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

[9314382] Staffeldt这个故事可以追溯到本世纪初黎曼（Riemann）、克莱因（Klein）和贝蒂（Betti）关于表面及其高维类似物（称为流形）的分类的基本工作。他们指出，对于每个曲面，都有一个几何上定义的自然数，称为其属，它表征了曲面的某些变换，即具有相同属的两个曲面是无法区分的。此外，给定任何自然数，都存在一个具有该属的模型曲面（为了可视化g属的曲面，想象一个带有g- 1个额外把手的咖啡杯的表面）。一个n维的流形是一个几何物体，它在局部看起来像n维空间，就像一个表面在一个足够小的区域里看起来像一个平面一样。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研究大于2维的流形。代数和微分拓扑、微分几何等领域的数学家和物理学家在本世纪投入了相当多的精力来发展三维及更高维流形的分类方案。虽然我们知道很多，但还没有建立这样的方案来成功地对三维流形进行分类。令人惊讶的是，在20世纪60年代，针对五维及以上的流形开发了一种多阶段分类方案。用于建立该方案的方法根本不适用于较低维度。在20世纪80年代早期，Michael Freedman证明了一些结果，这些结果导致了一个大的四维流形的自然族的分类，单连通的，直到拓扑类型，这是所寻求的最粗略的分类。他的部分工作致力于构造许多以前不知道存在的流形。弗里德曼提出他的顶级分类定理后不久，西蒙·唐纳森（Simon Donaldson）利用规范论的技术，也就是这次会议的主题，阐明了四维流形的一种更精细的分类：到光滑等价的分类，或到差分同态的分类。Donaldson的主要思想是通过分析可与单连通光滑流形相关联的辅助结构空间的几何形状来研究单连通光滑四流形上的精细结构。他的第一个结论是，弗里德曼的新流形几乎没有一个是光滑的，这是很难找到它们的主要原因。后来的工作产生了许多新的变体，用于在给定的粗糙、拓扑、弗里德曼类型中区分光滑流形。会议将回顾规范理论的性质，这些性质对理解辅助结构空间的几何形状至关重要，并将澄清它们之间的联系。然后，演讲者将转向一种情况，这种情况与唐纳森的规范理论具有相同的形式性质，但其分析是不完整的，并且有几个进一步研究的途径似乎是开放的。

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.