On geometric eigenvalue bounds

The aim of this talk is to introduce classical methods of dealing with eigenvalues on manifolds. We will begin with variational characterization via  minimax principles, then we will talk about topological restraints arising form Courant's nodal domain theorem. From there we turn towards isoperimetric problems introducing Cheeger constant and proving Cheeger's inequality - establishing a lower bound for smallest eigenvalue. If time allows we will show Buser's upper bound for $\lambda_1$ and provide some examples.