22 June 2017 from 15:30 to 16:30

The 11th Antonie F. Monna Memorial Lecture: Alan Sokal (New York University)

Title: Total positivity: a concept at the interface between algebra, analysis and combinatorics

Abstract:  A matrix M of real numbers is called totally positive if every minor of M is nonnegative. This somewhat bizarre concept from linear algebra has surprising connections with analysis --- notably polynomials and entire functions with real zeros, and the classical moment problem and continued fractions --- as well as combinatorics. I will explain briefly some of these connections, and then introduce a generalization: a matrix of polynomials (in some set of indeterminates) will be called coefficientwise totally positive if every minor of M is a polynomial with nonnegative coefficients. Also, a sequence (an)n≥0  of real numbers (or polynomials) will be called (coefficientwise) Hankel-totally positive if the Hankel matrix H = (ai+j)i,j ≥= 0 associated to (an) is (coefficientwise) totally positive. It turns out that many sequences of polynomials arising in enumerative combinatorics are (empirically) coefficientwise Hankel-totally positive; in some cases this can be proven using continued fractions, while in other cases it remains a conjecture.

Atlas Lecture Theater, Koningsberger building.

Refreshments 15h-15h30

Drinks 16h30-17h in mathematics library

Start date and time
22 June 2017 15:30
End date and time
22 June 2017 16:30