## Wolfgang Gaim : Semiclassical approximations to quantum mechanical equilibrium distributions

- Presentations ( 227 Views )In his 1932 paper, Eugene Wigner introduced the now famous Wigner function in order to compute quantum corrections to classical equilibrium distributions. We show how to extend this program and compute semiclassical approximations to quantum mechanical equilibrium distributions for slow, semiclassical degrees of freedom coupled to fast, quantum mechanical degrees of freedom. The main examples are molecules and electrons in crystalline solids. Where we will focus on the thermodynamics of the Hofstadter model as an application of the general results. The semiclassical formulas contain, in addition to quantum corrections similar to those of Wigner, also modifications of the classical Hamiltonian system used in the approximation: The classical energy and the Liouville measure on classical phase space turn out to have non-trivial-expansions in the semiclassical parameter. This talk is based on joint work with Stefan Teufel.

## Ellen Eischen : L-functions, congruences, and applications

- Presentations ( 179 Views )L-functions, certain meromorphic functions that include the Riemann zeta-function, encode important number-theoretic information. The first part of this talk will focus on some striking properties of special values of the Riemann zeta-function and certain other L-functions (namely, congruences modulo powers of a prime number). In the second part of the talk, I will introduce tools that are useful for studying these congruences. These tools have applications not only to number theory, but also to homotopy theory. This will be a colloquium style talk, intended for a broad audience.