群作用下的度量空间粗几何与高指标问题

Higher index theory is an active research field in operator K-theory and non-commutative geometry. In this project, we shall study the Baum-Connes conjecture and the coarse Baum-Connes conjecture in a unified frame work which is formulated on general non-compact metric spaces under proper isometric actions by discrete groups. We are going to prove a general coarse geometric higher index theorem under a much weaker condition that the metric spaces are coarsely embeddable but not necessarily equivariantly coarsely ebmeddable into Hilbert space or other geometric spaces. On the other hand, we also aim to prove an equivariant version of the Strong Novikov conjecture on low degree cohomology classes for locally compact groups acting on non compact topological spaces which are not necessarily Riemannian manifolds. In spirit, we wish to obtain certain higher index theorems without using any geometric conditions on the groups. Finally, we will study the coarse Baum-Connes conjecture on warped cones in order to find new counterexamples to it. We will also study the coarse Baum-Connes conjecture on a newly founded box spaces of free groups which are not coarsely embeddable into Hilbert space and at the same time do not coarsely contain expanders. These topics are of great interest at the forefront of current research in operator algebras and noncommutative geometry. They should have many important applications in geometry, topology and analysis.

高指标理论是算子代数与非交换几何的重要研究方向。本项目拟在离散群等距适当作用于非紧度量空间的统一框架下研究Baum-Connes猜想与粗Baum-Connes猜想，在度量空间非等变粗嵌入Hilbert空间或其他空间的更弱条件下，证明一般的粗几何等变高指标定理。另一方面，我们将研究局部紧群作用于非流形拓扑空间的关于低阶上同调类的等变Novikov猜想，把离散群作用推广到连续群作用的情形，希望寻求不依赖于群的几何条件的高指标定理。此外，我们将研究翘曲锥的粗几何以寻求粗Baum-Connes猜想新的反例，并研究最近发现的既不能粗嵌入Hibert空间但又不包含膨胀图的一类度量空间上粗Buam-Connes猜想是否成立。本项目的研究内容是目前算子代数与非交换几何国际前沿领域的热点问题，在几何、拓扑、分析等领域中具有重要应用。

高指标问题是算子代数与非交换几何的重要研究方向。本项目在离散群或拓扑群等距作用于非紧空间的一般框架下研究“等变Novikov猜想”与“粗Baum-Connes猜想”。本项目完成了预定的主要研究目标，取得了若干重要成果：在离散群作用于度量空间的统一框架下以及一个较弱条件下，证明了“等变粗Novikov猜想”；在局部紧群不必自由、不必余紧作用于拓扑流形的非常一般的条件下证明了“等变Novikov猜想”；证明了单连通非正曲率完备黎曼流形的子空间上的“Lp-粗Novikov猜想”;对一大类不能粗嵌入Hilbert空间的相对膨胀图证明了“粗Baum-Connes猜想”等。本项目5篇代表性论文分别发表在Communications in Mathematical Physics、Journal of Noncommutative Geometry（2篇）、Israel Journal of Mathematics、Kyoto Journal of Mathematics等国际著名数学期刊上。本项目的研究成果对算子代数与非交换几何领域的重大问题“粗Baum-Connes猜想”与“粗Novikov猜想”的成立范围做出了很大拓展，对拓扑学中的Novikov猜想、几何学中的Gromov-Lawson-Rosenberg正标量曲率猜想的研究具有重要科学意义。