Hans Georg Feichtinger: Function spaces for irregular sampling

Abstract: The role of Wiener Amalgam Spaces in the theory of irregular sampling Many results about regular sampling, such as the famous Shannon sampling theorem and its practical version variants valid for weighted $L^p$-spaces rely in variants of Poisson's formula, i.e. on the strict lattice structure of the sampling set. Perturbation results like the famous Kadec theorem allow small perturbations of such a situation, but do not cover a variety of cases where the sampling set is less structured, although stable recovery is still possible.

Here frame theory, or better the theory of Banach frames comes in, and one can develop iterative algorithms with guaranteed convergence (at a geometric rate), valid for families of Banach spaces of functions or distributions (so far beyond the usual Hilbert space setting of $L^2(R^d)$).

Dealing with infinite sums, taking samples at well spread sampling points, verifying robustness results (e.g. against jitter error) requires the use of a family of function spaces, called Wiener Amalgam Spaces, which are well suited when it comes to combine local with global considerations under circumstances much more general than those of perturbed lattices.

In this setting one also finds an appropriate notion of ``controlled point sets'' (the relatively separated ones), which in fact can be split into a finite union of (effectively) separated point sets. This situation was in fact relevant for the formualtion of the so-called Feichtinger conjecture (which has been answered positively as a consequence of the Kadison-Singer conjecture, in 2014).

Estimates and statements based mostly on the behaviour of pointwise multipliers and convolution operators on those Wiener Amalgam Spaces will be shown demonstrating the usefulness of this machinery in the context of irregular sampling (e.g. for band-limited functions or spline-type functions) as well as in the theory of irregular Gabor transforms.

Uwe Grimm: Recent progress in mathematical diffraction

Abstract: Diffraction methods continue to provide the main tool for the structure analysis of solids. The corresponding inverse problem of determining a structure from its diffraction pattern is difficult and, in general, does not define a structure uniquely. Kinematic diffraction, which is an approximation that is reasonable for X-ray diffraction where multiple scattering effects can be neglected, is well suited for a mathematical approach via measures. Measures provide a natural mathematical concept to quantify the distribution of matter in space as well as the distribution of scattering intensity. This approach has substantially been developed since the discovery of quasicrystals required an extension of the methods used to compute the diffraction of periodic crystals.
The need for further insight emerged from the question of which distributions of matter, beyond perfectly periodic crystals, lead to pure point diffraction patterns, hence to diffraction patterns comprising sharp Bragg peaks only. More recently, it has become apparent that one also has to study continuous diffraction in more detail, with a careful analysis of the different types of diffuse scattering involved.
In this mini-course, I will present an introduction to the theory and discuss general results on the basis of various characteristic and explicit examples, putting particular emphasis on the analysis of non-periodic structures such as mathematical quasicrystals.

Karlheinz Gröchenig: Deformation of Gabor frames

Abstract: We introduce a new notion for the deformation of Gabor frames. Such deformations are in general nonlinear and, in particular, include the standard jitter error and linear deformations of phase space. With this new notion we prove a strong deformation result for Gabor frames and Gabor Riesz sequences that covers the known perturbation and deformation results. Our proof of the deformation theorem requires a new characterization of Gabor frames and Gabor Riesz sequences.

Michael Kreisel: Gabor frames for quasicrystals

Abstract: When studying discrete Gabor systems, we often assume that the time-frequency shifts form a lattice. In these talks, we will look at Gabor systems where the time-frequency shifts are supported on a quasicrystal. Since quasicrystals do not generally form groups, we cannot study them using the same techniques as lattice Gabor frames. Instead we will study these Gabor frames by using a groupoid which is naturally associated to a quasicrystal. By studying certain twisted groupoid C*-algebras, we are able to show the connections between the physics of quasicrystals and Gabor theory. I will discuss the possibility of a Janssen representation for such frames, which is related to the (non)existence of tight multiwindow Gabor frames supported on a quasicrystal. I will also discuss potential applications to the twisted version of Bellissard’s gap labeling conjecture, and to the HRT conjecture.

Lorenzo Sadun: Tiling spaces and topology

Abstract: In a pair of lectures, we'll go over the topological structure of tiling spaces. In the first lecture, we'll go over examples of tiling spaces (e.g. substitutions, or cut-and-project), their local topology, and their realization as inverse limits. In the second lecture, we'll go over different notions of tiling cohomology, what this cohomology tells us about tiling spaces, and what it tells us about maps between tiling spaces.


Abstract: The first quasicrystals where discovered by Dan Shechtman in the year 1984. By a fast cooling down process of a Aluminium-Manganese composition new symmetries of the atomic positions where produced which where incompatible with a translation symmetry. Aluminium as well as Manganese have a crystal structure and so it is natural to ask whether related Schrödinger operators can be approximated by Schrödinger operators arising by such kind of crystals, i.e. periodic Schrödinger operator. We will adress this question to the case of discrete Schrödinger operators on 2 (Z). As it turns out under weak assumptions on the system the spectrum of a Schrödinger operator can be approximated as a set by periodic ones.
After stating the result the basic definitions will be provided. The important ingredient to guarantee the convergence of the spectrum with respect to the Hausdorff distance will be discussed as well as the existence of periodic approximants.

Jean Bellissard: Modeling Liquids and Bulk Metallic Glasses

Abstract: After a review of the known properties of bulk metallic glasses (BMG) and their liquid phase, the concept of anankeon, introduced by Egami will be explained. Liquids can be described thermodynamically as a free gas of anakeons, predicting the Dulong-Petit law abserved for the heat capacity. The shear transformation zone theory (STZ) due to Langer at al. using also the anankeon, explains the plasticity properties of the glass phase. Then a microscopic description based on the properties of Delone sets will be offered. The concept of Pachner moves, used in Computational Geometry, will be the key point to represent microscopically the anankeons. The Graph of Contiguity will be introduced and will serve as a basis for a Markov process liable to describe the dynamics of the anankeons. Agreements with numerical simulations will be presented. If the time allow, arguments will be given in favor of a description of the liquid-glass transition in terms of the Sherrington-Kirkpatrick mean field theory for spin glasses. Several open problems for mathematicians will be offered in conclusion.

Pierre-Henry Collin: C*-algebras of Penrose hyperbolic tilings and associated K-theory

Abstract: We will construct an hyperbolic tiling arising from an usual substitution tiling. Using several continuous hulls of this tiling we can then construct the C*-algebra of a Penrose hyperbolic tilings. We will finally show the link between the K-theory of the tiling with classical computation of K-theory in the case of substitution tilings.

Jean-Pierre Gabardo: Frames of exponentials on small sets

Abstract: If x_1, …, x_m are finitely many points in R^d, let E_\epsilon = \cup_{i=1}^m x_i + B_\epsilon, where B_\epsilon= \{ x \in R^d : |x| < \epsilon \} and let \hat{f} denote the Fourier transform of f. Given a positive Borel measure \mu on R^d, we provide a necessary and sufficient condition for the inequalities
A || f ||^2_2 \leq \int_{R^d} |\hat{f} (\xi) |^2 d\mu(\xi) \leq B || f ||^2_2
(f \in L^2(E_{\epsilon}), for some A,B>0 and for some \epsilon > 0 sufficiently small. If G is a (possibly dense) subgroup of R, we characterize those measures \mu for which the inequalities above hold whenever x_1, …, x_m are finitely many points in G (with \epsilon depending on those points but not A or B). We also point out an interesting connection between this problem and the notion of well-distributed sequence.
This is joint work with Chun-Kit Lai (San Francisco State U.).

Sigrid Grepstad: Universal sampling on quasicrystals

Abstract: Matei and Meyer have shown that simple quasicrystals are universal sampling sets. In this talk we look closer at the critical case when the average sampling rate equals the measure of the spectrum. We will see that in this case, an arithmetical condition on the quasicrystal determines whether it is a universal set of “stable and non-redundant” sampling. This is joint work with Nir Lev.

Deguan Han: Group representation frames: Multi-frames meet super-frames through duality

Abstract: There is an abstract version of the Gabor systems duality principle for group representations, and it has some connections with the classification of the free-group von Neumann algebras. In this talk I will revisit this general duality principle, and discuss some recent both trivial and nontrivial observations that establish some (hopefully new) connections between "super-frames" and "muti-frames" through a dual pair of group representations. In particular, this allows a generalization of the Gabor frame duality principle to subspace Gabor frames.

Johannes Kellendonk: Characterising Delone sets by means of their dynamical system

Abstract: We consider the dynamical system associated to a repetitive Delone set of finite complexity and show how the dynamical properties characterise geometrical properties of the set.

Mihalis Kolountzakis: Uniqueness of the Poisson Summation Formula, applications and limitations

Abstract: A "Uniqueness of the Poisson Summation Formula"-type theorem generally says, under varying assumptions, that whenever the Fourier Transform of a discrete measure is also a discrete measure then this Fourier pair can arise by finitely many applications of the PSF and elementary operations such as linear combiinations, dilations, translations and modulations. In the study of tilings by translations, such uniqueness theorems are used as a source of structure in tilings. We will show such examples in dimensions 1, 2 and 3. We will also discuss recent results according to which the assumptions of such a uniqueness theorem cannot be weakened too much. In other words, we will see examples of discrete Fourier pairs that cannot arise by applying the PSF finitely many times.

Chun Kit Lai: Exponential bases and Fourier frames on fractals

Abstract: We study the construction of exponential bases and Fourier frames on general L^2 space with different measures, particularly the measures supported on fractals. This problems date back to conjecture of Fuglede. It lies at the interface between analysis, geometry and number theory and it relates to translational tilings. The frequency set constructed is hardly periodic, but demonstrates certain self-similarity. In this talk, we give an introduction to this topic, and report on some of the recent advances.

Ian Putnam: Introduction to Smale spaces.

Abstract: I will start with a brief introduction to Smale's program for smooth dynamical systems and how Smale spaces, as defined by David Ruelle fit into the picture. Most of that talk will be devoted to describing examples of Smale spaces, which include substitution or self-similar tilings and some other fractal objects.

José Luis Romero: Localization operators and time-frequency covers

Abstract: Localization operators formalize the notion of acting on a function by cutting a piece of its time-frequency description, as much as permitted by the uncertainty principle. The talk will be an introduction to these operators and to the problem of quantifying the relation between a function and a sequence of time-frequency localized pieces. The expectation is that such a quantitative equivalence holds when the set of time-frequency localization regions forms a well-behaved cover. I will discuss results in this direction with special emphasis on irregular covers, i.e. covers not arising as lattice translates of a set.

2015-06-03, Antoine Julien