Monopoles and 3-Manifolds

论文题目：单极子和3流形。摘要孤立的复杂二维奇异点的拓扑结构具有内在的刚性，这种刚性在其他任何维度上都不表现出来。例如，在一个大而明确划分的二维奇点族中，人们可以通过执行鲁棒拓扑操作来确定奇点的脆弱解析不变量（如几何属、米尔诺纤维的签名、Minor数等）。这些都方便地用theseberg - witten不变量编码。研究者打算根据Seiberg-Witten方程的绝热变形，通过观察Seiberg-Witten单极子的基本结构来研究这一令人惊讶的现象背后的原因。本提案的目标是对奇点进行微观分析，奇点通常被称为“灾难”。我们可以把这些表面想象成除了在少数地方可以看到形成“尖峰”和“尖峰”，数学上称为“奇点”的表面之外，几乎在任何地方都很好。如果一个观察者坐在一个规则的点上环顾四周，他/她观察到的“地平线”是圆的，球形的。奇点的“视界”被数学家称为“奇点的连接点”，其形状与常规点的“视界”大不相同。事实上，这个连接的形状本身就包含了大量关于奇点附近空间如何扭曲、弯曲和折叠的信息。这种信息被物理学家称为“单极子”的某些物体大量携带。它们在许多方面与电磁波相似，但它们对所处宇宙的形状更为敏感。研究人员将通过“用高分辨率显微镜观察这些奇点”来分析这些单极子的结构，这一过程在数学上被称为空间的绝热变形。这项技术已经产生了令人鼓舞的初步结果。

PROPOSAL DMS-0303601PI: Liviu Nicolaescu, University of Notre DameTitle: Monopoles and 3-manifoldsABSTRACTThe topology of isolated complex two-dimensional singularities has abuilt-in rigidity which is not manifested in any other dimensions.For example, in a large and clearly delimited family oftwo dimensional singularities, one can determine fragile analyticinvariants of the singularity (such as the geometric genus, the signatureof the Milnor fibers, the Minor number etc.) by performing robusttopological manipulations. These are conveniently encoded by theSeiberg-Witten invariants. The investigator intends to investigate thereasons behind this surprising phenomenon by looking into the finerstructure of Seiberg-Witten monopoles relying on an adiabatic deformationof the Seiberg-Witten equations.The goal of the present proposal is to perform a micro-analysis ofsingularities, objects popularly known as `catastrophes'. One couldvisualise these as surfaces which are nice most everywhere except atfew places where one sees forming `cusps' and `spikes', mathematicallyreferred to as `singular points'. If an observer sits at a regular pointand looks around, the `horizon' he/she observes is round, spherical. The`horizon ' of a singular point is an object mathematicians referto as `the link of the singularity' and is shaped quite differentlythan the `horizon' of a regular point. In fact, the shape of this linkalone contains a wealth of information about how the space in thevicinity of the singular point twists, bends and folds. This kind ofinformation is carried in abundance by certain objects thephysicists refer to as `monopoles'. These are similar in many respects tothe electromagnetic waves, but they are more sensitive to the shape ofthe Universe they travel in. The investigator will analyze thestructure of these monopoles by `looking at these singularitiesthrough a high resolution microscope', a process mathematically know asadiabatic deformation of the space. This technique has already producedencouraging preliminary results.