Abstract: The Heisenberg group plays a very important role in NCG, quantum mechanics, representation theory, and number theory. I shall give a general introduction to the idea of the unitary dual of a locally compact group and shall then focus on Heisenberg group and a characterization of its unitary dual via Stone-von Neumann theorem. I shall then indicate an application of the Selberg trace formula when one tries to decompose the representation of H on L^2 (\Gamma \H), where \Gamma is the standard integral lattice in H.

Abstract: A classical theorem of Fatou states that a bounded holomorphic function in the unit disc $\Delta \subset \mathbb C$ has radial limits almost everywhere on $\partial \Delta$. Ever since, the problem of making sense of boundary values of holomorphic functions (in one or more variables) has been an active area of research, often yielding far-reaching theories (think Hardy spaces). In this talk I will give an overview of two classical approaches to the problem, and will outline the idea of a new construction of boundary values of holomorphic functions for domains with non-smooth boundary.