Student Analysis Seminar: Physical and geometric isoperimetric inequalities on Riemannian manifolds

I will be discussing some fun facts and ideas regarding physical and geometric isoperimetric inequalities on Riemannian manifolds. The former regards the first eigenvalue of Laplacian (equivalently the Rayleigh quotient, or Poincaré inequality) and the latter regards the perimeter of domain. Classical theories follow from first variation and comparison with model spaces with constant curvature etc.