Group Representations and the Baum-Connes Assembly Map

本项目将探索新兴的非交换几何领域与李群表示理论之间的新联系。项目的第一部分最初受到数学物理探索的启发。它恢复了George Mackey提出的将半单李群的表示与其Cartan运动群的表示相对应的建议，但它是根据非交换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Baum-Connes理论中表征理论的指数理论方法与Beilinson和Bernstein的几何表征理论之间的联系。该项目的第三部分旨在更深入地研究Baum-Connes理论和Langlands不可约表征分类之间的联系。该项目的最后一个主要部分的目标是在非交换几何中构造辛几何中的“量化交换与约简”现象。预计这将使人们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凑群体的表示理论，但该项目的长期目标是在单个组成部分的前景的指导下，更广泛地应用在表示理论中获得的见解。群表示理论是现代数学中一个反复出现的主题。它起源于代数，但这门学科与几何、微分方程和许多其他数学领域有着重要的联系。这个项目的重点是在数学上捕捉连续对称概念的群体（比如一个圆的连续旋转对称，它可以围绕自己旋转任何角度，而不是一个正方形的离散对称，它只能围绕自己旋转四分之一圈）。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。物理科学中基本的可观测量，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着时间的流逝而保持不变的期望的重申。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，相同的表示被发现与特殊函数理论、数论、微分方程以及其他领域密切相关。这个项目的各个组成部分的共同目标是为表示理论带来新的数学技术。长期目标是概念化和加深我们对一个领域的理解，这个领域在一百多年来一直是数学及其应用的核心。

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.