The ``manifold hypothesis'' posits tha t many high dimensional data sets that occur in the real world actually are actually scattered on or around a much lower dimensional manifold embedded in the high dimensional space. Estimating attributes of this "ground truth " manifold from finitely many samples (point cloud) is a problem of statist ical inference. Given such a point cloud that is modelled as an independent and identically distributed (i.i.d) sample from a (nice) density on a clos ed manifold, over the past decade there is a body of literature which consi ders the question: forming a random geometric (weighted) graph on the point cloud (by, for example, joining points that are within a threshold distanc e by a weighted edge) how well can one estimate the spectrum (eigenvalues, eigenfunctions) of the (weighted) Laplace-Beltrami operator on the ground t ruth manifold, from that of the graph laplacian associated with the random geometric graph?

After introducing the problem, we will show how this question is one of ``stochastic homogenization'', a traditio nally well-studied theme in partial differential equations originating in t he theory of composite materials. Warming up with results that are new even for the classical "periodic" homogenization problem, we will describe how one can obtain optimal convergence rates for the spectrum of the graph lapl acian using tools from the recent theory of quantitative stochastic homogen ization. Briefly: borrowing tools from percolation theory, the argument pro ceeds by ``coarse-graining'' the random geometry in the problem to lar ge scales, where the environment "appears Euclidean". Then, one adapts &nbs p;arguments from the more recent quantitative theory of homogenization.&nbs p;

This talk represents joint work with Scott Armstrong (Courant).