K-homology and Noncommutative Geometry of Brieskorn Varieties

Abstract: For each integer k=1,...,28, consider the zero set of the complex polynomial z_1^2+z_2^2+z_3^2+z_4^2+z_5^{6k-1} in C^5, intersected with a sufficiently small (standard) sphere centered at origin. Brieskorn showed that these intersections represent all possible differentiable structures (up to oriented diffeomorphism) on topological 7-dimensional sphere, one standard and 27 exotic. In this talk, I discuss K-homology invariants suggested by the Arveson-Douglas conjecture, why they are not enough (in their natural setting) to detect exotic spheres, and then report on an ongoing research with Xiang Tang and Guoliang Yu, on a noncommutative analogue of Milnor's monodromy map which could detect exotic spheres. 

Host: Yanli Song