Workshop in Analysis and Probability
July 5th - August 1st 2010
Schedule of Talks
As talks are scheduled, the information will be posted here. Please check back often.
Unless otherwise noted, all talks will be presented in Milner Hall, Room 317.
Refreshments will be provided 15 minutes before each talk.
| July 7 | 4pm-5pm | Karl Liechty |
Connections Between Orthogonal Polynomials, Random Matrices, and Statistical Mechanics
We will discuss exact solutions of the six-vertex model of statistical physics and their relations to orthogonal polynomials and random matrix models. This is joint work with Pavel Bleher. |
| July 12 | 4pm-5pm | Paulette Willis |
Group actions, labeled graphs and C*-algebras
A labeled graph (E, L) over an alphabet A consists of a directed graph E together with a labeling map L : E1 → A. One can associate a C*-algebra to a labeled graph (E, L) in such a way that if the labeling L is trivial then the resulting C*-algebra is the C*-algebra of the graph E. In this presentation, I will discuss joint work with Teresa Bates and David Pask concerning (discrete) group actions on labeled graphs and the resulting crossed product C*-algebras. In particular, I will discuss our main theorem which shows that the crossed product that arises when a group acts freely on a labeled graph is strongly Morita equivalent to the C*-algebra of the quotient graph of the action. I will focus on the two major ideas needed to prove this Morita equivalence. The first is a generalization of the so-called Gross-Tucker theorem, which shows that a free la- beled graph action is naturally equivariantly isomorphic to a skew product action obtained from the quotient labeled graph. The second is a generalization of a theorem of Kaliszewski, Quigg, and Raeburn to the effect that the C*-algebra of a skew product labeled graph is naturally isomorphic to a co-crossed product of a coaction of the group on the C*-algebra of the labeled graph. |
| July 14 | 4pm-5pm | Quanlei Fang |
Schatten class membership of Hankel operators on the unit sphere
Let Hf be a Hankel operator on the Hardy space of the unit sphere in ℂn, n ≥ 2. We determine the membership of Hf in the Schatten class Cp for all possible symbol functions f in the L2 of the sphere. In the case p > 2n, Hf ∈ Cp if and only if Hf maps the constant function 1 into the Besov space Bp. In the case p ≤ 2n, the membership Hf ∈ Cp implies Hf = 0. This is a joint work with Jingbo Xia. |
| July 15 | 4pm-5pm | Detelin Dosev |
Commutators on Lp
In this talk a classification of the commutators on lp and Lp, 1≤ p≤∞, will be given. We will show that the commutators on X for X=lp or X=Lp, $1≤ p≤∞, are the operators not of the form λ I + K where λ≠ 0 and K is an operator in the largest ideal in L(X). The main steps for proving these results will be outlined and applications to other Banach spaces will be presented as well. The results are based on a joint work with W. B. Johnson and G. Schechtman. |
| July 16 | 4pm-5pm | Piotr Nowak |
Diameters, distortion and eigenvalues
We study the relation between the diameter, the lp-distortion and the first eigenvalue of the p-Laplacian on a finite graph. We prove a general inequality relating these three notions and show some applications, in particular to Cayley and Schreier graphs arising from self-similar groups. |
| July 19 | 4pm-5pm | Dongyang Chen |
The bounded compact approximation property of pairs
In this talk, the notion of the bounded compact approximation property (BCAP) of a pair [Banach space and its subspace] is used to prove that if X is a closed subspace of L∞ with the bounded compact approximation property, then L∞/X has the bounded compact approximation property. An immediate consequence is that if X is a closed subspace of L1 so that X⊥ has the bounded compact approximation property, then X* has the bounded compact approximation property. We also show that X* has the λ-BCAP with conjugate operators if and only if the pair (X,Y) has the λ-BCAP for each finite codimensional subspace Y ⊆ X. |
| July 20 | 4pm-5pm | Alexey Popov |
Almost invariant subspaces of operators and algebras of operators
Let X be a Banach space and T a bounded operator on X. A subspace Y of X is said to be almost invariant under T if TY is contained in Y + F for some finite-dimensional subspace F of X. This notion is only of interest if Y is both of infinite dimension and of infinite codimension; we call such subspaces half-spaces. In this talk we will discuss various properties of operators with almost invariant subspaces. In particular, we show that: (i) there exist operators without invariant half-spaces that have almost invariant half-spaces; (ii) if a closed algebra A of operators has a common almost invariant half-space then the dimensions of errors corresponding to different operators in A are uniformly bounded; (iii) if A is generated by a finite number of commuting operators and has a common almost invariant half-space then it also has a common invariant half-space. |
| July 21 | 11am-12pm | Anna Skripka |
Commutators on II1 factors
Characterization of commutators has been obtained for all (von Neumann) factors but II1. In particular, it is unknown if every element with zero trace is a commutator in II1. Jointly with K. Dykema, we obtained that the set of commutators in a II1 factor contains all normal elements with purely atomic distribution and zero trace as well as all nilpotents and some other classes of non-normal elements. |
| July 21 | 4pm-5pm | Kevin Beanland |
Ordinal ranks on the space of strictly singular operators
Co-analytic ranks are fundamental tools in Descriptive Set Theory and have proven to be extremely useful in studying the geometry of Banach spaces. If X and Y are separable Banach spaces, then the set L(X,Y) of all bounded linear operators from X to Y carries a natural structure of a standard Borel space. It is easy to see that SS(X,Y), the strictly singular operators from X to Y, is a co-analytic subset of L(X,Y). In this talk we will define an ordinal rank p and a co-analytic rank r, on SS(X,Y) such that p does not exceed r. We will also give examples of spaces X and Y such that: (1) SS(X,Y) is co-analytic non-Borel and p is unbounded; (2) SS(X,Y) is Borel; (3) SS(X,Y) is co-analytic and the rank p is bounded. Finally, we will discuss some open problems. |
| July 22 | 11am-12pm | Piotr Nowak |
Bounded cohomology and exact groups
We prove a characterization of exact groups, or more generally topologically amenable actions, in terms of vanishing of the bounded cohomology of the group. This generalizes to this new context a classical theorem of B.E.Johnson on amenability and answers a question of Nigel Higson. |
| July 22 | 2:50pm-3:50pm | Michael Doré |
Fréchet differentiability of planar valued Lipschitz functions in Hilbert Spaces
We study Lipschitz functions and universal differentiability sets - that is, sets for which every Lipschitz function contains a point of Fréchet differentiability. It is an old result of Preiss that such sets may be null. We cover new work by the author and Maleva determining the possible Hausdorff dimension of such sets. We also discuss how the problem changes if one looks at Lipschitz functions whose codomain has dimension higher than one. |
| July 23 | 11am-12pm | Han Ju Lee |
On Banach spaces with absolute norm and polynomial numerical indices
In this talk, I briefly review some Basic properties of polynomial numerical index and properties of Banach spaces with absolute norms. Then we assume that X is a Banach space with absolute norm and show that if real Banach space X has Radon-Nikodým property or is Asplund, then its polynomial index n(2)(X) < 1 unless it is one-dimensional. On the other hand, if X is a complex Banach space with Radon-Nikodým property and n(2)(X) = 1, then X is finite-dimensional and isometrically isomorphic to l∞m. Also, the only Asplund complex space X with n(2)(X) = 1 is c0(Λ). |
| July 23 | 4pm-5pm | Gosia Czerwinska |
Some rotundity properties in symmetric spaces of measurable operators
The talk will be devoted to geometric structure of symmetric spaces of measurable operators E(M,τ) corresponding to a symmetric Banach function lattice E, a semifinite von Neumann algebra M acting in a Hilbert space H, and a faithful normal semifinite trace τ on M. We investigate several complex rotundity properties such that complex rotundity (C - R), complex local uniform rotundity (C - LU R) and (real) midpoint local uniform rotundity (MLUR). In fact we prove that if a singular value function μ(x) of x ∈ E(M,τ) is a complex extreme, complex locally uniformly rotund, or strongly extreme point of the unit ball of E then the operator x has the similar property in the space E(M,τ). The converse implication holds true whenever the von Neumann algebra M is either non-atomic or is the space of bounded operators B(H) and then E(M,τ) becomes a unitary matrix space CE. As a consequence we obtain analogous relations between E and E(M,τ) for global properties (C-R), (C-MLUR), (C-LUR) or (MLUR). Below there is a sample of our results on complex extreme points. Theorem Let M be a non-atomic von Neumann algebra. An operator x is a complex extreme point of BE(M,τ) if and only if μ(x) is a complex extreme point of BE and one of the following, not mutually exclusive, conditions holds: (i) μ∞(x) = 0 (ii) n(x)Mn(x*) = 0 and |x| ≥ μ∞(x)s(x). This work was motivated by earlier results on extreme points and (local) uniform rotundity in CE and E(M,τ) by Arazy and Chilin, Krygin and Sukochev. It is a joint work with Anna Kaminska from the University of Memphis. |
| August 2 | 11am-12pm | Jan Spakula |
Coarse amenability and coarse embeddings
A deep result of G. Yu states that if a metric space of bounded geometry coarsely embeds into a Hilbert space, it satisfies the Coarse Baum-Connes conjecture. Yu devised another property - coarse amenability, or "property A" - as a criterion for coarse embeddability into a Hilbert space. In this talk, I will describe a construction of a metric space with bounded geometry which coarsely embeds, but is not coarsely amenable. A non-bounded geometry example (quite different from ours) was found previously by P. Nowak. This is a joint work with G. Arzhantseva and E. Guentner. |
| August 2 | 4pm-5pm | Timur Oikhberg |
Automatic continuity of orthogonality preserving linear maps
Several known results assert that an orthogonality (resp. disjointness) preserving bijection between C*-algebras (function spaces) must be continuous. In this work, we establish the automatic continuity of bijections from a C*-algebra to a Banach space, provided the images of orthogonal elements satisfy a certain geometric orthogonality condition. Related results for vector-valued spaces of continuous functions are also obtained. This is joint work with A. M. Peralta and M. Ramirez. |
Last modified: 20 May 2014, Alejandro Chavez-Dominguez