Spectra of locally symmetric spaces and their asymptotic properties

The talk will give an introduction to the spectral theory of quotients of semisimple Lie groups by lattices and of the associated locally symmetric spaces. Classic examples are provided by the lattices ${\rm SL} (n, \mathbb{Z})$ (and their finite index subgroups) in the semisimple Lie groups ${\rm SL} (n, \mathbb{R})$. Of particular interest is the case of noncompact quotients (which includes this series of examples). Here, even the existence of infinitely many eigenvalues in the discrete spectrum is not clear (and has been established only in the last decade). There are (to a large part conjectural) structural relationships between these spectra for different groups, and connections to number theory. An important tool is the trace formula introduced by Selberg (1956) and developed by Arthur (since 1978), which in this case is the result of a regularization procedure. We will discuss asymptotic counting problems for the discrete spectrum (Weyl's law and limit multiplicities).