Seminars and Colloquia by Series

Wednesday, November 10, 2010 - 15:00 , Location: Skiles 269 , Nguyen Cong Phuc , LSU , Organizer: Michael Lacey
We discuss a global weighted estimate for a class of divergence form elliptic operators with BMO coefficients on Reifenbergflat domains. Such an estimate implies new global regularity results in Morrey, Lorentz, and H\"older spaces for solutionsof certain nonlinear elliptic equations. Moreover, it can also be used to obtain a capacitary estimate to treat a measuredatum quasilinear Riccati type equations with nonstandard growth in the gradient.
Wednesday, November 3, 2010 - 14:00 , Location: Skiles 269 , Andrej Zlatos , University of Wisconsin, Madison , andrej@math.wisc.edu , Organizer:
We consider the influence of an incompressible drift on the expected exit time of a diffusing particle from a bounded domain. Mixing resulting from an incompressible drift typically enhances diffusion so one might think it always decreases the expected exit time. Nevertheless, we show that in two dimensions, the only simply connected domains for which the expected exit time is maximized by zero drift are the discs.
Wednesday, October 27, 2010 - 14:00 , Location: Skiles 269 , Greg Knese , University of Alabama , Organizer:
The Schur-Agler class is a subclass of the bounded analytic functions on the polydisk with close ties to operator theory.  We shall describe our recent investigations into the properties of rational inner functions in this class. Non-minimality of transfer function realization, necessary and sufficient conditions for membership (in special cases), and low degree examples are among the topics we will discuss.
Wednesday, October 20, 2010 - 14:00 , Location: Skiles 269 , Mishko Mitkovski , Georgia Tech , Organizer:
A separated sequence of real numbers is called a Polya sequence if the only entire functions of zero type which are bounded on this sequence are the constants. The Polya-Levinson problem asks for a description of all Polya sequences. In this talk, I will present some points of the recently obtained solution. The approach is based on the use of Toeplitz operators and de Branges spaces of entire functions. I will also present some partial results about the related Beurling gap problem.
Wednesday, October 6, 2010 - 14:00 , Location: Skiles 269 , Miguel Pinar , Dpto. Matematica Aplicada, Universidad de Granada , Organizer: Jeff Geronimo
Sobolev orthogonal polynomials in two variables are defined via inner products involving gradients. Such a kind of inner product appears in connection with several physical and technical problems. Matrix second-order partial differential equations satisfied by Sobolev orthogonal polynomials are studied. In particular, we explore the connection between the coefficients of the second-order partial differential operator and the moment functionals defining the Sobolev inner product. Finally, some old and new examples are given.
Tuesday, October 5, 2010 - 13:00 , Location: Skiles 269 , Sasha Aptekarev , Keldish Institute for Applied Mathematics , Organizer: Jeff Geronimo
The asymptotic theory is developed for polynomial sequences that are generated by the three-term higher-order recurrence Q_{n+1} = zQ_n - a_{n-p+1}Q_{n-p}, p \in \mathbb{N}, n\geq p, where z is a complex variable and the coefficients a_k are positive and satisfy the perturbation condition \sum_{n=1}^\infty |a_n-a|<\infty . Our results generalize known results for p = 1, that is, for orthogonal polynomial sequences on the real line that belong to the Blumenthal-Nevai class. As is known, for p\geq 2, the role of the interval is replaced by a starlike set S of p+1 rays emanating from the origin on which the Q_n satisfy a multiple orthogonality condition involving p measures. Here we obtain strong asymptotics for the Q_n in the complex plane outside the common support of these measures as well as on the (finite) open rays of their support. In so doing, we obtain an extension of Weyl's famous theorem dealing with compact perturbations of bounded self-adjoint operators. Furthermore, we derive generalizations of the classical Szeg\"o functions, and we show that there is an underlying Nikishin system hierarchy for the orthogonality measures that is related to the Weyl functions. Our results also have application to Hermite-Pad\'e approximants as well as to vector continued fractions.
Wednesday, September 29, 2010 - 16:30 , Location: Skiles 269 , Dmitriy Bilyk , University of South Carolina , Organizer: Michael Lacey
Low discrepancy point distributions play an important role in many applications that require numerical integration. The methods of harmonic analysis are often used to produce new or de-randomize known probabilistic constructions. We discuss some recent results in this direction.
Monday, September 27, 2010 - 14:00 , Location: Skiles 255 , Michael Barnsley , Department of Mathematics, Australian National University , Organizer: Jeff Geronimo
Let A and B be attractors of two point-fibred iterated function systems with coding maps f and g. A transformations from A into B can be constructed by composing a branch of the inverse of f with g. I will outline the shape of the theory of such transformations, which are termed "fractal" because their graphs are typically of non-integer dimension. I will also describe the remarkable geometry of these transformations when the generating iterated functions systems are projective. Finally, I will show how they can be used to provide new insights into dynamical systems and also how they can be used to manipulate, filter, process and efficiently store digital images, and how they can be used in image synthesis, leading to applications in the visual arts.
Wednesday, September 22, 2010 - 14:00 , Location: Skiles 269 , Michael Goldberg , University of Cincinnati , michael.goldberg@uc.edu , Organizer:
We prove an extension of the Wiener inversion theorem for convolution of summable series, allowing the terms to take values in a space of bounded linear operators.  The resulting algebra is no longer commutative due to the composition of operators.  Inversion theorems arise naturally in the context of proving dispersive estimates for the Schr\"odinger and wave equation and lead to scale-invariant conditions for the class of admissible potentials. All results are joint work with Marius Beceanu.
Wednesday, September 15, 2010 - 14:00 , Location: Skiles 269 , Alexander Volberg , Michigan State , Organizer: Michael Lacey
A2 conjecture asked to have a linear estimate for simplest weighted singular operators in terms of the measure of goodness of the weight in question.We will show how the paradigm of non-homogeneous Harmonic Analysis (and especially its brainchild, the randomized BCR) was used to eventually solve this conjecture.

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