AEC 2019

Invited Speakers

• Nicolas Broutin (Sorbonne Université, France)

  Scaling limits of random trees and random graphs:
  Abstract: Since the pioneering work of Aldous in the 90s about the asymptotic shape of large random trees, the connection between combinatorial structures and stochastic processes such as Brownian motion have been proven very fruitful. I will present some essential aspects of the theory for random trees, and random graphs that allow to prove convergence of these as measured metric spaces. I will also try to point in the directions of recent developments and applications.

• George Labahn (University of Waterloo, Canada)

  Order Bases: Applications and Computation
  Abstract: Order Bases takes as input a vector or matrix of power series F and describes all solutions (as a module) for approximation problems of the form F p = O(z^ω) with ω a scalar or a vector. These approximation problems date back to the work of Hermite and his student Padé and later contributions for Order bases were given by Mahler. More recently applications of Order bases to problems in Combinatorics have appeared through the work of Salvy and Bostan. In these lectures we give the history (basically coming from rational approximation and interpolation problems), fast algorithms for computation and applications. The applications will include fast computation of problems with matrix polynomial arithmetic, matrix normal forms in addition to the problems arising in Combinatorics.

• Alan Sokal (University College London, U.K. and New York University, U.S.A.)

  Continued fractions and Hankel-total positivity
  Abstract: The expansion of power series into continued fractions goes back at least to Euler in 1746, but it gained impetus following Flajolet's seminal 1980 discovery that any Stieltjes-type (resp. Jacobi-type) continued fraction can be interpreted combinatorially as a generating function for Dyck (resp. Motzkin) paths with specified height-dependent weights. There are now literally dozens of sequences (a_n)_n≥0 of combinatorial numbers or polynomials for which a continued-fraction expansion is explicitly known. I will discuss some of these, as well as some methods for proving them. I will also discuss a very recent generalization of these ideas to branched continued fractions.
  A matrix of real numbers is called totally positive if all its minors are nonnegative; total positivity has applications to many areas of pure and applied mathematics. In particular, a Hankel matrix (a_i+j)_i,j≥0 is totally positive if and only if its underlying sequence (a_n)_n≥0 is a Stieltjes moment sequence. I will extend these ideas to discuss the coefficientwise total positivity of matrices of polynomials. It turns out that many sequences of combinatorial polynomials are (or at least appear empirically to be) coefficientwise Hankel-totally positive; and in some cases this can be proven using continued fractions.