**Sugata Mondal**

Indiana University, Bloomington, USA

August 3, 2017

**Small Eigenvalues of Riemannian surfaces**:
Any eigenvalue of the Laplace operator, acting on the function space of a hyperbolic surface $S$, below $1/4$ is called a {it small or exceptional} eigenvalue of $S$. Historically, Selberg's $1/4$-conjecture was a motivation for the study of these eigenvalues.
Existence of hyperbolic surfaces having small eigenvalues was first obtained by B. Randol Later P. Buser found a simpler construction of such surfaces. He also found initial bounds on the number of small eigenvalues of a given surface depending on the topology of the surface.
Later P. Schumtz sharpened the methods developed by Buser and from his bounds he (and later Buser also) conjectured that the number of these eigenvalues of a closed hyperbolic surface is at most the Euler characteristic of the surface. An extended version of this conjecture was proved by Otal and Rosas. In their paper Otal-Rosas asked if their result can be extended to all smooth surfaces. In a series of three papers, joint with Werner Ballmann and Henri