The goal of these lectures is to discuss hoe methods of coarse geometry can be applied to estimate analytic quantities such as the eigenvalues of the Laplacian. As applications I will for instance discuss (1) the recurrence properties of the simple random walk on planar graphs, or (2) prove that every Riemannian manifold whose fundamental group surjects onto a free group admits a tower of covers which forms an expander. To a large extent the methods are rather elementary, although some familiarity with Riemannian geometry should be helpful.

The goal of these lectures is to discuss hoe methods of coarse geometry can be applied to estimate analytic quantities such as the eigenvalues of the Laplacian. As applications I will for instance discuss (1) the recurrence properties of the simple random walk on planar graphs, or (2) prove that every Riemannian manifold whose fundamental group surjects onto a free group admits a tower of covers which forms an expander. To a large extent the methods are rather elementary, although some familiarity with Riemannian geometry should be helpful