Algebraic Geometry of Difference Operators and Real Bundles

DMS-0401448大卫本-兹维几何朗兰兹计划提出了一个非凡的组织原则表示理论的代数曲线，灵感来自朗兰兹哲学，结合谐波分析和伽罗瓦理论在数域。也就是说，它描述了一个几何模拟的谱理论上的模空间的丛。此外，在贝林森和德林费尔德的著作中，复代数曲线（黎曼曲面）的整体理论源自局部到整体原理，其局部成分是共形场论（它本身就是弦论的基础）的代数结构。我的建议，强调与D。纳德勒，旨在发展一个真实的版本的几何朗兰兹计划，描述了调和分析的模空间的真实的丛的真实的代数曲线，扩大了范围的相互作用与经典表示论和弦理论。一个强大的推动力，这种扩展是其潜在的作用，作为一个增强的经典表示理论的真实的半单李群。具体来说，我们建议，朗兰兹分类的表示出现基本上是我们的程序的特殊情况下，代数曲线是投影线。另一个动机来自于它在有边界的黎曼曲面上描述的局部特征。在这里，我们发现了与迅速兴起的D-膜物理理论、共形场论的边界条件之间的强烈相互作用。现代数学的一个基本主题是利用对称性作为一种组织原则，在一个优雅的总体框架中将各种各样的和潜在的令人困惑的现象联系起来。也许这种趋势的最好例子是朗兰兹纲领，它确定了数论中对称性出现的一般模式，并将费马大定理的解算在其成功之列。近年来，朗兰兹哲学被应用于新的、更多的几何学方向，特别是曲面几何学，在那里，它与物理学中令人兴奋的弦理论发展背后的对称性原理发生了联系。我目前的研究计划（与D。Nadler）建议对这个几何朗兰兹程序进行扩展，以组织与具有边界或端部的表面相关的对称性。这个扩展有两个主要的新颖的吸引力。一方面，它试图涵盖，从而摆脱新的几何光，在研究对称性涉及真实的数字的经典话题。当所涉及的表面仅仅是一个圆盘时，这个主题自然出现。另一方面，它也暗示了与弦理论中最活跃的研究领域之一--弦可以附着或终止的膜的研究--的一种新的密切联系。因此，这一建议提供了一个新的例子，说明朗兰兹哲学的多功能性和统一性。

DMS-0401448 David Ben-ZviThe geometric Langlands program proposes an extraordinary organizing principle for representation theory over algebraic curves, inspired by the Langlands philosophy which binds harmonic analysis and Galois theory over number fields. Namely, it describes a geometric analog of spectral theory on moduli spaces of bundles. Moreover in the work of Beilinson and Drinfeld this global theory for complex algebraic curves (Riemann surfaces) arises from a local-to-global principle, whose local components are algebraic structures underlying conformal field theory (which itself underlies string theory). My proposal, highlighting joint work with D. Nadler, seeks to develop a real version of the geometric Langlands program, describing a harmonic analysis on moduli spaces of real bundles on real algebraic curves, widening the scope of interactions with both classical representation theory and string theory. A strong impetus for this extension is its potential role as an enhancement of the classical representation theory of real semisimple Lie groups. Specifically, we propose that the Langlands classification of representations appears essentially as the special case of our program when the algebraic curve is the projective line. Another motivation arises from the local features of its description on Riemann surfaces with boundary. Here we find a strong interplay with the rapidly emerging physical theory of D-branes, the boundary conditions of conformal field theory.A fundamental theme of modern mathematics is the exploitation of symmetry as an organizing principle, linking diverse and potentially baffling phenomena in an elegant overarching framework. Perhaps the prime example of this trend is the Langlands program, which identifies a general pattern in the appearance of symmetry in number theory, and counts among its successes the solution of Fermat's Last Theorem. In recent years, the Langlands philosophy has been applied in new and more geometric directions, in particular the geometry of surfaces, where it makes contact with the symmetry principles that underly the exciting developments of string theory in physics. My current research proposal (with D. Nadler) suggests an extension of this geometric Langlands program to organize the symmetries associated to surfaces with boundaries or ends. This extension has two main novel attractions. On the one hand, it seeks to encompass, and thereby shed new geometric light on, a classical topic in the study of symmetries involving real numbers. This topic appears naturally in the case when the surface involved is simply a disc. On the other hand, it suggests an intimate new link with one of the most active areas of interest in string theory, the study of the membranes where strings can attach themselves, or end. Thus this proposal provides a new instance of the versatility and unifying appeal of the Langlands philosophy.