Analysis Seminar Fall 2019

Time:Friday Sept. 20, 2019, Period 9 = 4:05 PM-4:55 PM
Location: Little 368
Speaker: Jury

Title: Columns and rows of noncommutative functions

Abstract: Bounded noncommutative functions in the row ball act as (left) multipliers of the Fock space, much as bounded analytic functions in the disk act as multipliers of the Hardy space. We give  a scheme for constructing explicit examples of pairs of nc rational functions such that for the corresponding multiplication operators $A,B$, the operator $A^*A+B^*B$ is a contraction, but $AA^*+BB^*$ is not. (This says that the “true column-row property” fails for bounded nc functions.) I will explain the motivation for the problem (coming from commutative function theory in several variables), construct the examples, and explain “what is really going on” behind the construction.  (This is part of joint work with Robert T.W. Martin.)

Time:Friday August 30, 2019, Period 9 = 4:05 PM-4:55 PM
Location: Little 368
Speaker: Pascoe
Universal monodromy in free analysis
The talk will serve two purposes: 1) to introduce free analysis, 2) to prove that for any connected free set, the monodromy theorem holds. The first example of a free function, a central object of study, is polynomial in several noncommuting variables. Free analysis is the full matricial (quantized) analogue of classical analysis. It finds applications beyond mathematics in engineering systems theory, quantum information theory, and convex optimization and semi-definite programming. Within mathematics it is intimately connected to the theories of completely positive maps and operator systems and spaces. In complex variables, the monodromy theorem states that if a function defined locally can be analytically continued along any path in a simply connected domain, then function has a single-valued analytic extension to the whole domain. We show that in the free analytic setting, the simply connected assumption can be dropped. This leads to a couple of surprising consequences, including a robust inverse function theorem and the universal existence of pluriharmonic conjugates.