Georgia Institute of Technology College of Sciences

We consider permanental and multivariate negative binomial distributions. We give sim- ple necessary and sufficient conditions on their kernel for infinite divisibility, without symmetry hypothesis. For existence of permanental distributions, conditions had been given by Kogan and Marcus in the case of a 3 × 3 matrix kernel: they had showed that such distributions exist only for two types of kernels (up to diagonal similarity): symmet- ric positive-definite matrices and inverse M-matrices. They asked whether there existed other classes of kernels in dimensions higher than 3. We give an affirmative answer to this question, by exhibiting (in any finite dimension higher than 3) a family of matrices which are kernels of permanental distributions but are neither symmetric, nor inverse M-matrices (up to diagonal similarity). Analog properties (by replacing inverse M-matrices by entrywise non-negative matrices) are given for multivariate negative binomial distribu- tions. These notions are also linked with the study of inverse power series of determinant. This is a joint work with N. Eisenbaum.

Dehn surgery is a fundamental tool for constructing oriented 3-Manifolds. If we fix a knot K in an oriented 3-manifold Y and do surgeries with distinct slopes r and s, we can ask under which conditions the resulting oriented manifold Y(r) and Y(s) might be orientation preserving homeomorphic. The cosmetic surgery conjecture state that if the knot exterior is boundary irreducible then this can't happen. My talk will be about the case where Y is an homology sphere and K is an hyperbolic knot.

This is an informal get-together of the Joint Meetings participants and locals interested in various aspects of Numerical Algebraic Geometry. This area combines numerical analysis and nonlinear algebra in algorithms that found various applications in other parts of mathematics and outside. (If interested in joining, email leykin@math.gatech.edu)

I will present a discrete family of multiple orthogonal polynomials defined by a set of orthogonality conditions over a non-uniform lattice with respect to different q-analogues of Pascal distributions. I will obtain some algebraic properties for these polynomials (q-difference equation and recurrence relation, among others) aimed to discuss a connection with an infinite Lie algebra realized in terms of the creation and annihilation operators for a collection of independent ascillators. Moreover, if time allows, some vector equilibrium problem with constraint for the nth root asymptotics of these multiple orthogonal polynomials will be discussed.