Seminars and Colloquia by Series

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.

Non-homogeneous Harmonic Analysis and randomized Beylkin--Coifman--Rokhlin algorithm (BCR): an application for the solutions of A2 conjecture.

Series
Analysis Seminar
Time
Wednesday, September 15, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Alexander VolbergMichigan State
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.

The Point Mass Problem on the Real Line

Series
Analysis Seminar
Time
Wednesday, September 8, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Manwah WongGeorgia Tech
In this talk, I will talk about recent developments on the point mass problem on the real line. Starting from the point mass formula for orthogonal polynomials on the real line, I will present new methods employed to compute the asymptotic formulae for the orthogonal polynomials and how these formulae can be applied to solve the point mass problem when the recurrence coefficients are asymptotically identical. The technical difficulties involved in the computation will also be discussed.

A Variational Estimate for Paraproducts

Series
Analysis Seminar
Time
Wednesday, September 1, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Yen DoGeorgia Tech
We show variational estimates for paraproducts, which can be viewed as bilinear generalizations of L\'epingle’s variational estimates for martingale averages or scaled families of convolution operators. The heart of the matter is the case of low variation exponents. Joint work with Camil Muscalu and Christoph Thiele.

On complex orthogonal polynomials related with Gaussian quadrature of oscillatory integrals

Series
Analysis Seminar
Time
Wednesday, April 28, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Alfredo DeañoUniversidad Carlos III de Madrid (Spain)
We present results on the asymptotic behavior of a family of polynomials which are orthogonal with respect to an exponential weight on certain contours of the complex plane. Our motivation comes from the fact that the zeros of these polynomials are the nodes for complex Gaussian quadrature of an oscillatory integral defined on the real axis and having a high order stationary point. The limit distribution of these zeros is also analyzed, and we show that they accumulate along a contour in the complex plane that has the S-property in the presence of an external field. Additionally, the strong asymptotics of the orthogonal polynomials is obtained by applying the nonlinear Deift--Zhou steepest descent method to the corresponding Riemann--Hilbert problem. This is joint work with D. Huybrechs and A. Kuijlaars, Katholieke Universiteit Leuven (Belgium).

Interpretation of some integrable systems via multiple orthogonal polynomials

Series
Analysis Seminar
Time
Wednesday, April 21, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Dolores BarriosPolytechnical University of Madrid
Some discrete dynamical systems defined by a Lax pair are considered. The method of investigation is based on the analysis of the matrical moments for the main operator of the pair. The solutions of these systems are studied in terms of properties of this operator, giving, under some conditions, explicit expressions for the resolvent function.

Tangent cones and regularity of real hypersurfaces

Series
Analysis Seminar
Time
Wednesday, April 14, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Mohammad GhomiGeorgia Tech
The tangent cone of a set X in R^n at a point p of X is the limit of all rays which emanate from p and pass through sequences of points p_i of X as p_i converges to p. In this talk we discuss how C^1 regular hypersurfaces of R^n may be characterized in terms of their tangent cones. Further using the real nullstellensatz we prove that convex real analytic hypersurfaces are C^1, and will also discuss some applications to real algebraic geometry.

Extrapolation of Carleson measures

Series
Analysis Seminar
Time
Monday, April 5, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Steven HofmannUniversity of Missouri
We discuss joint work with J.-M. Martell, in which werevisit the ``extrapolation method" for Carleson measures, originallyintroduced by John Lewis to proveA_\infty estimates for certain caloric measures, and we present a purely real variable version of the method. Our main result is a general criterion fordeducing that a weight satisfies a ReverseHolder estimate, given appropriate control by a Carleson measure.To illustrate the useof this technique,we reprove a well known theorem of R. Fefferman, Kenig and Pipherconcerning the solvability of the Dirichlet problem with data in some L^p space.

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