Small Scale Index Theory, Scalar Curvature, and Gromov's Simplicial Norm

Explore the relationship between scalar curvature and topological constraints in Riemannian manifolds through this mathematical lecture by Qiaochu Ma from Texas A&M University. Examine how scalar curvature, which encodes volume information of small geodesic balls and represents the weakest curvature invariant, imposes specific topological limitations on manifolds. Learn about groundbreaking research demonstrating that for manifolds with scalar curvature lower bounds (including negative values), the simplicial norm of the Poincaré dual of the A-hat class can be controlled. Discover the connections between small scale index theory, scalar curvature analysis, and Gromov's simplicial norm through this collaborative work with Guoliang Yu. Gain insights into advanced differential geometry concepts and their applications in understanding the geometric-topological interplay in Riemannian manifolds.