Index Theory and K-Theory for Operator Algebras

小行星9704001 本研究项目围绕P. Baum和A.康纳斯 如果这个猜想成立，那么它就回答了理解和计算群C*-代数的K-理论的问题。 猜想的有效性意味着诺维科夫猜想（高阶签名的同伦不变性）、稳定的格罗莫夫-劳森-罗森伯格猜想（闭自旋流形允许正标量曲率的黎曼度量的充分必要条件）和凯迪森-卡普兰斯基猜想（无挠离散群的约化C*-代数中不存在幂等元）的有效性。 当应用于李群时，该猜想精确地将半单李群G的调和表示理论与Mackey和Wigner关联到G的半直积李群的表示理论之间的Mackey-Wigner类比。 因此，这个猜想是不寻常的，因为它跨越了几个不同的数学领域，并统一了许多以前似乎不相关的问题和问题。 世纪数学的一个主题是数学系统的某些特征乍一看似乎是分析的（即，基于诸如微分和积分的微积分方法）实际上是拓扑的（即，只依赖于初等连续几何）。 该研究项目在非交换几何的新背景下发展了这一主题。 一个猜想（由P. Baum和A. Connes）将研究局部紧群的解析和拓扑不变量。 如果是真的，这个猜想将是一个潜在的原则，揭示了一个意想不到的统一在一些数学问题和问题。 ***

9704001 Baum This research project centers on a conjecture formulated by P. Baum and A. Connes. If true, this conjecture gives an answer to the problem of understanding and calculating the K-theory of group C*-algebras. Validity of the conjecture implies validity for the Novikov conjecture (homotopy invariance of higher signatures), the stable Gromov-Lawson-Rosenberg conjecture (necessary and sufficient conditions for a closed Spin manifold to admit a Riemannian metric of positive scalar curvature), and the Kadison-Kaplansky conjecture (non-existence of idempotents in the reduced C*-algebra of a torsion-free discrete group). When applied to Lie groups, the conjecture makes precise the Mackey-Wigner analogy between the tempered representation theory of a semi-simple Lie group G and the representation theory of the semi-direct product Lie group that Mackey and Wigner associate to G. Thus the conjecture is unusual in that it cuts across several different areas of mathematics and unifies a number of problems and issues that previously appeared to be unrelated. A theme in nineteenth and twentieth century mathematics has been that certain features of mathematical systems that at first glance seem to be analytical (i.e., based on methods of calculus such as differentiation and integration) in fact are topological (i.e., depend only on elementary continuous geometry). This research project develops this theme within the new context of non-commutative geometry. A conjecture (formulated by P. Baum and A. Connes) will be studied that relates analytic and topological invariants of locally compact groups. If true, the conjecture will be an underlying principle revealing an unexpected unity in a number of mathematical problems and issues. ***