QLunch: Jurij Volčič

Speaker: Jurij Volčič, Drexel University, Philadelphia

Title: Polynomial moment inequalities and optimization

Abstract: Polynomial inequalities in multivariate moments are ubiquitous in probability, statistics, economics, partial differential equations, and quantum information theory. This talk presents some results on positivity and optimization of moment polynomials, which are polynomial expressions in variables and their formal mixed moments. An analog of Hilbert’s 17th problem holds for pseudo-moments; on the other hand, moment polynomials positive on actual measures are shown to be sums of squares and formal moments of squares up to arbitrarily small perturbation of their coefficients. When only measures supported on a bounded semialgebraic set are considered, a stronger positivity certificate is given, suitable for optimization. As an application, certain nonlinear Bell inequalities are settled.