg-Witten方程中的一些应用已经得到了发展，从而为接触几何提供了新的视角。物理理论是由构形空间和验证特定方程的对象（例如度量）定义的。物理系统的状态受其在空间边界或无穷远处的构型的约束。因此，问一个理论是丰富的还是空洞的问题是很自然的：是否有可能在边界上找到一个具有给定行为的物理系统，如果有，我们有很多解决方案吗？换句话说，系统是软的还是硬的？除了物理意义之外，这种被称为场论的方法在数学上有着深刻的含义。首先，需要详细阐述分析中的原始工具，称为模空间理论，它有自己的优点。其次，该方法涉及到非常不同的问题。为了说明这一点，我们提到以下情况：我们考虑一个三维空间（可以是三维球体）包围一个四维空间（比如球体内的球）。然后，我们观察边界上的结（它们可能是DNA链）。这些结可以被认为是四维空间中表面的边界。由此可以看出，三维空间中的结理论与四维空间中的曲面理论之间存在着微妙的联系。

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.