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Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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).

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.