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THE ABEL SYMPOSIUM 2012

 

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Abstracts

Artur Avila

Global theory of one-frequency Schrödinger operators

Ingrid Daubechies

Sparsity in data analysis and computation

Laszlo Erdős

Universality of spectral statistics for random matrices

Abstract: Random matrices appear in many different branches of mathematics, with applications ranging from statistics, combinatorics and communication networks to quantum mechanics and even number theory. Despite many facets of random matrices, their spectral statistics exhibit a remarkably universal behavior. The celebrated Wigner-Gaudin-Mehta-Dyson conjecture asserts that the local eigenvalue statistics depend only on the symmetry class of the matrix and is independent of the detailed structure of the matrix ensemble. We have recently proved this conjecture by embedding the random matrix into a stochastic flow of matrices and analyzing the relaxation mechanism of the coupled stochastic differential equation for the eigenvalues. This approach has revealed the intrinsic underlying mechanism behind matrix universality. The talk will be an overview of these developments for non-experts, summarizing our recent joint works with P. Bourgade, A. Knowles, B. Schlein, H.-T. Yau and J. Yin.

Hans Feichtinger

Postmodern Fourier Analysis:
Reconsidering Classical Fourier Analysis from a Time-Frequency View-Point

Abstract: Classical Fourier Analysis has developed from Fourier series through Fourier transform in one and several variables. In both situations Lebesgue integration plays a fundamental role, e.g. for Fourier inversion or the verification of Plancherel's theorem. The existence of the Haar measure and Pontrjagin's duality theory for LCA groups have laid the foundation for Harmonic Analysis over LCA groups, as promoted in the book by A.Weil, providing the appropriate natural framework for an abstract Fourier transform, convolution theorems or Plancherel's theorem. Finally the theory of Schwartz tempered distributions has allowed to extend the Fourier transform beyond the setting of functions and has made it a crucial tool for Hörmander's approach to (pseudo-) differential operators. A rich variety of fast codes for the DFT, the discrete Fourier transform (FFT, FFTW, etc.) plays a crucial role in many electronic devices of our daily life, allowing the efficient realization of algorithms for digital signal and image processing.

Modern time-frequency analysis is providing an appropriate framework for signals, functions, or distributions which cannot be analyzed by any of the classical tools, because they assume typically either periodicity of decay at infinity. The very natural way out of this problem is the use of a local Fourier transforms, with a “sliding window”, the so-called STFT (short-time Fourier transform). As a result the objects under consideration are (continuous) functions over the so-called time-frequency plane of phase space. Among the most important windows one finds of course the Gauss function, due to its optimal TF-concentration and its Fourier invariance. In this way a very natural link to Fock spaces (the range of the transform) and to coherent states is given. It is one of the crucial observations that for any decent window the STFT can be completely recovered from its samples over sufficiently dense (in a geometric sense) lattices. The corresponding theory is Gabor analysis, referring to Gabor's seminal paper of 1946.

The mathematical analysis of Gabor expansions is an ongoing enterprize with many interesting and deep mathematical results, with many interesting links between group representation theory (mostly of the Heisenberg group), the theory of function spaces and complex analysis methods. Robustness of Gabor expansions, the study of Gabor multipliers, the localization theory of regular Gabor frames and many other results require typically some mild properties on the windows under consideration, and not just square integrability (which is OK for the continuous STFT). The (minimal TF-invariant) Segal algebra $S_0(R^d)$ appears to be the most appropriate universal space for many of these questions. It is also in the center of a so-called Banach Gelfand triple which allows to describe mapping properties of classical linear mappings, e.g. the Fourier transform, or the correspondence between linear operators and corresponding spreading or Kohn-Nirenberg symbols and have by now an important role in the theory of pseudo-differential operators (cf. the so-called Sjöstrand class).

As we will point out these Banach spaces (of function resp. distributions) are not only appropriate for the description of questions arising in Gabor analysis or TF-analysis in general, but form also a quite appropriate setting for questions of classical Fourier analysis. Functions from $S_0(R^d)$ are appropriate summability kernels, the dual space can be used to discuss questions of spectral analysis, and one can base a relatively elementary theory of generalized stochastic processes on this frame-work. Finally this setting is also well suited for a description of the problem of numerical approximation of continuous questions (e.g. by sampling and periodization) of continuous problems, both in Fourier analysis or TF-analysis.

Stéphane Jaffard

Recent challenges in multifractal analysis

Abstract: The purpose of multifractal analysis is to describe the pointwise singularities of functions and measures, and, in particular, to determine their Hausdorff dimensions. We will present the basic principles of the subject, and show how it allowed to develop interplays between several areas of mathematical analysis, such as harmonic and functional analysis, geometric measure theory, stochastic processes and analytic number theory. We will also present recent applications in signal and image processing and expose some mathematical open problems which are motivated by these applications.

Izabella Laba

Buffon's needle estimates for rational product Cantor sets

Abstract: The Favard length of a planar set E is the average length of its one-dimensional projections. In a joint paper with Bond and Volberg, we prove new upper bounds on the decay of the Favard length of finite iterations of 1-dimensional planar Cantor sets with a rational product structure. This improves on the earlier work of Nazarov-Peres-Volberg, Bond-Volberg, and Laba-Zhai, and introduces new algebraic and number-theoretic methods to this area of research. The estimates are of interest in geometric measure theory, ergodic theory and analytic function theory.

Nikolai Makarov

Eigenvalues of random normal matrices

Abstract: I will give an overview of various aspects of the random matrix model. A special emphasis will be made on recent results (joint with Seung Yeop Lee) concerning topology of the “droplet”, the region in the plane occupied by the eigenvalues when the size of the random matrix is very large. The structure of the random matrix theory will be illustrated in a picture where SLE curves, Julia sets, and Kleinian groups all play a role.

Clément Mouhot

Propagation of chaos and return to equilibrium for Kac's random walks

Abstract: Kac proposed in 1956 to study the derivation of the (spatially homogeneous) Boltzmann equation from a many-particle jump processes with random binary collisions (“Kac's walk”). This limit is closely connected to the notion of propagation of chaos, i.e. product-like structure of the many-particle distribution in the many-particle limit. Several questions were raised: propagation of chaos for the unbounded collision rates encountered in physics, estimations of rates of relaxation uniform in the many-particle limit, propagation of entropic chaos and derivation of the H-theorem. We shall present answers to these questions, obtained in a joint work with Stephane Mischler. They are based on a new quantitative stability approach for many-particle limits, that allows to take advantage of dissipativity of the level of the limit PDE. If time allows we shall discuss connexions with the BBGKY hierarchy, and how this method applies to other many-particle limits.

Alexander Olevskii

Singular distributions and symmetry of the spectrum

Abstract: We discuss the “Fourier symmetry” of measures and distributions on the circle in relation with the size of their support. The talk is based on joint work with Gady Kozma.

Alexei Poltoratskii

Spectral gaps and oscillations

Abstract: How to estimate the size of a gap in the Fourier spectrum of a measure (function, distribution)? How fast should a function with a spectral gap osscillage near infinity? A number of classical problems of Harmonic Analysis, posted by Beurling, Kolmogorov, Krein, Wiener and others can be reformulated in these terms. In my talk I will discuss several of such problems along with their recent solutions.

Eero Saksman

Complex interpolation and rotation of quasiconformal maps

Abstract: We introduce interpolation on Lp-spaces with complex exponents, and apply it to obtain optimal estimates for rotation of quasiconformal maps. The talk is based on joint work with K. Astala, T. Iwaniec and I. Prause.

Eric Sawyer

The indicator/interval testing characterization of the two weight inequality for the Hilbert transform

Abstract: We present joint work with M. Lacey. C.-Y. Shen and I. Uriarte-Tuero that characterizes when the Hilbert transform H_u maps one locally finite weighted space $L^2(u)$ to another $L^2(v)$. Namely, when the strong $A_2$ condition holds together with the pair of dual indicator/interval testing conditions: the $L^2(v)$ norm of $H_u(1_E)$ is at most a fixed multiple of the $L^2(u)$ norm of $1_I$ for all closed subsets $E$ of a finite interval $I$. This provides a first real variable characterization of the two weight inequality, but leaves open the Nazarov-Treil-Volberg conjecture that it suffices to test over only the cases $E=I$ is an interval.

Johannes Sjöstrand

Return to equilibrium, non-self-adjointness and symmetries

Abstract: In this talk we review some results for non-self-adjoint differential operators, where symmetries play an important role. In the case of the Kramers–Fokker–Planck operator we explain how a certain supersymmetry and PT-type symmetry lead to precise results about the reality and the size of the exponentially small eigenvalues in the low temperature limit. This also applies sometimes to coupled oscillators with heat baths, but a recent result says that when the temperatures of the baths are different, the supersymmetry will sometimes break down and would have to be replaced by more direct methods of semi-classical analysis.

The talk is mainly based on joint works with Frédéric Hérau and Michael Hitrik, but uses results and ideas of Witten, Helffer, Nier, Héerau, Bismut, Tailleur – Tanase-Nicola – Kurchan and other people.

Sergei Treil

Commutators, paraproducts and BMO in non-homogeneous martingale harmonic analysis

Abstract: The talk will be devoted to the basic objects of the martingale harmonic analysis in the non-homogeneous settings, in particular the paraproducts and the commutators. These operators appear naturally and play an important role in harmonic analysis.

As it is well known, in the classical homogeneous situation one can characterize BMO by the boundedness of the paraproducts or commutators. It turns out that unlike the classical case, the general situation is much more interesting. For example, unlike the homogeneous case, the boundedness of a paraproduct does not mean that the symbol is in BMO: moreover, the $L^p$-boundedness of the paraproduct does depend on $p$.

However, the so-called extended paraproduct also appears naturally, and the boundedness of this operator characterizes BMO. This implies that in the non-homogeneous case the space BMO also can be characterized by the boundedness of commutators, provided that the underlying martingale transform satisfies some mixing properties.

We will also discuss the geometry of the martingale difference bases in $L^p$.

Yosef Yomdin

Turan and Remez-type inequalities, moments, and Taylor domination for some classes of functions

Abstract: Let a family of analytic functions $f_{\lambda}(z)=\sum_{k=0}^{\infty}a_{k}(\lambda)z^{k}$ be given, and let $R(\lambda)$ be the radius of convergence of $f_\lambda$. The family $f_\lambda$ possesses a property of an $(N,C)$-uniform Taylor domination if $$ |a_{k}(\lambda)|R^{k}(\lambda) \leq C\max_{i=0,\dots,N}|a_{i}(\lambda)|R^{i}(\lambda),\qquad k=N,N+1,\dots, $$ with $N$ and $C$ not depending on $\lambda$. Taylor domination provides, in particular, a uniform bound on the number of zeroes of $f_\lambda$ in each disk strictly contained in the disk of convergence.

One can hope to use Taylor domination in study of the Poincaré first return mapping (i.e. in counting limit cycles) of plane polynomial vector fields. Taylor coefficients in this case satisfy a certain linear differential-difference recurrence relation. We present some results on Taylor domination for simpler, but still important, recurrence relations.

The starting example is the family $R^d_\lambda$ of all rational functions of degree $d$. Here uniform Taylor domination follows from the classical Turan lemma. Equivalently, Taylor domination holds for solutions of a linear recurrence relation with constant coefficients. Next, we discuss linear recurrence relations with the coefficients polynomially depending on the index (Taylor coefficients of solutions of linear polynomial ODE's). In this case we prove a weaker version of Taylor domination, where $C$ in (1) is replaced by $C^k$.

Next we consider Taylor domination for Stieltjes transforms, i.e. for sequences of the consecutive moments of a given function $f$. Here a key tool is counting sign changes of $f$ and a validity of a Remez-type inequality for this function.

If time allows, we shall discuss also a remarkable connection, discovered by Bautin in 1935, of the uniform Taylor domination property to the commutative algebra of the Taylor coefficients, as functions of the parameter $\lambda$.

This is a joint work with D. Batenkov.

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