Abstract: Spectral theory on locally symmetric spaces is closely related to the theory of automorphic forms and has deep connections with number theory. Spectra of canonical differential operators on quotients of symmetric spaces by discrete groups, which are defined arithmetically, are expected to carry fundamental arithmetic information. In this talk I will give an introduction to the subject. Then I will discuss some aspects of spectral theory on locally symmetric spaces, especially the Weyl law and the limit multiplicity problem. The analytic properties of automorphic L-functions have a significant impact on the structure of the spectrum. This is another issue which I will discuss in some detail.
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