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

小行星9401440 BC（= Baum-Connes）猜想是与拓扑学（即连续几何学）和分析学（即基于多元微积分的数学分支）有关的一个非常一般的等式。 这个猜想是不寻常的，因为它跨越了几个不同的数学领域，从而揭示了一个潜在的统一性，这是以前完全未知的。 这里的研究集中在证明这一猜想的某些类别的例子，然后获得推论。 更精确地说，设G是一个Hausdorff、局部紧且第二可数的拓扑群。 主要的例子是李群，p-adic群和离散群。 C * G表示G的约化C *-代数，KC * G是它的K-理论。 BC猜想如果成立，则给出了计算KC * G问题的答案。 该猜想的有效性在几何拓扑学和表示论中的著名问题上有应用。 ***

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. ***