Mathematical Sciences: Analytical Gauge Theory; January 5-9, 1994; Las Cruces, New Mexico

小行星9314382 这个故事可以追溯到基本工作的黎曼，克莱因和贝蒂在本世纪之交，分类的表面和他们的高维类似物，所谓的流形。 松散的峰值，他们表明，对于每个表面都有一个几何定义的自然数称为它的亏格，它表征了某些变换的表面，即两个具有相同亏格的表面是不可区分的。 此外，给定任何自然数，都有一个具有该亏格的模型曲面（为了形象化亏格为g的曲面，想象一个咖啡杯的表面有g- 1个额外的把手）。 一个n维流形是一个局部看起来像n维空间的几何对象，就像一个曲面在一个足够小的区域看起来像一个平面一样。 阿尔伯特·爱因斯坦的相对论强调时空的四维流形，使科学界相信有必要接受大于二维的流形。 数学家在代数和微分拓扑学，微分几何等领域，物理学家已经投入了相当大的努力，在本世纪发展分类计划的流形的三维和更高。 虽然有很多是已知的，没有这样的计划已经建立，成功地分类流形的三维。 令人惊讶的是，在20世纪60年代，一个多级分类方案被开发用于五维及更大维的流形。 用来建立这种模式的方法根本不适用于较低的维度。 在20世纪80年代初迈克尔弗里德曼证明的结果，导致分类的一个大的自然家庭的四维流形，简单连接的，拓扑类型，coarshopping分类，一直在寻求。 他的部分工作是致力于建设许多流形以前不知道存在。 不久后，弗里德曼提出了他的顶级理论分类定理，西蒙唐纳森使用技术规范理论，主题的会议，以阐明一个更精细的分类四维流形：分类到光滑等价，或到超同态。 唐纳森的主要思想是通过分析可以与单连通光滑流形相关联的辅助结构空间的几何来研究单连通光滑四流形上的精细结构。 他的第一个结果是，几乎没有任何弗里德曼的新流形可以顺利，主要原因是它是如此难以找到他们。 后来的工作产生了一系列新的变种，用于区分给定粗糙，拓扑，弗里德曼类型内的光滑流形。 会议将回顾规范理论的性质，这对理解辅助结构空间的几何形状至关重要，并将澄清连接。 然后，演讲者将转向一种情况，它具有与唐纳森的规范理论相同的形式性质，但它的分析是不完整的，而且似乎有几条进一步研究的途径。

9314382 Staffeldt The story can be traced back to the fundamental work of Riemann, Klein, and Betti at the turn of this century, on the classification of surfaces and their higher dimensional analogues, called manifolds. Loosely peaking, they showed that for each surface there is a geometrically defined natural number called its genus, that characterizes the surface up to certain transformations, i.e. two surfaces with the same genus are indistinguishable. Moreover, given any natural number there is a model surface with that genus (to visualize a surface of genus g, imagine the surface of a coffee cup with g- 1 extra handles). A manifold of dimension n is a geometric object that looks locally like n dimensional space, in the same way that a surface looks in a small enough region, like a plane. Albert Einstein's relativity theory, with its emphasis on the four dimensional manifold of space-time, convinced the scientific world of the necessity to come to terms with manifolds of dimension greater than two. Mathematicians in the fields of algebraic and differential topology, differential geometry, among others, and physicists have devoted considerable effort in this century to developing classification schemes for manifolds of dimension three and higher. Although a great deal is known, no such schemes have been established that successfully classify manifolds of dimension three. Amazingly, in the 1960s a multistage classification scheme was developed for manifolds of dimension five and greater. The methods used to establish that scheme simply don't apply in the lower dimensions. In the early 1980s Michael Freedman proved results that led to the classification of a large natural family of four dimensional manifolds, the simply-connected ones, up to topological type, the coarsest classification that had been sought. Part of his work was devoted to constructing many manifolds not previously know to exist. Soon after Freedman produced his top ological classification theorem, Simon Donaldson used techniques of gauge theory, the subject of the conference, to shed light on a finer classification of four dimensional manifolds: the classification up to smooth equivalence, or up to diffeomorphism. Donaldson's main idea is to study the finer structure on simply connected smooth four manifolds by analyzing the geometry of spaces of auxiliary structures that can be associated with simply connected smooth manifolds. His first result was that hardly any of Freedman's new manifolds could be smooth, the main reason why it was so hard to find them. Later work has produced a host of new variants for distinguishing smooth manifolds within a given coarse, topological, Freedman type. The conference will review the properties of gauge theory which are crucial for understanding the geometry of the spaces of auxiliary structure and will clarify the connections. Then the speaker will turn to a situation which has the same formal properties of gauge theory a la Donaldson, but whose analysis is incomplete and for which several avenues for further research seem to be open.