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SUMIRFAS 2016
Abstracts
Radoslaw Adamczak, University of Warsaw
Uncertainty relations for high dimensional random unitary matrices
I will discuss various types of uncertainty principles that hold with high probability for quantum measurements described by Haar distributed random unitary matrices in high dimensions. If time permits I will also present some applications to the problem of locking classical information in quantum states and to Euclidean embeddings into ℓ1n(ℓ2m).
Kelly Bickel, Bucknell University
Compressions of the shift on two-variable model spaces
There are many classical results about operator-theoretic properties of the compressed shift on one-variable model spaces, especially spaces associated to finite Blaschke products. In this talk, we will discuss generalizations of such results to the setting of two-variable model spaces associated to rational inner functions on the bidisk. Among other things, we will discuss characterizations and properties of the numerical range and radius of compressed shifts on two variable model spaces as well as when the commutator of a compressed shift with its adjoint has finite rank. This is joint work with Constanze Liaw and Pam Gorkin.
Mehrdad Kalantar, University of Houston
On irreducibility of boundary representations
Boundary actions of groups were defined and studied by Furstenberg in 60’s and 70’s as a powerful tool to prove various rigidity results. These actions can be considered as the opposite, within the class of stationary actions, to probability measure preserving actions. It has been conjectured that the Koopman representation of any boundary action is irreducible. This has been confirmed for several classes of groups, including lattices of Lie groups and word hyperbolic groups. In this talk we discuss generalizations of some of those results. This is joint work in progress with Yair Hartman (Northwestern).
Pavlos Motakis, Texas A&M University
A metric interpretation of reflexivity for Banach spaces
This lecture mainly concerns the question as to whether the property of a separable Banach space to be reflexive can be characterized metrically. Broadly speaking, a metric characterization of a property of Banach spaces is an equivalent formulation of this property that refers only to the metric structure of the space and not to its linear structure.
On each Schreier family Sα, α<ω1 we define two metrics d∞,α and d1,α and we study the separable Banach spaces X for which there exists a map Φ : Sα→X and two positive constants c, C so that, for all A,B∈Sα, (∗) cd∞,α(A,B) ≤ ‖Φ(A) - Φ(B)‖ ≤ Cd1,α(A,B). As it turns out, such maps can always be constructed on non-reflexive Banach spaces. However, within the class of separable reflexive Banach spaces the existence of a map with property (∗) is closely linked to the Szlenk index of that space, an index measuring the "size" of the space's dual. We draw two main conclusions. The first one concerns a metric interpretation of reflexivity, namely the following: a separable Banach space is reflexive if and only if for every α<ω1 there exists a map satisfying (∗). The second one concerns a metric characterization of the Szlenk index of a reflexive Banach space. More precisely, for countable ordinal numbers α with the property α = ωα it follows that for a separable reflexive Banach space X, (Sα,d1,α) bi-Lipschitzly embeds into X is and only if max{Sz(X),Sz(X*)} > α. This is joint work with Thomas Schlumprecht.
Wonhee Na, Texas A&M University
Principal functions for bi-free central limit distributions
The completely non-normal operator l(v1)+l(v1)*+i(r(v2)+r(v2)*) on a subspace of the full Fock space F(H) arises from a bi-free central limit distribution. We find the principal function of this operator, and as an application we find its spectrum and essential spectrum. This is joint work with Ken Dykema.
Mark Rudelson, University of Michigan
Delocalization of eigenvectors of random matrices
Heuristically, delocalization for a random matrix means that its normalized eigenvectors look like the vectors uniformly distributed over the unit sphere. This can be made precise in a number of different ways. We will consider two complimentary approaches to delocalization. For a matrix with independent entries, we show that with high probability, the largest coordinate of a normalized eigenvector is of the same order as for a uniform random unit vector. This means that the Euclidean norm of an eigenvector cannot be concentrated on a few coordinates.
On the other hand, we show that with high probability, any sufficiently large set of coordinates of an eigenvector carries a non-negligible portion of its Euclidean norm. The latter result pertains to a large class of random matrices including the ones with independent entries, symmetric, skew-symmetric matrices, as well as more general ensembles.
Jaydeb Sarkar, Indian Statistical Institute, Bangalore
Module approach to operator theory
Let {T1,...,Tn} be a set of n commuting bounded linear operators on a Hilbert space H. Then the n-tuple (T1,...,Tn) turns H into a module over C[z1,...,zn] in the following sense: C[z1,...,zn] × H → H, (p,h) ↦ p(T1,...,Tn)h where p∈C[z1,...,zn] and h∈H. The above module is usually called the Hilbert module over C[z1,...,zn]. Hilbert modules over C[z1,...,zn] (or natural function algebras) were first introduced by R. G. Douglas and C. Foias in 1976. The two main driving forces were the algebraic and complex geometric views to multivariable operator theory.
In this talk, we start with an introduction of Hilbert modules over function algebras and survey some recent developments. The list of topics that will be covered include the following: model theory from the Hilbert module point of view, Hilbert modules of holomorphic functions, module tensor products, localizations, dilations, submodules and quotient modules, free resolutions, reproducing kernel Hilbert spaces, factorizations of kernel functions.
Robin Tucker-Drob, Texas A&M University
Ergodic hyperfinite subgraphs and primitive subrelations
We show that every ergodic p.m.p. graph has an ergodic hyperfinite subgraph. We use this to show that every p.m.p. ergodic treeable equivalence relation has an ergodic hyperfinite primitive subrelation, and that every free p.m.p. ergodic action of the free group F_n is orbit equivalent to an action in which one of the generators acts ergodically. This also leads to a new proof of Hjorth's Lemma on cost attained.
Brett Wick, Washington University in St. Louis
Commutators and BMO
In this talk we will discuss the connection between functions with bounded mean oscillation (BMO) and commutators of Calderon-Zygmund operators. In particular, we will discuss how to characterize certain BMO spaces related to second order differential operators in terms of Riesz transforms adapted to the operator and how to characterize commutators when acting on weighted Lebesgue spaces.