Mathematical Sciences: K-Theory for Operator Algebras, Index Theory, Riemann-Roch

BC （= Baum- connes）猜想是一个与拓扑（即连续几何）和分析（即基于多变量微积分的数学分支）相关的极其一般的等式。这个猜想是不寻常的，因为它跨越了数学的几个不同领域，从而揭示了一个以前完全未知的潜在统一。这里的研究集中在对某些类别的例子证明这个猜想，然后得到推论。更精确地说，设G是一个Hausdorff，局部紧，二次可数的拓扑群。主要的例子是李群、p进群和离散群。C*G表示G的约简C*代数，KC*G表示它的k理论。BC猜想，如果成立，给出了计算KC*G问题的答案。该猜想的有效性可以应用于几何拓扑学和表示理论中的一些著名问题。＊＊＊

9401440 Baum The BC (= Baum-Connes) conjecture is an extremely general equality relating topology, i.e. continuous geometry, and analysis, i.e. the branch of mathematics based on multi-variable calculus. This conjecture is unusual in that it cuts across several different areas of mathematics and thus reveals an underlying unity which previously was completely unknown. The research pursued here centers on proving this conjecture for certain classes of examples and then obtaining corollaries. More precisely, let G be a topological group which is Hausdorff, locally compact and second countable. The main examples are Lie groups, p-adic groups and discrete groups. C*G denotes the reduced C*-algebra of G, and KC*G is its K-theory. The BC conjecture, if true, gives an answer to the problem of calculating KC*G. Validity of the conjecture has applications to well-known problems in geometry-topology and representation theory. ***