Algebraic Geometry of Difference Operators and Real Bundles

几何朗兰兹程序为代数曲线上的表示理论提出了一个非凡的组织原则，受到朗兰兹哲学的启发，该哲学将谐波分析和伽罗瓦理论结合在数域上。也就是说，它描述了谱理论在束的模空间上的一个几何类比。此外，在Beilinson和Drinfeld的工作中，复杂代数曲线（黎曼曲面）的全局理论源于局部到全局原理，其局部分量是共形场理论（它本身是弦理论的基础）的代数结构。我的建议，突出了与D. Nadler的合作，寻求发展一个几何朗兰兹程序的真实版本，描述了真实代数曲线上真实束的模空间的调和分析，扩大了与经典表示理论和弦理论相互作用的范围。这一扩展的一个强大推动力是它作为实半单李群的经典表示理论的增强的潜在作用。具体地说，我们提出，当代数曲线是射影线时，朗兰兹表示分类本质上是我们的程序的特殊情况。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，我们发现与快速出现的d膜物理理论，共形场论的边界条件有很强的相互作用。现代数学的一个基本主题是利用对称作为一种组织原则，在一个优雅的总体框架中将各种各样的、可能令人困惑的现象联系起来。也许这一趋势的主要例子是朗兰兹程序，它确定了数论中对称现象的一般模式，并将费马大定理的解列为其成功之一。近年来，朗兰兹哲学被应用于新的和更多的几何方向，特别是曲面的几何，在那里它与对称原理联系在一起，而对称原理是物理学中弦理论令人兴奋发展的基础。我目前的研究计划（与D. Nadler一起）建议对这种几何朗兰兹程序进行扩展，以组织与边界或端点曲面相关的对称性。这个扩展有两个主要的新颖的吸引力。一方面，它试图包含，并因此提供新的几何光，在涉及实数的对称性研究的经典主题。当涉及的表面只是一个圆盘时，这个主题自然出现。另一方面，它表明了与弦理论中最活跃的兴趣领域之一密切的新联系，弦可以附着或结束的膜的研究。因此，这一建议提供了朗兰兹哲学的多功能性和统一吸引力的新实例。

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.