at the

American Institute of Mathematics, San Jose, California

organized by

Igor Klep, Scott McCullough, and Jurij Volcic

- 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?
- 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.
- Progress on matrix and tracial convexity, operator systems and quantum channels.
- 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.

The workshop schedule.

A report on the workshop activities.