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Series: Analysis Seminar

In his celebrated paper on area distortion under planar quasiconformal mappings (Acta 1994), K. Astala proved that a compact set E of Hausdorff dimension d is mapped under a K-quasiconformal map f to a set fE of Hausdorff dimension at most d' = \frac{2Kd}{2+(K-1)d}, and he proved that this result is sharp. He conjectured (Question 4.4) that if the Hausdorff measure \mathcal{H}^d (E)=0, then \mathcal{H}^{d'} (fE)=0. This conjecture was known to be true if d'=0 (obvious), d'=2 (Ahlfors), and more recently d'=1 (Astala, Clop, Mateu, Orobitg and UT, Duke 2008.) The approach in the last mentioned paper does not generalize to other dimensions. Astala's conjecture was shown to be sharp (if it was true) in the class of all Hausdorff gauge functions in work of UT (IMRN, 2008). Finally, we (Lacey, Sawyer and UT) jointly proved completely Astala's conjecture in all dimensions. The ingredients of the proof come from Astala's original approach, geometric measure theory, and some new weighted norm inequalities for Calderon-Zygmund singular integral operators which cannot be deduced from the classical Muckenhoupt A_p theory. These results are intimately related to (not yet fully understood) removability problems for various classes of quasiregular maps. The talk will be self-contained.

Series: Analysis Seminar

Note change in time.

The theory of geometric discrepancy studies different variations of the following question: how well can one approximate a uniform distribution by a discrete one, and what are the limitations that necessarily arise in such approximations. Historically, the methods of harmonic analysis (Fourier transform, Fourier series, wavelets, Riesz products etc) have played a pivotal role in the subject. I will give an overview of the problems, methods, and results in the field and discuss some latest developments.

Series: Analysis Seminar

It is easy to ask for the number T(g,n) of (rooted) graphs with n edges on a surface of genus g. Bender et al gave an asymptotic expansion for fixed g and large n. The contant t_g remained missing for over 20 years, although it satisfied a complicated nonlinear recursion relation. The relation was vastly simplified last year. But a further simplification was made possible last week, thus arriving to Painleve I. I will review many trivialities and lies about this famous non-linear differential equation, from a post modern point of view.

Series: Analysis Seminar

Pseudodifferential operators and affine pseudodifferential operators arise naturally in the study of wireless communications. We discuss the origins of these operators and give new conditions on the kernels and symbols of pseudodifferential and affine pseudodifferential operators which ensure the operators are trace class (and more generally, Schatten p-class).

Series: Analysis Seminar

The Dirichlet space is the set of analytic functions on the disc that have a square integrable derivative. In this talk we will discuss necessary and sufficient conditions in order to have a bilinear form on the Dirichlet space be bounded. This condition will be expressed in terms of a Carleson measure condition for the Dirichlet space. One can view this result as the Dirichlet space analogue of Nehari's Theorem for the classical Hardy space on the disc. This talk is based on joint work with N. Arcozzi, R. Rochberg, and E. Sawyer

Series: Analysis Seminar

A mapping F between metric spaces is called quasisymmetric (QS) if for every triple of points it distorts their relative distances in a controlled fashion. This is a natural generalization of conformality from the plane to metric spaces. In recent times much work has been devoted to the classification of metric spaces up to quasisymmetries. One of the main QS invariants of a space X is the conformal dimension, i.e the infimum of the Hausdorff dimensions of all spaces QS isomorphic to X. This invariant is hard to find and there are many classical fractals such as the standard Sierpinski carpet for which conformal dimension is not known. Tyson proved that if a metric space has sufficiently many curves then there is a lower bound for the conformal dimension. We will show that if there are sufficiently many thick Cantor sets in the space then there is a lower bound as well. "Sufficiently many" here is in terms of a modulus of a system of measures due to Fuglede, which is a generalization of the classical conformal modulus of Ahlfors and Beurling. As an application we obtain a new lower bound for the conformal dimension of self affine McMullen carpets.

Series: Analysis Seminar

The Horn inequalities give a characterization of eigenvalues of self-adjoint n by n matrices A, B, C with A+B+C=0. The proof requires powerful tools from algebraic geometry. In this talk I will talk about our recent result of these inequalities that are indeed valid for self-adjoint operators of an arbitrary finite factors. Since in this setting there is no readily available machinery from algebraic geometry, we are forced to look for an analysts friendly proof. A (complete) matricial form of our result is known to imply an affirmative answer to the Connes' embedding problem. Geometers especially welcome!

Series: Analysis Seminar

The Horn inequalities give a characterization of eigenvalues of self-adjoint n by n matrices A, B, C with A+B+C=0. The proof requires powerful tools from algebraic geometry. In this talk I will talk about our recent result of these inequalities that are indeed valid for self-adjoint operators of an arbitrary finite factors. Since in this setting there is no readily available machinery from algebraic geometry, we are forced to look for an analysts friendly proof. A (complete) matricial form of our result is known to imply an affirmative answer to the Connes' embedding problem. Geometers in town especially welcome!

Series: Analysis Seminar

Variable transformations are used to enhance the normally poor performance of trapezoidal rule approximations of finite-range integrals I[f]=\int^1_0f(x)dx. Letting x=\psi(t), where \psi(t) is an increasing function for 0 < t < 1 and \psi(0)=0 and \psi(1)=1, the trapezoidal rule is applied to the transformed integral I[f]=\int^1_0f(\psi(t))\psi'(t)dt. By choosing \psi(t) appropriately, approximations of very high accuracy can be obtained for I[f] via this approach. In this talk, we survey the various transformations that exist in the literature. In view of recent generalizations of the classical Euler-Maclaurin expansion, we show how some of these transformations can be tuned to optimize the numerical results. If time permits, we will also discuss some recent asymptotic expansions for Gauss-Legendre integration rules in the presence of endpoint singularities and show how their performance can be optimized by tuning variable transformations. The variable transformation approach presents a very flexible device that enables one to write his/her own high-accuracy numerical integration code in a simple way without the need to look up tables of abscissas and weights for special Gaussian integration formulas.

Series: Analysis Seminar

In any standard course of Analytical Mechanics students are indoctrinated that a Lagrangian have a profound physical meaning (kinetic energy minus potential energy) and that Lagrangians do not exist in the case of nonconservative system. We present an old and regretfully forgotten method by Jacobi which allows to find many nonphysical Lagrangians of simple physical models (e.g., the harmonic oscillator) and also of nonconservative systems (e.g., the damped oscillator). The same method can be applied to any equation of second-order, and extended to fourth-order equations as well as systems of second and first order. Examples from Physics, Number Theory and Biology will be provided.