Group Representations and the Baum-Connes Assembly Map

这个项目将探索非交换几何的新兴领域和李群的表示理论之间的新联系。 该项目的第一部分最初受到数学物理探索的启发。它恢复了乔治麦基的一项建议，即把半单李群的表象与其嘉当运动群的表象对应起来，但它是根据非对易几何的最新发展来这样做的。该项目的第二个组成部分，旨在发展之间的联系，指数理论的方法，表征理论纳入鲍姆-康纳斯理论和几何表征理论的贝林森和伯恩斯坦。该项目的第三部分旨在更深入地研究鲍姆-康纳斯理论和不可约表示的朗兰兹分类之间的联系。该项目的最后一个主要部分的目标是在非对易几何中的辛几何中构建“量子化与还原交换”现象。 预计这将有助于更清楚地了解这一现象的范围。虽然该项目最初涉及相对较好理解的紧致群表示理论，但该项目的长期目标是在各个组成部分的观点指导下，更广泛地应用表示理论中获得的见解。群表示理论是现代数学中反复出现的主题。 它的起源在于代数，但该主题与几何，微分方程和许多其他数学领域有重要联系。 这个项目的重点是在数学上捕捉连续对称概念的群体（例如圆的连续旋转对称，它可以绕自身旋转任何角度，而不是正方形的离散对称，它只能绕自身旋转四分之一圈）。由于物理空间和时间固有的连续对称性（旋转、平移等），这些群是物理定律数学表达的基础。物理科学中的基本可观测量，如能量和动量，都与这些对称性配对。例如，能量守恒定律是对物理定律随时间推移保持不变的期望的重述。在量子理论中，空间和时间的对称性意味着基本粒子对应于所谓的不可约酉群表示，确定这些表示成为一个有趣和重要的问题。在数学中，同样的表示被发现与特殊函数理论、数论和微分方程等领域密切相关。这是该项目的各个组成部分的共同目标，使新的数学技术承担表示论。长期目标是概念化和加深我们对一个一百多年来一直是数学及其应用中心的领域的理解。

This project will explore new connections between the emerging area of noncommutative geometry and the representation theory of Lie groups. The first part of the project was inspired initially by explorations in mathematical physics. It revives a proposal of George Mackey to correspond representations of a semisimple Lie group with those of its Cartan motion group, but it does so in the light of more recent developments in noncommutative geometry. The second component of the project seeks to develop links between the index-theoretic approach to representation theory incorporated into the Baum-Connes theory and the geometric representation theory of Beilinson and Bernstein. A third segment of the project aims to investigate more deeply the connection between the Baum-Connes theory and the Langlands classification of irreducible representations. The goal of the final major portion of the project is to frame the "quantization commutes with reduction" phenomenon in symplectic geometry within noncommutative geometry. It is expected that this will lead to a clearer understanding of the range of the phenomenon. Although the project initially involves the relatively well-understood representation theory of compact groups, a long-term aim of the project is to apply insights gained more broadly within representation theory, guided by the outlooks of the individual components.Group representation theory is a recurring theme in modern mathematics. Its origins lie within algebra, but the subject has important ties to geometry, to differential equations, and to many other mathematical areas. This project focuses on the groups that capture mathematically the concept of continuous symmetry (such as the continuous, rotational symmetry of a circle, which may be rotated about itself by any angle, as opposed to the discrete symmetry of a square, which may be rotated about itself only by quarter turns). These groups are basic to the mathematical expression of the laws of physics, thanks to the continuous symmetries (rotations, translations, and others) intrinsic to physical space and time. The fundamental observable quantities in physical science such as energy and momentum are paired with these symmetries. For example, the law of conservation of energy is a restatement of the expectation that the laws of physics remain unchanged as time passes. In quantum theory, the symmetries of space and time imply that fundamental particles correspond to so-called irreducible unitary group representations, and it becomes a matter of interest and importance to determine these representations. Within mathematics, the same representations have been found to be intimately linked to the theory of special functions, to number theory, and to differential equations, among other areas. It is the common objective of the various components of this project to bring new mathematical techniques to bear on representation theory. The long-term goal is to conceptualize and deepen our understanding of an area that has been central to mathematics and its applications for more than a hundred years.