Pure Math Grad Colloquium

Yuming Zhao, Department of Pure Mathematics, University of Waterloo

"Positivity and sum-of-squares"

A multivariate polynomial is said to positive if it takes only non-negative values over the reals. Hilbert's 17th problem concerns whether any positive polynomial can be expressed as a sum of squares of other polynomials. In general, we say an element of a *-algebra is positive if it is a positive operator in all *-representations. Many problems in math and computer science are closely connected with deciding whether a given element is positive and finding certificate of positivity. In particular, we are interested in sum-of-squares certificate.

In this talk, I'll discuss the computational complexity of determining positivity and some applications of sum-of-squares in quantum information theory. I will also present joint work with Arthur Mehta and William Slofstra, in which we show that positivity is undecidable in tensor products of free algebras. As a consequence, there are no certificates of positivity in such algebras.

MC 5479