We distinguish a class of random point processes which we call
Giambelli compatible point processes. Our definition was partly
inspired by determinantal identities for averages of products and
ratios of characteristic polynomials for random matrices.
It is closely related to the classical Giambelli formula for Schur symmetric functions.

We show that orthogonal polynomial ensembles, z-measures on
partitions, and spectral measures of characters of generalized
regular representations of the infinite symmetric group generate
Giambelli compatible point processes. In particular, we prove
determinantal identities for averages of analogs of characteristic
polynomials for partitions.

Our approach provides a direct derivation of determinantal
formulas for correlation functions

Phase transitions in classical spin systems are well understood,
however phase transitions in quantum spin systems are not ... at least
that is what most mathematicians would say. (If anything, they might
question how well we even understand classical spin systems.) Physicists,
on the other hand, say that if the classical model has a phase transition,
then the quantum model does as well. We will prove that, for some special
models. The main tools are reflection positivity, coherent states, and a
new result which generalizes the Berezin-Lieb inequality to the level of
matrix elements.

The subject of the talk is the spectrum of a two-dimensional
Schrodinger operator with constant magnetic field and a compactly supported electric potential. The eigenvalues of such an operator form clusters around the Landau levels.
The eigenvalues in these clusters accumulate towards the Landau levels super-exponentially fast. It appears that these eigenvalues can be related to a certain sequence of orthogonal polynomials in the complex domain. This allows one to accurately describe the rate of accumulation of eigenvalues towards the Landau levels. This description involves the logarithmic capacity of the support of the electric potential. The talk is based on a joint work with Nikolai FIlonov from St.Petersburg.