Title: Stochastic OptimizationSpeaker: Ricardo Fukaswaw Affiliation: University of Waterloo Location MC 6029 or please contact Rian Neogi for Zoom link
Abstract: While deterministic optimization problems are very useful in practice, often times the assumption that all data is known in advance does not hold true. One possible way to relax this assumption is to assume that the data depends on random variables.
Title: Positivity and sums of squares in products of free algebrasSpeaker: William Slofstra Affiliation: University of Waterloo Location MC 5501 or please contact Melissa Cambridge for Zoom link
Abstract: A noncommutative polynomial is said to be positive relative to some constraints if plugging matrices (or more generally, operators on a Hilbert space) satisfying the constraints into the polynomial always yields a positive operator. It is a natural problem to determine whether or not a given polynomial is positive, and if it is, to find some certificate of positivity. This problem is closely connected with noncommutative polynomial optimization, where we want to find matrices or operators that maximize the operator norm of some polynomial, subject to the constraint that some other polynomials in the operators are positive or vanish. When the algebra cut out by the constraints is a free algebra, free group algebra, or similar algebra, it's well-known that a polynomial is positive on operators satisfying the constraints if and only if it's a sum of Hermitian squares in the algebra.