September 5 |
Kristin Courtney, UVaNilpotent approximations of universal operators and some conjecturesBecause of its elegance and utility, von Neumann's inequality has become canon in operator theory, and its extensions to various contexts are still the subject of a wide range of research. The inequality says that, given any polynomial p in one variable, the maximal norm of the operator p(T), as T ranges over all contractive Hilbert space operators, can be determined by considering only contractive operators on a one dimensional Hilbert space, i.e. elements of the unit disk in the complex plane.Using universal C*-algebras, one can readily show a von Neumann-type inequality for non-commutative *-polynomials, which says that, given a non-commutative *-polynomial q, the maximal norm of the operator q(T), as T ranges over contractive Hilbert space operators, can be determined by considering only contractive operators on finite-dimensional Hilbert spaces, i.e. matrices of norm at most 1. Our first goal in this talk is to show why it actually suffices to consider only nilpotent matrices of norm at most 1. Moving to polynomials in two variables, von Neumann's inequality notably extends when the argument ranges over pairs of commuting contractive Hilbert space operators. Can we again look to matrices for a bound for the norm of any noncommutative *-polynomial in two variables whose inputs are two doubly commuting contractive operators? Does it suffice to consider only nilpotent matrices? Surely these questions are not too difficult, are they? |
September 12 |
Brian Lins, Hampden-Sydney CollegeThe Illumination Conjecture and fixed points of nonexpansive mapsThe Illumination Conjecture is a famous unsolved conjecture in combinatorial geometry. It predicts that the surface of any convex body in R^{n} can be completely illuminated by 2^{n} floodlights. Surprisingly, it has not been proven, even in 3-dimensions! This talk will focus on a new connection between this famous conjecture and the fixed points of nonexpansive maps in finite dimensional normed spaces. |
September 19 |
no meeting |
September 26 |
no meeting |
October 3 |
READING DAY (no meeting) |
October 10 |
Ben Hayes, UVaAlgebraic actions of sofic groups and their entropy I: general backgroundI will discuss the notion of an algebraic action, which is an action of a discrete group by automorphisms on a compact group. I will be particularly focused on the discussed of the entropy of such an action for the case when the acting group is sofic (soficity will be defined in the talk). |
October 17 |
Ben Hayes, UVaAlgebraic actions of sofic groups and their entropy II: Fuglede-Kadison determinants and entropyContinuing on the previous lecture, I will discuss entropy for a large class of algebraic actions and how this relates to von Neumann algebra invariants. |
October 24 |
Geoff Price, US Naval AcademyOn the classification of binary shiftsA spin system is a sequence of Hermitian unitary operators u_1, u_2,... that pairwise commute or anticommute. Under a mild condition the von Neumann algebra generated by a spin system is the hyperfinite factor R. Our focus is on spin systems with translation-invariant commutation relations, allowing for the construction of unital *-homomorphisms on R. These binary shifts are uniquely determined by the conditions alpha(u_i)=u_{i+1}. We discuss the classification problem of binary shifts, and show how this problem is related to results on the ranks of sequences of Toeplitz matrices over GF(2).The Virginia Operator Theory and Complex Analysis Meeting (VOTCAM) will be held at UVa on Saturday October 28. Website |
October 31 |
maybe too spooky for a meeting |
November 7 |
no seminar |
November 14 |
no seminar |
November 21 |
Last day before Thanksgiving break (no meeting) |
November 28 |
Sarah Browne, Penn StateE-theory spectrumE-theory is an invariant of C*-algebras and in particular is a collection of abelian groups defined in terms of homotopy classes of certain morphisms of C*-algebras. This makes it a natural object to define in terms of stable homotopy groups. In my talk I will detail the notion of E-theory and the framework we require, namely a spectrum of quasi-topological spaces, to represent the E-theory groups as a stable homotopy theory. Then I will highlight how we encode E-theory properties into this construction. |
December 5 |