Past Seminars
academic year 2020 - 2021
APDE Webinar
Thursday, July 15th, 2021, 17:00 -18:00
What is an adequate fractional Sobolev seminorm in arbitrary bounded domains?
Irene Drelichman (Universidad Nacional de la Plata)
Abstract
The non-local character of the Gagliardo seminorm has some undesirable consequences in irregular domains. For instance, the inclusion W^{1,p} \subset W^{s,p} (0<s<1) may not hold. In this talk I will discuss how this seminorm can be replaced by a modified one, and some connections to fractional Sobolev-Poincaré inequalities, interpolation theory and the Bourgain-Brézis-Mironescu limit theorem when s\to 1^-.
The results of this talk are joint work with R. G. Durán
Link to the recorded seminar:
https://drive.google.com/file/d/1JrCzgTzbkcdpEId7hntkkrGhwN70yLrn/view?usp=sharing
APDE Webinar
Thursday, July 8th, 2021, 12:00 -13:00
Sharp L^p estimates for oscillatory integral operators of arbitrary signature
Marina Iliopoulou (University of Kent)
Abstract
The restriction problem in harmonic analysis asks for L^p bounds on the Fourier transform of functions defined on curved surfaces. In this talk, we will present improved restriction estimates for hyperbolic paraboloids, that depend on the signature of the paraboloids. These estimates still hold, and are sharp, in the variable coefficient regime. This is joint work with Jonathan Hickman.
Link to the recorded seminar:
https://drive.google.com/file/d/1FB-N4L8hxQdohmrWK_lv_oZDsVX28vvw/view?usp=sharing
APDE Webinar
Thursday, July 1st, 2021, 17:00 -18:00
On classical inequalities for autocorrelations and autoconvolutions.
Abstract
We will discuss some convolution inequalities on the real line, the study of these problems is motivated by a classical problem in additive combinatorics about estimating the size of Sidon sets. We will also discuss many related open problems. This talk will be accessible for a broad audience.
Link to the recorded seminar:
https://drive.google.com/file/d/1mCRn8pNQa6vuriNoxAR_rYow8U77aP1e/view?usp=sharing
APDE Webinar
Thursday, June 24th, 2021, 12:00 - 13:00
Friedlander comparison theorem for the eigenvalues of the Stokes operator
Clément Denis (Aix-Marseille Université)
Abstract
Link to the recorded seminar:
https://drive.google.com/file/d/1bwD-Ynm83xUxWpnKJJJIBh84CTbB4KEI/view?usp=sharing
APDE Webinar
Thursday, June 17th, 2021, 12:00 - 13:00
Ground State approximation in Quasi-Classical regime for Quantum Fields models
Abstract
When quantum matter interacts with an intense radiation field the microscopic analysis given by Quantum Field Theories (QFT) can be simplified by deriving an effective model by means of semi-classical techniques. In this regime, that we called Quasi-Classical, the field is so intense that it can be considered like a macroscopic object and play the role of a classical environment influencing the subsystem of particles. In this talk we show how the minimization problem for the energy of the microscopic QFT Hamiltonian is equivalent to the minimization of a non-linear Pekar-type functional. Furthermore, we prove that, in the semi-classical limit for the field, the ground state energy and ground state of the QFT models converge to the relative ones of suitable effective Schrödinger operators.
From a joint work with M. Correggi and M. Falconi.
Link to the recorded seminar:
https://drive.google.com/file/d/1SfjfrfEokMHn-wAmtI7UMNk-GloLLzNb/view?usp=sharing
APDE Webinar
Thursday, June 10th, 2021, 17:00 - 18:00
Bulk-edge correspondence at positive temperature
Horia Cornean (Aalborg University)
Abstract
We consider a quantum Hall effect setting, involving unbounded random Schrödinger operators with long range magnetic fields.
Our main result is a general formula, which states that the derivative with respect to the external magnetic field of a given physical bulk quantity equals the expectation of an edge velocity operator associated to the same observable. No spectral (mobility) gaps are required.
In particular, we show that the grand canonical bulk magnetization is -up to a universal constant- equal to an averaged edge charge current flowing parallel to the cut, at all temperatures. By taking the zero temperature limit and imposing a gap condition, we recover the usual equality between bulk and edge Hall conductivities.
This is joint ongoing work with M. Moscolari and S. Teufel.
Link to the recorded seminar:
https://drive.google.com/file/d/1Dwc9MH5YzLRHtZ6ijRSadg43zKB67zCe/view?usp=sharing
APDE Webinar
Thursday, June 3th, 2021, 17:00 - 18:00
The matrix-weighted Hardy-Littlewood maximal function is unbounded
Kristina Škreb (University of Zagreb)
Abstract
The convex body maximal operator is a natural generalization of the Hardy-Littlewood maximal operator. In this work we are considering its dyadic version in the presence of a matrix weight. Surprisingly, it turns out that this operator is not bounded, which is in a sharp contrast to the boundedness of a Doob's inequality in this context. First, it will be discussed how to interpret these operators in a space with matrix weight. For this, we will use convex bodies to replace absolute values (equivalent to the more familiar Christ-Goldberg type definition). We will also discuss the Carleson Embedding Theorems that are the natural partners of these maximal operators and observe a different behaviour as well.
This is a joint work with F. Nazarov, S. Petermichl and S. Treil.
Link to the recorded seminar:
https://drive.google.com/file/d/1LP2HMXHowk9Ird0zDovPvEKbex68theH/view?usp=sharing
APDE Webinar
Thursday, May 27th, 2021, 12:00 - 13:00
On Montgomery's pair correlation conjecture: a tale of three integrals
Abstract
A classic Montgomery conjecture (1973) is closely related to the behavior of the zeros of the Riemann zeta function. The works of Gallagher, Mueller, Goldston, Gonek and Montgomery establish equivalences of the Montgomery conjecture with the asymptotic behavior of 3 integrals. In this talk, we will discuss how we can substantially improve the known upper and lower bounds for these integrals using Fourier analysis. This is a work joint Emanuel Carneiro (ICTP-IMPA), Vorrapan Chandee (Kansas State University) and Micah B. Milinovich (University of Mississippi).
Link to the recorded seminar:
https://drive.google.com/file/d/1AcwZip5a7dRO3DnokOLuAzpiAOW4FvfK/view?usp=sharing
APDE Webinar
Thursday, May 20th, 2021, 12:00 - 13:00
Discrete magnetic Laplacians, covering graphs and spectral gaps.
Abstract
A periodic graph G is an infinite graph on which a finitely generated group H acts and such that the quotient graph G/H is finite. In this talk we will analyze the conditions under which the spectrum of the Laplacian on G has gaps, i.e., its spectrum does not reach all possible values. To address this question we will study the discrete magnetic Laplacian on the finite quotient. A basic tool for the analysis is the definition of a partial order on the class of finite graphs which controls the spectral spreading of eigenvalues under elementary perturbation of the graph (e.g., edge and vertex virtualisation). As a corollary we will prove the Higuchi-Shirai conjecture for Z-periodic trees. Time periming I will mention other possible applications of the preorder (spectral classification of graphs, construction of isospectral magnetic graphs, etc.)
Link to the recorded seminar:
https://drive.google.com/file/d/1HAcsWMHVUQssa6bUH7BzY8ZvqYn57DTz/view?usp=sharing
APDE Webinar
Thursday, May 13th, 2021, 12:00 - 13:00
Static and Dynamical, Fractional Uncertainty Principles
Abstract
How does the mean value of an observable evolve under the action of the linear Schrödinger equation? I will present some results when the observable, or weight, is a fractional power. One of the main tools is the static, fractional uncertainty principle, from which we can deduce a dynamical analogue. Motivated by the Talbot effect, I will show what happens when the initial datum is periodic and, in particular, the Dirac comb. In the latter case the evolution resembles a realization of a Lévy process, and the fluctuations concentrate around rational times and exhibit multifractality. This is a joint work with Sandeep Kumar and Luis Vega.
Link to the recorded seminar:
https://drive.google.com/file/d/1yE3MlsYEMLCaAZXvOWSZ9eYgf4OZHp5T/view?usp=sharing
Bilbao - Bordeaux - Toulouse Rotating Seminar
Tuesday, May 11th, 2021, 15:00 - 16:00
On Dirichlet Laplace eigenfunctions in Lipschitz domains with small Lipschitz constant.
Eugenia Mallinnikova (Stanford)
Abstract
We consider bounded domains in the Euclidean space with Lipschitz boundary and locally small Lipschitz constant. We proof the sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions in such domains. One of our tools is the analysis of the frequency function of a harmonic function vanishing on a part of the boundary.
The talk is based on a joint work with A. Logunov, N. Nadirashvili, and F. Nazarov.
Link to the recorded seminar:
https://drive.google.com/file/d/1U0Cc6UlkL0-uHGkqXyLOGOpUNpYizhXo/view?usp=sharing
APDE Webinar
Thursday, May 6th, 2021, 12:00 - 13:00
Global maximizers for spherical restriction
Diogo Oliveira e Silva (University of Birmingham)
Abstract
We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler--Lagrange equation being smooth, a fact of independent interest which we discuss. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions $d\geq 2$ and general even exponents.
This talk is based on results obtained with René Quilodrán.
Link to the recorded seminar:
https://drive.google.com/file/d/11Wj1ETRoS2QCxmhQbmM9h5o1rJtNnpqa/view?usp=sharing
APDE Webinar
Thursday, April 29th, 2021, 17:00 - 18:00
Pointwise ergodic theorems for bilinear polynomial averages
Abstract
We shall discuss the proof of pointwise almost everywhere convergence for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages. This is my recent work with Ben Krause and Terry Tao. I will also talk about recent progress in this area.
Link to the recorded seminar:
https://drive.google.com/file/d/1rA5AAwvTlmhM_iLiVMhvmvQUvdmxsX82/view?usp=sharing
APDE Webinar
Thursday, April 22th, 2021, 12:00 - 13:00
Fourier interpolation and time-frequency localization
Aleksei Kulikov (NTNU)
Abstract
Link to the recorded seminar:
https://drive.google.com/file/d/1PfS0iIqn4yjx5o9ILQBvBpCVNEugAyYO/view?usp=sharing
APDE Webinar
Thursday, April 15th, 2021, 12:00 - 13:00
Cost of null controllability for parabolic equations with vanishing viscosity
Abstract
Link to the recorded seminar:
https://drive.google.com/file/d/1JC4erOtOcQndsWZvpLmczYtbUiH3EEKR/view?usp=sharing
APDE Webinar
Thursday, March 25th, 2021, 12:00 - 13:00
Weak type Gagliardo seminorms via maximal inequalities
Abstract
The celebrated Bourgain—Brezis—Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. In this talk we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities and sequences of operators. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to the PDE's, ergodic theory, Fourier series, etc. This is joint work with Mario Milman.
Link to the recorded seminar:
https://drive.google.com/file/d/1ONuTIJ5s71IfA6WgpLPPnLxVd6UaKC-N/view?usp=sharing
APDE Webinar
Thursday, March 18th, 2021, 12:00 - 13:00
Half-integer planar point defects in nematic liquid crystals: a qualitative study
Abstract
Nematic liquid crystals are precisely the material appearing in liquid crystal displays, televisions and most displays overall. Despite impressive technological applications, the mathematical theories of liquid crystal still present formidable challenges, related to the interactions of topology, analysis and algebra.
Some of most fascinating and challenging aspects of the mathematical theory of liquid crystals concern the defect patterns and the extent to which the mathematical theory is capable of predicting these patterns. We will survey some recent progress in this direction. Half-integer planar point defects in nematic liquid crystals: a qualitative study
Link to the recorded seminar:
https://drive.google.com/file/d/1CwpSa3ohBO0mfyzbYFqV8yuwh_Z56O2B/view?usp=sharing
APDE Webinar
Thursday, March 11th, 2021, 12:00 - 13:00
Uniqueness and numerical reconstruction for inverse problems dealing with interval size search.
Jone Apraiz (UPV/EHU)
Abstract
Link to the recorded seminar:
https://drive.google.com/file/d/1884_YCWlRKRoejw3Co9gMDpJcxYib8Bt/view?usp=sharing
APDE Webinar
Thursday, March 4th, 2021, 12:00 - 13:00
New endpoint estimates for Calderón-Zygmund operators on von Neumann algebras
José Conde-Alonso (UAM and ICMAT)
Abstract
The classical Calderón-Zygmund decomposition is a fundamental tool that helps one study endpoint estimates for numerous operators near L1. In this talk, we will discuss an extension of the decomposition to a particular operator valued setting where noncommutativity makes its appearance. Noncommutativity will allow us to get rid of the -usually necessary- UMD property of the Banach space where functions take values.
Based on joint work with L. Cadilhac and J. Parcet.
Link to the recorded seminar:
https://drive.google.com/file/d/1hYKlw80Co6-yzkqgRW0n9re_XH-5KAdI/view?usp=sharing
APDE Webinar
Thursday, February 25th, 2021, 12:00 - 13:00
A limiting absorption principle for time-harmonic isotropic Maxwell’s equations
Abstract
Abstract: In this seminar we investigate the L^p-L^q mapping properties of the resolvent associated with the time-harmonic isotropic Maxwell operator. As spectral parameters close to the spectrum are also covered by our analysis, we establish an L^p-L^q type limiting absorption principle for this operator. Our analysis relies on new results for Helmholtz systems with zero order non-Hermitian perturbations.
The talk is based on a joint work with R. Mandel.
Link to the recorded seminar:
https://drive.google.com/file/d/1tx9IX6EnNLZ92cnzGsC-b4enjfDAg67h/view?usp=sharing
APDE Webinar
Thursday, February 18th, 2021, 17:00 - 18:00
A heat equation approach to some problems in conformal CR geometry.
Nicola Garofalo (University of Padova)
Abstract
In this talk I present an approach, based on the heat equation and some of its variants, to various formulas arising in conformal CR geometry. These formulas play a pivotal role in inverting the relevant nonlocal operators as well as in constructing explicit solutions of the fractional CR Yamabe problem.
This is a joint work with G. Tralli.
Link to the recorded seminar:
https://drive.google.com/file/d/1MGnzTl_PEKn2N61BCGWxpPh0GSLQy-yy/view?usp=sharing
APDE Webinar
Thursday, February 11th, 2021, 12:00 - 13:00
Riemann's Non-differentiable function and the binormal curvature flow
Abstract
We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object has a non-obvious non- linear geometric interpretation. We recall that the binormal flow is a standard model for the evolution of vortex filaments. We prove the existence of solutions of the binormal flow with smooth trajectories that are as close as desired to curves with a multifractal behavior. Finally, we show that this behavior falls within the multifractal formalism of Frisch and Parisi, which is conjectured to govern turbulent fluids.
This is a joint work with Valeria Banica
Link to the recorded seminar:
https://drive.google.com/file/d/114L9V3z9Fwt-yu1q3CnZznPZi128sWLh/view?usp=sharing
APDE Webinar
Thursday, February 4th, 2021, 12:00 - 13:00
Anderson localization and regularity of the integrated density of states for random Dirac operators
Sylvain Zalczer (BCAM)
Abstract
Originally coming from the relativistic quantum theory, the Dirac operator is a first-order differential operator which is of great interest in the study of graphene models. After having introduced these models, I will focus on the case of disordered graphene, which is modeled by a random operator. I will prove that we still have the well-known property of disordered system called "Anderson localization", which is the fact that a conducting material becomes an insulator. In a last part, I will deal with the regularity of the integrated density of states for the same model.
This is a joint work with J.-M. Barbaroux and H. D. Cornean.
Link to the recorded seminar:
https://drive.google.com/file/d/1hshg44MV_LSo_b7okf428JKQIsmza-Lq/view?usp=sharing
APDE Webinar
Thursday, January 28th, 2021, 12:00 - 13:00
The small island problem for the degenerate lake equations
Lars Eric Hientzsch (Institut Fourier, University of Grenoble Alpes)
Abstract
The lake equations are introduced as a 2D model for the vertically averaged horizontal component of a 3D incompressible fluid. A lake is described by a 2D domain, the surface of the lake, and a non-negative topography function. The 2D velocity is subject to an anelastic constraint depending on the topography. The equations are degenerate if the topography is allowed to vanish. Indeed, vorticity and velocity are then related through non-uniformly elliptic equations.
In this talk, we prove two stability results for the lake equations for singular geometries and degenerated topographies. First, motivated by natural phenomena such as flooding or erosion we consider a sequence of lakes with an island of vanishing topography that shrinks to a point. In the limit, a vortex-wave type solution is obtained. We highlight key differences between the problem under consideration and the respective problem for the incompressible 2D Euler equations (flat topography). The degeneracy of the topography crucially alters the behavior of solutions.
Second, we address the stability for a sequence of lakes without island for which an island appears in the limit, e.g. due to a decreasing level of water in the lake.
This is joint work with C. Lacave and E. Miot.
Link to the seminar:
https://zoom.us/j/97395120947?pwd=MnRpd3NUcUJidit6bEFJUEx2TUZqUT09
APDE Webinar
Thursday, January 14th, 2021, 12:00 - 13:00
Hölder continuous turbulent weak solutions of the incompressible Euler equations
Abstract
In this talk I will give a general overview on the available results for Hölder continuous wild weak solutions of the incompressible Euler equations. Starting from the celebrated Onsager's conjecture I will explain how some deep regularity of the kinetic energy can be derived even in the range in which it is not necessarily conserved. This is linked to a conjecture by Isett and Oh from 2016, which we recently proved to be true in a joint work with Riccardo Tione, and also to the construction of wild weak solutions of Euler that are smooth outside a set of singular times of quantifiable Hausdorff dimension. The latter is a forthcoming joint work with Silja Haffter in which we also state a conjecture on what the sharp Hausdorff dimension of the singular set of such turbulent solutions could be.
Link to the talk:
https://zoom.us/j/91533421483?pwd=YyttaTNpV0E0QlRXNlZDazJEM2FJZz09
APDE Webinar
Thursday, December 17th, 2020, 12:00 - 13:00
About Schrödinger and Dirac operators with scaling critical potentials
Luca Fanelli (IKERBASQUE and UPV/EHU)
Abstract
Lower-order perturbations of the free Hamiltonians usually appear in Quantum Mechanics, ad models describing the interaction of a free particle with an external field. In some cases, the perturbation lies at the same level as the free Hamiltonian, and the resulting conflict can generate interesting phenomena. We will introduce the Inverse Square and Coulomb potentials as toy models, and describe the main features of the complete Hamiltonians from the point of view of Fourier Analysis, Spectral Theory, and dispersive evolutions.
Link to the recorded seminar:
APDE Webinar
Thursday, December 10th, 2020, 17:00 - 18:00
Sharp constants for maximal operators on finite graphs
Cristian González-Riquelme (IMPA)
Abstract
Let $G$ be a finite graph and $M_{G}$ be its centered Hardy-Littlewood maximal operator defined with respect to the counting measure and the metric induced by the edges. In this talk we will present some recent results concerning the norm $\|M_{G}\|_{p}$ for $p\ge 1$ when $G=K_n$ or $S_n$. Also, we will discuss some sharp constants for the $p-$variation of $M_{G}$ when acting on such graphs. Moreover, we will discuss some open questions that arise in this setting. This talk is based on joint work with José Madrid (UCLA)
Link to the recorded seminar:
https://drive.google.com/file/d/1ZqEmTCuJkQjCW4pJWcmk4lfQSABpn5oe/view
APDE Webinar
Thursday, December 3rd, 2020, 12:00 - 13:00
Sharp estimates in certain theorems of Marcinkiewicz, Sjögren and Sjölin
Odysseas Bakas (Lund University)
Abstract
Extending earlier work of Littlewood-Paley and Marcinkiewicz, Sjögren and Sjölin showed in 1981 that Littlewood-Paley operators formed with respect to lacunary sets of finite order are $L^p$-bounded for all $1<p<\infty$.
Recently, Lerner established sharp weighted $A_2$-estimates for classical Littlewood-Paley operators formed with respect to lacunary sequences, i.e. lacunary sets of order $N=1$. The main aim of this talk is to show how one can use and suitably adapt Lerner’s arguments to obtain sharp weighted $A_2$-estimates for Littlewood-Paley operators formed with respect to lacunary sets of any finite order.
If time permits, some related problems will also be discussed.
Link to the recorded seminar:
APDE Webinar
Thursday, November 26th, 2020, 17:00 - 18:00
Vector-valued extensions of multilinear operators and a multilinear UMD condition
Zoe Nieraeth (BCAM)
Abstract
Vector-valued extensions of important operators in harmonic analysis have been actively studied in the past decades. A centerpoint of the theory is the result of Burkholder and Bourgain that the Hilbert transform extends to a bounded operator on $L^p(\mathbb{R};X)$ if and only if the Banach space $X$ has the so-called UMD property. In the specific case where $X$ is a Banach function space, it is a deep result of Bourgain and Rubio de Francia that this UMD property is equivalent to having bounded vector-valued extensions of the Hardy-Littlewood maximal operator to both $X$ and to its dual $X^\ast$. In this talk I will place these ideas in the context of the modern technique of domination by sparse forms. These forms are intimately related to Muckenhoupt weight classes and the multilsubinear Hardy-Littlewood maximal operator.
Moreover, I will discuss the current progress in extending the UMD property to a multilinear setting. In joint work with Emiel Lorist, we first considered such a condition for tuples of Banach function spaces. Using this notion, we developed a way of obtaining vector-valued sparse domination from scalar-valued sparse domination. I will talk about how this technique provides new quantitative weighted vector-valued bounds for a class of operators including multilinear Calderón-Zygmund operators and the bilinear Hilbert transform.
APDE Webinar
Thursday, November 19th, 2020, 12:00 - 13:00
Construction of quasimodes for non-selfadjoint operators using propagation of wave-packets
Victor Arnaiz (Laboratoire de Mathématiques d’Orsay)
Abstract
In this talk, I will present new results on the study of pseudospectra of non-selfadjoint operators. As it is well known, the precise knowledge of the high-energy distribution of the spectrum of a non-selfadjoint Schrödinger operator is not always enough to obtain proper estimates on the resolvent (contrary to the self-adjoint case), which is essential, for instance, to obtain optimal decay rates for the energy of solutions to the evolution problem governed by such an operator. A more suitable object of study in this framework is the pseudo-spectrum, given by the set of points in the complex plane for which it is possible to find an approximate solution to the eigenvalue problem (which are not necessarily asymptotically close to true eigenvalues). Using recent developments on the description of the propagation of Hagedorn wave-packets by non-Hermitian quadratic operators, we give new constructions of quasimodes which test resolvent estimates for certain semiclassical non-selfadjoint operators satisfying suitable dynamical assumptions.
APDE Webinar
Thursday, November 12th, 2020, 17:00 - 18:00
Approximation and Coincidence: Corona decompositions vs. Big Pieces
Simon Bortz (University of Alabama)
Abstract
Corona-type decompositions are ubiquitous in harmonic analysis. Given a dyadic grid these decompositions say in a quantitative way that some `bad behavior’ does not occur frequently and even more that some `good behavior’ persists in a strong, quantitative manner. This talk will focus on the role of corona decompositions in quantitative (uniform) rectifiability and, more generally, quantitative geometry. In this context, the ideas began with the remarkable work of David and Semmes, who used corona-type decompositions to characterize the boundedness of `nice’ Calderón-Zygmund operators on d-dimensional sets in R^n. Here their corona decomposition is with respect to approximation by Lipschitz graphs.
David and Semmes introduced a related notion called `big pieces’. Here we fix a collection of sets S and say a set ``E is big pieces of S” if at every scale and location (on E) there is a set from S that coincides with E in an `ample’ way. In many ways this coincidence is much easier to work with. For instance, if `nice’ Calderón-Zygmund operators are (uniformly) bounded on S then the same is true for E. The main result presented is that if E admits a corona-type approximation by a family of sets S, then E is `big pieces squared of S’. While this was already shown by Azzam and Schul in the context of uniform rectifiability, we demonstrate that this is a very general phenomenon (in metric spaces, with possibly non-integer dimension, an arbitrary family of approximating sets…). In particular, this theorem has direct applications to parabolic uniform rectifiability. Time permitting I will discuss some future work on parabolic uniform rectifiability and some (perhaps difficult) open problems
APDE Webinar
Thursday, November 5th, 2020, 12:00 - 13:00
Connections between Bombieri-type inequalities and equidistribution of points on Riemannian manifolds
Abstract
APDE Webinar
Thursday, October 29th, 2020, 17:00 - 18:00
Fourier interpolation with the zeros of the Riemann zeta function
Abstract
Originating in work of Radchenko and Viazovska, a new kind of Fourier analytic duality, known as Fourier interpolation, has recently been developed. I will discuss the underlying general duality principle and present a new construction associated with the nontrivial zeros of the Riemann zeta function, obtained in joint work Andriy Bondarenko and Danylo Radchenko. I will emphasize how the latter construction fits into the theory of the Riemann zeta function.
Link to the recorded seminar:
https://eu-lti.bbcollab.com/recording/8a6f8f1c011e484d849d55477f47921c
APDE Webinar
Thursday, October 22nd, 2020, 17:00 - 18:00
On the Boundary Harnack Principle
Daniela De Silva (Barnard College - Columbia University)
Abstract
In this talk we discuss the classical Boundary Harnack Principle and some of its applications to Free Boundary problems. We then present a recent direct analytic proof of this result, for solutions to linear uniformly elliptic equations in either divergence or non-divergence form.
Link to the recorded seminar:
https://eu-lti.bbcollab.com/recording/a0767ba7e3d442d4b9400fa8b831acbb
APDE Webinar
Thursday, October 15th, 2020, 17:00 - 18:00
Fourier interpolation formulas via modular forms
Abstract
In recent years, various results in the field of Fourier interpolation and uncertainty principles have been obtained using the theory of modular forms. I will talk about an extension of an interpolation theorem by D. Radchenko and M. Viazovska to higher dimensions and sketch the main ideas in the proof.
Link to the recorded seminar:
https://eu-lti.bbcollab.com/recording/296885e72d084ffb8384d24a2e4f305d
APDE Webinar
Thursday, October 8th, 2020, 12:00 - 13:00
Stable solutions to semilinear elliptic equations are smooth up to dimension 9
Xavier Cabré (ICREA - Institució Catalana de Recerca i Estudis Avançats &
Universitat Politècnica de Catalunya)
Abstract
The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist singular stable energy solutions. In this talk I will describe a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems.
APDE Webinar
Thursday, October 1st, 2020, 12:00 - 13:00
Gradient of the single layer potential and rectifiability
Carmelo Puliatti (UPV/EHU)
Abstract
APDE Webinar
Thursday, September 24th, 2020, 12:00 - 13:00
Uncertainty principles, interpolation formulas and sphere packing problems
Abstract
In this talk we will discuss how uncertainty principles and interpolation formulas are connected to sphere packing problems and talk about some recent developments on these fronts
APDE Webinar
Thursday, September 17th, 2020, 12:00 - 13:00
On some properties for an incompressible, non-viscous in-out flow in a 2D domain
Marco Bravin (BCAM)
Link: https://eu.bbcollab.com/guest/fbebf6301b4f49f18d7285610e1e6f10
The link to the session will be active from 11:30.
Abstract
APDE Webinar
Thursday, September 10th, 2020, 17:00 - 18:00
Construction of group invariant spaces for approximating functional
data, with applications to digital images
Davide Barbieri (Universidad Autónoma de Madrid)
Link: https://eu.bbcollab.com/guest/fbebf6301b4f49f18d7285610e1e6f10
The link to the session will be active from 16:30.
Abstract
Suppose we are given a finite, typically large, dataset of L^2 functions with domain the Euclidean space or any LCA group, and a semidirect product group G of discrete translations and automorphisms/linear applications acting on such a domain. We consider the problem of approximating the dataset by its projection onto the subspace spanned by the action of G on a finite, ideally small, set of functions, called generators. In this seminar, we will first discuss a constructive proof that provides the generators of the optimal subspace for the approximation, and then see the results of this construction on common datasets of natural images.
This is a joint work with C. Cabrelli, E. Hernández and U. Molter.
Thesis defense
Thursday, September 2nd, 16:00
The Schrödinger equation and uncertainty principles
Mikel Aguirre (Mondragon Unibertsitatea)
Link: https://eu.bbcollab.com/collab/ui/session/guest/8a92de55f8f64f1bba169a2758a2e895
The link to the session will be active from 15:30.