Monopoles and 3-Manifolds

提案DMS-0303601 PI：Liviu Nicolaescu，圣母大学标题：单极和3-流形摘要孤立复杂二维奇点的拓扑 具有在任何其他维度上都没有表现出来的固有刚性，例如，在一个大的且明确界定的二维奇点族中， 可以确定奇点的脆弱解析不变量（如几何亏格、Milnor纤维的特征、Minor数等）。by performing执行robusttopological鲁棒topological拓扑manipulations操作.这些都是方便的编码theSeiberg-Witten不变量。 调查员打算调查背后的原因 这一惊人的现象， 研究Seiberg-Witten单极子的精细结构依赖于Seiberg-Witten方程的绝热变形。本建议的目标是 对奇异点进行微观分析，这些物体通常被称为“灾难”。人们可以把这些想象成表面，除了少数地方， 在那里人们看到形成“尖点”和“尖峰”，通常被称为“奇点”。 如果一个观察者坐在一个固定的点上环顾四周， 是圆的，球形的 奇点的“视界”是数学家们把它看作“奇点的纽带”的对象。 and is shaped形状quite相当differently不同than the 'horizon地平线' of a regular定期point点. 事实上，这个链接的形状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 被大量携带， 物理学家们称之为“天体”的物体。这些在许多方面与电磁波相似，但它们对电磁波更敏感。 宇宙的形状。 研究人员将通过以下方式分析这些单极子的结构： “通过高分辨率显微镜观察这些奇异点”，这个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.