Noncommutative inequalities

June 14 to June 18, 2021

at the

American Institute of Mathematics, San Jose, California

organized by

Igor Klep, Scott McCullough, and Jurij Volcic

Original Announcement

This workshop will be devoted to equations and inequalities for functions in matrix variables. In the last two decades, positivity of matrix and operator evaluations of noncommutative polynomials, rational functions, or trace polynomials has been intensively and systematically studied using techniques from real algebraic geometry and operator theory. Several important results on sums-of-squares positivity certificates and convexity have been established. On the other hand, major breakthroughs in this topic were recently made by computational complexity, random matrix theory and quantum information theory, such as refutation of the multifaceted Connes' embedding conjecture. This workshop will bring together the most dynamic members of these communities, that have so far intersected very little at conferences. The expected outcomes include: progress on major problems already of interest across communities, such as Hilbert's 17th problem for trace polynomials; identification of further questions of mutual interest and alignment of research efforts; and the development of a collaborative research infrastructure.
  1. Tracial inequalities post Connes' embedding conjecture. For example, due to the latter's recent refutation, there exists a noncommutative polynomial that has nonnegative trace on all matrix contractions, but not on all operator contractions from von Neumann algebras. What is a concrete example? Are there easily identifiable large classes of polynomials that are not such examples, and prominently appear in quantum information and quantum computation?
  2. Advances on noncommutative rational functions, such as rational sums-of-squares certificates for positive trace polynomials, eigenvalues of noncommutative rational functions, centralizers in the free skew field.
  3. Progress on matrix and tracial convexity, operator systems and quantum channels.
  4. Applications of invariant theory and computational complexity to noncommutative functions, such as similarity problem for matrix tuples and (non)-existence of a sum-of-squares certificate on \(4\times4\) matrices.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Workshop Videos

Papers arising from the workshop:

Convexity of a certain operator trace functional
by  Eric Evert, Scott McCullough, Tea \vStrekelj, and Anna Vershynina,  Linear Algebra Appl. 643 (2022), 218–234