In this talk I will give a reasonably self-contained overview of the C*-algebraic approach to non-equilibrium quantum statistical mechanics with emphasis on the recent developments.

We present a two compartment model for tumor dormancy based on an idea of Zetter to wit: The vascularization of a secondary (daughter) tumor can be suppressed by inhibitor originating from a larger primary (mother) tumor. We apply this idea at the avascular level to develop a model for the remote suppression of secondary avascular tumors via the secretion of primary avascular tumor inhibitors. The model gives good agreement with experimental observation (Derm. Surg. 29(2003) 664-667). The authors reported on the emergence of a polypoid melanoma at a site remote from a primary polypoid melanoma after excision of the latter . The authors observed no recurrence of the melanoma at the primary site, but did observe secondary tumors at secondary sites five to seven centimeters from the primary site within a period of one month after the excision of the primary site. We attempt to provide a reasonable biochemical/cell biological model for this phenomenon. We show that when the tumors are sufficiently remote, the primary tumor will not influence the secondary tumor while, if they are too close together, the primary tumor can effectively prevent the growth of the secondary tumor, even after it is removed. It should be possible to use the model as the basis for a testable hypothesis which could be checked in a controlled in vitro experiment.

Einstein's kinetic theory of the Brownian motion, based upon light
water molecules continuously bombarding the heavy pollen, provided an explanation of diffusion from the Newtonian mechanics. Since the discovery of quantum mechanics it has been a challenge to verify the emergence of diffusion from the Schrodinger equation. In this talk I will report on a mathematically rigorous derivation of a diffusion equation
as a long time scaling limit of a random Schr\"odinger equation in a weak, uncorrelated disorder potential. This is a joint work with M. Salmhofer and H.T. Yau.

I will try to give a broad review of the amasing
developments in the theory of infinite groups during the last 25 years. These include the
emergence of Monsters and flourishing of the Asymptotic Theory of Finite Groups. We will focus on important examples and formulate some open problems.