Analysis and Probability Seminar

Welcome to Analysis and Probability Seminar at UF!

UPCOMING SEMINARS

SPRING 2024

Location: LIT 201

Time: Tuesdays Period 9 (4:05 – 4:55 P.M.)

Date	Speaker	Description
January 30	Austin Jacobs	Cross correlation is an integral transform strongly related to convolution that is commonly used in motion tracking. Here we will explain this basic process, giving illustrations along the way, and explain how exactly this works and how this can help inform our intuition for what exactly is happening in related integral transforms.
February 6	Dr. Mike Jury	We will recall the Strong Szego Limit Theorem from last semester’s talk, and I will present a relatively modern proof due to Bump and Diaconis (2002). Along the way we will observe connections to random matrix theory and the theory of symmetric functions.
February 13	George Roman	We examine moments of multivariate trace polynomials over the unitary group, generalizing the work of Diaconis and Shahshahani (1994). In the process, we extend the notion of certain symmetric polynomials.
February 20	Hrvoje Šikić	Dual Integrable Representations: Studies of various reproducing function systems emphasized the role of translations and the Fourier periodization function. These influenced the development of the concept of dual integrable representations, a large and important class of unitary representations on LCA groups. The key ingredient is the bracket function that enables the explicit description of corresponding cyclic spaces. Since its introduction, the notion was extended to some specific classes of non-abelian groups, and a natural problem emerged, i.e., whether it can be extended to the entire class of (including non-abelian) locally compact groups. In recent paper with Ivana Slamić we solved this problem.
February 27	Igor Klep	The talk will discuss state polynomials, i.e., polynomials in noncommuting variables and formal states of their products. The motivation behind this theory arises from the study of Bell inequalities and correlations in quantum networks. We will give a state analog of Artin’s solution to Hilbert’s 17th problem showing that state polynomials, positive over all matrices and matricial states, are sums of squares with denominators. Further, archimedean Positivstellensätze in the spirit of Helton-McCullough are presented leading to a hierarchy of efficiently computable semidefinite relaxations converging monotonically to the optimum of a state polynomial subject to state constraints.
March 5	Tea Štrekelj	In this talk we discuss several notions of a face of a matrix convex set. We will introduce fixed-level as well as multicomponent matrix faces (matrix exposed faces). The aim is to extend the concepts of a matrix extreme point and a matrix exposed point, respectively. Their properties resemble those of (exposed) faces in the classical sense, e.g., we will show that the C*-extreme (matrix extreme) points of a matrix face (matrix multiface) of a matrix convex set K are matrix extreme in K. As in the case of extreme points, any fixed-level matrix face is ordinary exposed if and only if it is a matrix exposed face. From this it follows that every fixed-level matrix face of a free spectrahedron is matrix exposed. On the other hand, matrix multifaces give rise to the noncommutative counterpart of the classical theory connecting (archimedean) faces of compact convex sets and (archimedean) order ideals of the corresponding function systems.
March 19	No speaker
March 26	George Roman	Earlier in the semester, we discussed a theorem on the limiting behavior of the mixed moments of tracial polynomials over the unitary group. In this talk, we go through the proof, examining many useful techniques in matrix algebra, as well as a version of Schur-Weyl duality.
April 2	Arya Memana	In this talk we will recall the classical Szegö’s theorem about asymptotic eigenvalue distribution of Toeplitz operators on Hardy space and understand it through William Arveson’s C* algebra approach.
April 9	Scott Mccullough	We will review the classic Caratheodory interpolation problem from complex analysis and then discuss operator theoretic variations both old and new.