| 14:50 | Noemí Wolanski
(Universidad de Buenos Aires, Argentina) Introduction |
| 14:50 | Stéphane Jaffard
(Université Paris-Est Créteil Val de Marne, France) Wavelets on the hunt for gravitational waves On September 14th 2015, LIGO (Laser Interferometer Gravitational-Wave Observatory) in USA, performed the first detection of a gravitational wave generated by the coalescence of two black holes. The signal processing algorithm which allowed this detection uses in a crucial way a variant of wavelet bases called ''Wilson bases'' (it is an orthonormal ''time-frequency'' decomposition, as opposed to standard wavelets which are of ''time-scale'' type). We will mention the origin of such basses, which starts with the seminal work of Gabor in the 50s, and was made more precise by K. Wilson at the beginning of the 80s (motivated by renormalization theory). We will then show why such bases are particularly well adapted to such gravitational waves, and which technical choices were made in the detection algorithm. Finally, we will mention the perspectives opened by this new type of astronomy which, for the first time, is not based on light or electromagnetic waves detection, and the role that such bases, or variants are expected to play in it. |
| 15:20 | Eugenio Hernández
(Universidad Autónoma de Madrid, Spain) Greedy algorithm and embeddings The greedy algorithm is a way to approximate elements of a Banach space by using the biggest coefficients of the representation of the element in a given basis. We will show how to obtain general embeddings between a Banach space and weighted Lorentz spaces and use them to quantify how good is this algorithm in comparison with the best approximation. Several examples will be presented. Joint work with P. Berná, O. Blasco, G. Garrigós and T. Oikhberg. |
| 16:00 | Chris Heil
(Georgia Institute of Technology, USA) Wavelets, Self-Similarity, and the Joint Spectral Radius: A Retrospective Three decades ago, the "Wavelet Revolution" unexpectedly brought us the construction wavelet orthonormal bases generated by smooth functions. Central to these constructions is the theory of multiresolution analyses and the existence of refinable \({\it scaling\,\,functions}\), whose graphs exhibit a certain type of Self-similarity. In this talk we will examine wavelets, refinablity, and self-similarity from the point of view of the \({\it joint\,\,spectral\,\,radius}\) of families of matrices, following the work of Cabrelli and Molter over time in the development of generalized self-similarity, multiwavelets, and multiwavelets in higher dimensions. |
| 16:30 | Sheldy Ombrosi
(Universidad Nacional del Sur, Argentina) Weighted endpoint estimates for commutators of Calderón-Zygmund operators. In this talk we will show recent advances in borderline weighted estimates for the Coifman, Rochberg and Weiss commutator. Given a symbol \(b\in{\bf BMO}\) and a Calderón-Zygmund operator \(T\) we study two-weight estimates for the commutator \([b, T]\). We also obtain quantitative estimates in the case of the weights satisfy certain additional geometric conditions. This talk is based in a joint work with Andrei Lerner and Israel Rivera-Ríos. |
| 17:30 | Ricardo Durán
(Universidad de Buenos Aires, Argentina) Improved Poincaré inequalities in fractional Sobolev spaces. First we recall the classic Poincaré and Sobolev-Poincaré inequalities for functions with vanishing mean value in a bounded domain. Then we present the so-called improved versions of these inequalities and mention some important consequences. Finally, we present generalizations of these inequalities for functions in fractional order spaces for the cases of John, s-John, and H\"older $\alpha$ domains, and we discuss the optimality of our results. The results presented were obtained in collaboration with Irene Drelichman. |
| 9:40 | John Benedetto
(University of Maryland, USA) Super-resolution by means of Beurling minimal extrapolation We address the super-resolution question: Given spectral data defined on a finite set of $d$-dimensional multi-integers; of all complex Radon measures on the $d$-dimensional torus, whose Fourier transform equals this data, does there exist exactly one with minimal total variation? We first note that this is a mathematical formulation of a large class of super-resolution problems that arises in image processing, that it generalizes some fundamental problems in compressed sensing, and that it has wide ranging applications in other fields. We prove a theorem that has quantitative implications about the possibility and impossibility of constructing such a unique measure. Our method introduces the notion of an admissibility range that fundamentally connects Beurling's theory of minimal extrapolation with the Candes and Fernandez-Granda theory of super-resolution. The method is also well-suited for the construction of explicit examples. This is part of an on-going collaboration with Weilin Li. |
| 10:20 | Luis Caffarelli
(University of Texas, USA) A problem of interacting obstacles Originally considered by Chipot and Vergara-Caffarelli this family of problems concerns "deformable" functions, pressing against each other like an elastic body against a membrane or the income and cost of an asset. We will discuss the existence and geometry of solutions. |
| 11:30 | Irene Martínez Gamba
(University of Texas, USA) Existence and uniqueness theory for binary collisional kinetic models We focus on Chapman-Kolmogorov type equations for kinetic evolution models binary interactions and develop an existence and uniqueness theory by posing the problems as solving an ODE in a suitable Banach space. These models range from the classical Boltzmann equation for rarefied elastic gases with hard potentials as transition probability rates, to quantum Boltzmann models for Bose-Einstein Condensates (BEC) at very low temperature, to wave-turbulence models in stratified flows such as the Zakharov equation. We show they share a common framework as being evolution problems for non-local multi-linear flows, whose solutions are continuous probability densities. There natural solutions spaces are those of observables, that is, integrable functions with polynomial weights (i.e. finite polynomial moments or expectations). We show that solutions are constructed by means of solving ODEs in a convex, bounded subspace of positive functions in the Banach space with suitable weighted integrable functions, where polynomial moments estimates and interpolation tools are enough to show that a Holder, sub-tangent and one-sided Lipschitz conditions hold. This work is in collaboration with Ricardo Alonso, Leslie Smith and Minh Binh Tran. |
| 12:00 | Eleonor Harboure
(IMAL - CONICET, Argentina) Local Calderón-Zygmund theory on proper open subsets Inspired on the local Hardy-Littlewood and the Hilbert transform on \(\Omega = (0,\infty)\), a proper open subset of \(\mathbb{R}\), we develop a local Calderón-Zygmund theory in a quite general context. We start with \((X,d)\), a metric space having the PHD property, a purely geometric condition, and a proper open subset \(\Omega\subset X\). We further assume that a Borel measure is defined in \(\Omega\) having the doubling property only for a family of balls that ''stay away'' from \(\partial\Omega\). We construct a Hardy-Littlewood maximal operator on this family and establish a theory of weights for the \(L^p\)-spaces. We introduce also the corresponding singular integrals that, to some extent, are controlled by the maximal function. All these operators are local in the sense that when applied to a function \(f\), their values at a point \(x\) only depend on the values of \(f\) on some neighborhood of \(x\). These results are part of joint work with O. Salinas and B. Viviani. |
| 12:30 | Akram Aldroubi
(Vanderbilt University, USA) Who is Ursula Molter? |