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.

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.

Milnor conjectured in 1970 that each etale cohomology group of a field (mod 2 coefficients) should have a presentation with units as generators and simple quadratic relations (the ring with this presentation is now called the "Milnor K-theory" of the field). This was proven by Voevodsky, but the odd version (mod p coefficients for other primes) has been open until very recently, and had been known as the Bloch-Kato Conjecture.

Using certain norm varieties, constructed by Rost, and techniques from motivic cohomology, we now know that this conjecture is true. This talk will be a non-technical overview of the ingredients that go into the proof, and why this conjecture matters to non-specialists

Homological structure of finite modules over regular local rings is fairly well understood. The classical results date back to Serre, Auslander and Buchsbaum. On the other hand, little is known, in general, about the structure of infinite modules. In this talk, we will consider a class of such modules most important from the geometric point of view, namely those that arise as stalks of coherent sheaves over the source of a morphism with regular target. We will sketch the idea how to generalize the classical theory to the case of those modules, by a kind of fibre dimension reduction argument.