We characterize linear operators preserving zero-restrictions on entire functions in weighted Bargmann-Fock spaces. This extends the characterization of linear operators on polynomials preserving stability (due to Borcea and the author) to the realm of entire functions, and translates into an optimal, albeit formal, Lee-Yang theorem.

We study the asymptotic distribution of the number Z N of zeros of random trigonometric polynomials of degree N as N →∞. It is known that as N grows to infinity, the expected number of the zeros is asymptotic to N. The asymptotic form of the variance was predicted by Bogomolny, Bohigas and Leboeuf to be cN for some c > 0. We prove that converges to the standard Gaussian. In addition, we find that the analogous result is applicable for the number of zeros in short intervals.

Let F be a finitely generated field of characteristic zero and Gamma <= GL(n) (F) a finitely generated subgroup. For gamma is an element of Gamma, let Gal(F(gamma)/F) be the Galois group of the splitting field of the characteristic polynomial of gamma over F. We show that the structure of Gal(F(gamma)/F) has a typical behavior depending on F, and on the geometry of the Zariski closure off Gamma (but not on Gamma).

4. Petrosyan, Arshak

et al.

Shahgholian, Henrik

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

We consider an obstacle-type problem Delta u = f(x)chi(Omega) in D, u = vertical bar del u vertical bar = 0 on D\Omega, where D is a given open set in R-n and Omega is an unknown open subset of D. The problem originates in potential theory, in connection with harmonic continuation of potentials. The qualitative difference between this problem and the classical obstacle problem is that the solutions here are allowed to change sign. Using geometric and energetic criteria in delicate combination we show the C-1,C-1 regularity of the solutions, and the regularity of the free boundary, below the Lipschitz threshold for the right-hand side.