Cheeger constant, spectral clustering and eigenvalue ratios of Laplacian

In this talk, I will explain an optimal dimension-free upper bound for eigenvalue ratios $\lambda_k/\lambda_1$ of the Laplacian on a closed Riemannian manifold with nonnegative Ricci curvature. This is achieved by borrowing tools from theoretical spectral clustering algorithm analysis in computer science. I will further discuss several of its applications, including improving higher-order Buser inequality, higher-order Gromov-Milman inequality and multi-way isoperimetric constant ratios estimate. Its extension to compact finite-dimensional Alexandrov spaces with nonnegative curvature affirms a recent conjecture of Funano and Shioya. On finite discrete graphs, it also has very interesting applications.