at the

American Institute of Mathematics, San Jose, California

organized by

Julius Borcea, Petter Branden, George Csordas, and Victor Vinnikov

- Polya-Schur problems: classification of linear preservers of polynomials and entire functions in one or several variables with prescribed zero sets.
- Lax-type problems: determinantal representations of multivariate Gårding-hyperbolic polynomials and related objects.
- Properties and applications of stable polynomials and polynomials with the half-plane property.

The recently established Lax conjecture has already proved to be very fruitful and is expected to have many more far-reaching consequences as we are yet to fully explore it. In particular, appropriate analogs of the Lax conjecture for multivariate (real) stable polynomials - i.e., polynomials which are non-vanishing whenever the imaginary parts of its variables are all positive - would be most useful. Indeed, in just a few years these polynomials have become an important tool in several apparently unrelated areas (matroid theory, quantum statistical mechanics, the theory of Hermitian matrices, negative dependence/probability theory) and seem to provide an appropriate framework for studying a number of difficult problems in these areas.

The workshop schedule.

A report on the workshop activities.

Group photo and another group photo

Papers arising from the workshop:

On the stability of Taylor sections of a function $\sum_k=0^\inftyz^k/a^k^2, a > 1$

by Olga M. Katkova and Anna M. Vishnyakova

A sufficient condition for a polynomial to be stable

by Olga M. Katkova and Anna M. Vishnyakova

Negative dependence and the geometry of polynomials

by Julius Borcea, Petter Brändén, and Thomas M. Liggett

Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products

by Julius Borcea and Petter Brändén

Polya-Schur master theorems for circular domains and their boundaries

by Julius Borcea and Petter Brändén

Multivariate Polya-Schur classification problems in the Weyl algebra

by Julius Borcea and Petter Brändén

Additive invariants on quantum channels and applications to regularized minimum entropy

by Shmuel Friedland

Remarks on BMV conjecture

by Shmuel Friedland

Matrices Totally Positive Relative to a Tree

by Charles R. Johnson, Roberto S. Costas-Santos, and Boris Tadchiev