Brown University home page Math Department Home Page Sergei Treil home page Brown University Department of Mathematics Analysis seminar, Spring 2005 Wednesdays 4:00-5:00 pm Kassar House 105 Hit "reload" often for the latest version Date  Speaker Title of the talk       2/2 Sergei Treil Brown Title: A new proof of the Carleson Embedding Theorem  Abstract: It is well known that if $\Phi$ is a bounded subharmonic function, then the measure $(1-\z|^2) \Delta \phi dxdy$ is Carleson. This fact can be easily proved using the Green's formula. I will use this result to give an alternative proof of the classical Carleson Embedding Theorem.  2/9   Special colloquium by K. Zombrun, Indiana University 2/16 Brett Wick,  Brown University Title: Embedding theorem in the unit ball in C^n 2/23   Special colloquium by V. Kaloshin, Caltech       3/2 David Cruz-Uribe,  Trinity College Title: Extrapolation on variable L^p Spaces Abstract: Click here to see the abstract 3/9 Michael Lauzon,  Brown University  Title: A Calderon-Zygmund decomposition for vector-valued functions with operator weights. Abstract: Reverse Holder condition was introduced by M. Goldberg as a simple sufficient condition for the boundednes of the maximal function in the weighted L2 space with operator weights. It is possible to show that the reverse Holder condition also implies the L^p boundednes of the maximal function in the most general setting of the norm-valued weights. However, as a simple example shows, no reverse Holder condition can guarantee the boundedness of the Hilbert or Martingale transforms. Situation is a bit different for the operator-valued weights, when all local norms are Hilbertian. Recently Sandra Pott proved the boundedness of the Martingale transform (and so of the Hilbert Transform) on L2(W), where W(t) is an operator-valued weight satisfying a reverse Holder condition. A variant of the Calderon-Zygmund decomposition will be presented and then used to extend Pott's results to the L^p case  3/16 John Wermer,  Brown University  Title: The Ahlfors Function on Plane Domains. Abstract: This will be an expository talk about certain extremal problems on finitely connected plane domains, treated by Ahlfors and Garabedian in the 40's, and later by Royden and many other authors. 3/23 Special seminar and colloquim W. Schlag, Caltech 3/30   Spring Break       4/6 Vladimir Fock,  Visiting Brown Title: Minimal surfaces in 3D hyperbolic spaces, Cosh-Gordon equation, and  quasifuchsian groups. Abstract: Fuchsian group is a discrete subgroup of real 2x2 matrices. Quasifuchsian group is a deformation of a fuchsian one within the class of discrete subgroups of complex 2x2 matrices. Hyperkaehler structure on a Riemann manifold is a triple of complex structures satisfying the algebra of quaternions and such that the manifold is Kaehler wit respect to all three of them. Cosh-Gordon equation is an equation which determines a canonical Hermitean metric on a Riemann surface with given holomorphic quadratic differential. If the differential is vanishing the canonical metric turns out to be the constant negative curvature one given by the Poincare uniformization theorem. The aim of the talk is to give an explicit description of the space of quasifuchsian groups in terms of the Cosh-Gordon equation providing it with hyperkaehler structure. The main tool for this relation is the consideraton of minimal surfaces in 3D hyperbolic spaces.  4/13 Daniela De Silva,  MIT Title: A singular energy minimizing free boundary Abstract: In this talk, we will exhibit the first example of a singular energy minimizing free boundary. This example occurs in dimension 7, which is conjectured to be the optimal dimension for free boundary regularity. This is analogous to the 8-dimensional Simons cone, in the theory of minimal surfaces. We will highlight the similarities between the two theories, explaining how the proof of the minimality of the Simons cone, was the inspiration for our proof. This is a joint work with David Jerison. 4/20           4/29 S.M. Verduyn Lunel, Universiteit Leiden (Netherlands) Title: New completeness theorems for classes of compact operators 5/4 Brian Cole, Brown University Title: Approximation problems on an open Riemann surface Abstract: We lift solutions to problems involving analytic functions in the complex plane to an open Riemann surface. We look at theorems of Davie and Mergelyan among others. 5/11 Norm Levenberg,  Auckland University  Title: A Hilbert Lemniscate Theorem in C^2