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Charles Fefferman
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document.writeln('Princeton University
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document.writeln('Whitney\'s Interpolation Problem and Interpolation I and IIWed. March 5, 10:00 a.m. (I) and 11:30 a.m. (II)
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'); document.writeln('Abstract: Fix positive integers m, n. Let f : E → R be given, with E an arbitrary given subset of Rn. How can we decide whether f extends to a Cm function F on the whole Rn? If F exists, how small can we take its Cm norm? What can we say about the derivatives of F at a given point? Can we take F to depend linearly on f ? What if we require only that F agree approximately with f on E? Suppose E is finite. Can we compute an F whose Cm norm is close to smallest possible? How many computer operations does it take? What if we are allowed to delete a few points from E? The first talk states results, the second talk gives some ideas from the proofs. Many of the results are joint work with Bo\'az Klartag.
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document.writeln('Conference Main Speakers
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document.writeln('Xiaoman Chen, Fudan University
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document.writeln('On Operator Norm Localization Property
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document.writeln('Abstract: A metric space X is said to have operator norm localization property if there exists c > 0, such that for every r > 0, there is R > 0 for which, if ν is a positive locally finite Borel measure on X, H is a separable infinite dimensional Hilbert space and T is a bounded linear operator acting on
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document.writeln('with propagation r, then there exists an unit vector
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document.writeln('satisfying the diameter(Supp(ξ)) ≤ R and ||T|| ≤ c||Tξ||. We prove that a finitely generated group Γ which is strongly hyperbolic with respect to a collection of finitely generated subgroups {H1,···,Hn} has operator norm localization property if and only if each Hi , i = 1, 2, ···, n has operator norm localization property. Furthermore we prove the following result. Let π be the fundamental group of a connected finite graph of groups with finitely generated vertex groups GP. If GP has operator norm localization property for all vertices P then π has operator norm localization property.
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document.writeln('Ronald G. Douglas, Texas A & M University
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document.writeln('Isomorphic Submodules are Rare
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document.writeln('Abstract: While Beurling\'s Theorem implies that each nonzero submodule of the Hardy module on the disk D is isometrically isomorphic to the Hardy module itself, a result of Richter states that for the Bergman module, the only such submodule is the Bergman module itself. In this talk, I discuss results on this phenomenon for quasi-free Hilbert modules in the multivariate case for bounded domains in Ck showing that the phenomenon is closely related to Hardy-like modules. Among results discussed are: (1) If the dimension of the quotient is finite, then k = 1 and only the Hardy module is possible if the domain is D. (2) If the module is essentially reductive and has an isomorphic submodule, then it is subnormal.
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document.writeln('Guihua Gong, University of Puerto Rico
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document.writeln('Positive Scalar Curvature and Non Commutative Geometry
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document.writeln('Abstract: Gromov conjectured that "A uniformly contractible complete Riemannian manifold can not have uniformly positive scalar curvature." Scalar curvature is a local invariant of a smooth manifold. This conjecture gives the information about global behavior (contractibility) of the manifold from the local invariant (scalar curvature) of the manifold. In a joint work with G. Yu, we proved that the Gromov conjecture holds for manifolds with subexponential volume growth. In this talk, we will present this result with explanation of why non commutative geometry is involved.
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document.writeln('Michael T Lacey, Georgia Institute of Technology
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document.writeln('The Small Ball Inequality in all Dimensions
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document.writeln('Abstract: The Small Ball Inequality concerns a lower bound on the L∞ norm of sums of Haar functions adapted to rectangles of a fixed volume. The relevant conjecture is improvement of the average case lower bound by an amount that is the square-root log of the volume of the rectangles. We obtain the first non-trivial improvement over the average case bound in dimensions four and higher. The conjecture is known in dimension 2, a result due to Wolfgang Schmidt and Michel Talagrand, with important contributions from Halasz and Temlyakov. Jozef Beck established a prior result in three dimensions, which argument we extend and simplify. This question is related to (1) Irregularities of Distribution, (2) Probability and (3) Approximation Theory.
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document.writeln('John McCarthy, Washington University
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document.writeln('Matrices and Varieties
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document.writeln('Abstract: For any pair T = (T1,T2) of commuting matrices, normalized so that both have norm one, there are many polynomials p(z1,z2) that annihilate the pair. There is a special choice with the property that the set V = {(z1,z2) : |z1| ≤ 1, |z2| ≤ 1, p(z1,z2) = 0 } is a spectral set for T, i.e. for any other polynomial q the inequality
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document.writeln(' ||q(T1, T2)|| ≤ ||q||v
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document.writeln('holds. I shall discuss how these bordered varieties V arise, and, more generally, some connections between the geometry of varieties and properties of function algebras.
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document.writeln('Vladimir Peller, Michigan State University
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document.writeln('Approximation by analytic matrix functions in Lp
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document.writeln('Abstract: I am going to speak about recent results obtained jointly with L. Baratchart and F. Nazarov on approximation in Lp of matrix functions on the unit circle by analytic matrix functions. We have obtained quite unexpected results that are quite different from the results in the case of approximation in L∞.
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document.writeln('Stefan Richter, University of Tennessee
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document.writeln('Some Remarks on the Dilation Theory for Commuting D-Contractions
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document.writeln('Abstract: A family in the sense of J. Agler\'s model theory is a collection of Hilbert space operators which is uniformly bounded in operator norm and closed under the formation of direct sums, restrictions to invariant subspaces, and unital *-representations. An extremal in a family is an operator that can only be extended to another operator in the family by taking direct sums. It is Agler\'s theorem that every operator in a family can be extended to an extremal for that family. Based on this, one can give quick proofs of basic extension and dilation theorems. For example every isometric operator has a unitary extension, every contraction has a co-isometric extension, etc. In this talk I will discuss some applications of the model theory in the multi-variable context.
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document.writeln('Richard Rochberg, Washington University
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document.writeln('The Dixmier Trace of Bergman Space Hankel Operators
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document.writeln('Abstract: If Hf is a (big) Hankel operator on the Bergman space of the disk with a smooth symbol f then we have the following formula for the Dixmier trace of its modulus:
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document.writeln('I will describe some related results, say a bit about the proof of this result and about the range of f for which it holds. I will also describe some extensions and variations of the result.
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document.writeln('Sergei Treil, Brown University.
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document.writeln('Singular Integrals and Perturbation Theory.
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document.writeln('Abstract: The theory of singular integral operators, in particular, a theorem by Arocena - Cotlar - Sadosky about two weight estimates of Hilbert transform is applied to the investigation of delicate spectral properties in perturbation theory of self-adjoint operators. As an application new result about the absence of the embedded singular spectrum for rank one perturbations is obtained. The talk is based on a joint work with C. Liaw.
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document.writeln('Alexander Volberg, Michigan State University and University of Edinburgh
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document.writeln('The Cauchy Integral, Its Friends and Family: Analysis, Geometry and Combinatorics
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document.writeln('Abstract: In 1733 count de Buffon asked the question: What is the probability for a needle of length L < 1 to intersect a grid of parallel lines on the plane having distance 1 between each other? In 1898 Paul Painlevé asked another question: How can one describe geometrically the compact sets on the plane such that the only functions analytic and bounded in the complement of these sets are constants? At the end of the 20th century it became clear that these two questions are closely related. Moreover, they are closely related to a wide variety of problems, from percolation on graphs to electrostatics. The key words here are "Calderón-Zygmund capacities". Capacities with positive kernels, even their non-linear counterparts, are well understood. But, recently the focus has been on capacities with signed, complex or vector-valued kernels. It is usually quite difficult to prove that they are even "capacities". In particular, this was the essence of Vitushkin\'s question and Tolsa\'s answer about one of them, analytic capacity assigned to the Cauchy kernel on the complex plane. Tolsa\'s proof does not work in three dimensions, however the corresponding capacity does exist in three dimensions. It is related to the gradient of the fundamental solution for the Lapalace equation, this is an exact counterpart of Cauchy kernel on the plane. We explain the universal approach to proving subadditivity of such new capacities. We will mention also a pretty amazing connection between Calderón-Zygmund capacities and the usual (but non-linear) capacities. Already Vitushkin explored a very enigmatic connection between analytic capacity and Geometric Measure Theory. He put forward the question about the "equivalence" of analytic capacity and the so-called Buffon needle probability. We will describe the status of this problem today. This will bring us naturally to a seemingly elementary problem of estimating the Buffon needle probability of a finite collection of disks located in a Cantor pattern. We will indicate the relation of this problem with percolation on graphs, tiling, the Besicovitch projection theorem, and the Kakeya problem.
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document.writeln('Jingbo Xia, SUNY at Buffalo
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document.writeln('Boundedness and Compactness of Hankel Operators on the Sphere
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document.writeln('Abstract: Let S be the unit sphere in Cn and let dσ be the spherical measure on S. Recall that the Hankel operator Hf is defined by the formula Hf = (1 - P)Mf|H2(S), where H2(S) is the Hardy space on S and P : L2(S,dσ) → H2(S) is the orthogonal projection. We show that a large amount of information about the function f - Pf can be recovered from the properties of the Hankel operator Hf. For example, if Hf is compact, then the function f - Pf is necessarily in VMO.
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Important Information for
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document.writeln('International Participants
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'); document.writeln('The purpose of the conferences is to bring together both experienced researchers and younger people, including graduate students, to discuss recent work and advances in classical analysis and modern analysis.
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'); document.writeln('Sponsored by the Shanks Endowment, the National Science Foundation, and Vanderbilt University.
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