Seminars and Colloquia by Series

Discrete Littlewood-Paley analysis and multiparameter Hardy spaces

Series
Analysis Seminar
Time
Wednesday, November 17, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Guozhen LuWayne State
In this talk, we will discuss the theory of Hardy spacesassociated with a number of different multiparamter structures andboundedness of singular integral operators on such spaces. Thesemultiparameter structures include those arising from the Zygmunddilations, Marcinkiewcz multiplier. Duality and interpolation theoremsare also discussed. These are joint works with Y. Han, E. Sawyer.

Weighted estimates for quasilinear equations with BMO coefficients on Reifenberg flat domains and their applications

Series
Analysis Seminar
Time
Wednesday, November 10, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Nguyen Cong PhucLSU
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.

Exit times of diffusions with incompressible drifts

Series
Analysis Seminar
Time
Wednesday, November 3, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Andrej ZlatosUniversity of Wisconsin, Madison
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.

Rational Inner Functions in the Schur-Agler Class

Series
Analysis Seminar
Time
Wednesday, October 27, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Greg KneseUniversity of Alabama
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.

Polya sequences, gap theorems, and Toeplitz kernels

Series
Analysis Seminar
Time
Wednesday, October 20, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Mishko MitkovskiGeorgia Tech
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.

Sobolev orthogonal polynomials in two variables and partial differential equations

Series
Analysis Seminar
Time
Wednesday, October 6, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Miguel PinarDpto. Matematica Aplicada, Universidad de Granada
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.

Higher-Order Three-Term Recurrences and Asymptotics of Multiple Orthogonal Polynomials

Series
Analysis Seminar
Time
Tuesday, October 5, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Sasha AptekarevKeldish Institute for Applied Mathematics
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.

Analysis in constructions of low discrepancy sets

Series
Analysis Seminar
Time
Wednesday, September 29, 2010 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Dmitriy BilykUniversity of South Carolina
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.

Theory and applications of fractal transformations

Series
Analysis Seminar
Time
Monday, September 27, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Michael BarnsleyDepartment of Mathematics, Australian National University
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.

A non-commutative Wiener Inversion Theorem and Schroedinger dispersive estimates

Series
Analysis Seminar
Time
Wednesday, September 22, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Michael GoldbergUniversity of Cincinnati
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.

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