Counting function and multiplicity of the Laplace eigenvalues

In this talk, we study geometric upper bounds for the multiplicity and counting function of the Laplacian eigenvalues. We discuss some of classical upper bounds due to Cheng, Gromov and Buser. Then we extend these results to domains with Dirichlet and Neumann boundary conditions. The idea of the proof and some interesting open questions will be discussed. This talk is based on joint work with G. Kokarev and I. Polterovich.