Workshop on Positivity
May 21-24, 2019
University of New Mexico, Albuquerque, NM, USA
Organizer: Anna Skripka

Mini-courses

• Dominique Guillot
  University of Delaware
  "Positivity preservers: theory and applications"
  List of prerequisites,   Lectures
  Abstract:
  Matrix positivity is a notion that has attracted the attention of many great minds. Yet, after at least two centuries of research, positive matrices still hide enigmas and raise challenges for the working mathematician. This mini-course will explore matrix positivity with an emphasis on operations that preserve it. Motivation and applications will be presented in several areas including classical analysis, combinatorics, and statistics. The course will be mostly self-contained, and will bring together techniques from many fields such as harmonic analysis, function theory, operator theory, statistics, combinatorics, and group representations.


• Chris Fraser
  University of Minnesota
  "Introduction to total positivity and cluster algebras"
  List of prerequisites
  Abstract:
  A real square matrix is totally positive if all of its minors are positive. This simple notion, and its generalizations, arises in several disparate parts of mathematics and physics. I will give an introduction to this class of matrices from several viewpoints (analytic, algebraic, and combinatorial), focusing especially on the problem of efficiently deciding when a matrix is totally positive. Solving this problem leads naturally to Fomin-Zelevinsky's theory of cluster algebras, a theory which has subsequently found deep connections with other areas of math. I will mention some key examples and highlights of this theory (e.g. finite type classification, Laurentness, positivity), and discuss some open problems.


• Clément Coine
  Central South University / Université de Franche-Comté, Besançon
  "Norms of Schur multipliers and applications"
  List of prerequisites
  Abstract

Lectures

• Bala Rajaratnam
  University of California, Davis / University of Sydney (visiting)
  "Positivity: a basic introduction"
  Abstract:
  In this talk, we will cover the basics that are required to follow more advanced topics on matrix positivity. First, we will consider various definitions of positive definiteness. Second, we will cover how positive definite matrices naturally arise in classical probability and statistics. We will then introduce how the need to understand positive definiteness arises in modern data science and big data applications. Finally, we will provide a gentle introduction to some basic results on maintaining positivity for symmetric matrices.


• Shaun Fallat
  University of Regina
  "Total positivity and the Cauchon algorithm: connections to rank and the bidiagonal factorization"
  List of prerequisites
  Abstract:
  A matrix is called totally nonnegative (TN) (resp. totally positive (TP)) if all of its minors are nonnegative (resp. positive). For this class of matrices, the Cauchon (reduction) algorithm was used in conjunction with characterizing TN cells. In addition, this algorithm, along with the related combinatorial object known as a Cauchon diagram, was used to describe an efficient recognition test for detecting TN matrices via sets of vanishing minors. In this talk, I will review some basic properties of TN matrices, including bidiagonal factorizations, and survey the Cauchon algorithm and Cauchon diagrams. From there, I will describe connections between the Cauchon algorithm and the bidiagonal factorization, and, as a consequence, I will discuss a mechanism for detecting the rank of a TN matrix via the Cauchon algorithm.


• Anna Skripka
  University of New Mexico
  "On positivity of multilinear Schur multipliers"
  Abstract:
  Multilinear Schur multipliers are fundamental objects in higher order perturbation theory, which are defined as intricate transformations on tuples of matrices generalizing both the entrywise and usual products. We will discuss open problems and partial results on positivity of these transformations.

The workshop is sponsored by
National Science Foundation
CAREER grant DMS-1554456