BCAM, Tuesday, July 23th, 2024, 17:00 - 18:00
Title: A different approach to endpoint weak-type estimates for Calderón-Zygmund operators
The weak-type (1,1) estimate for Calderón-Zygmund operators is fundamental in harmonic analysis. We investigate weak-type inequalities for Calderón-Zygmund singular integral operators using the Calderón-Zygmund decomposition and ideas inspired by Nazarov, Treil, and Volberg. We discuss applications of these techniques in the Euclidean setting, in weighted settings, for multilinear operators, for operators with weakened smoothness assumptions, and in studying the dimensional dependence of the Riesz transforms.
UPV/EHU, Thursday, June 20th, 2024, 12:00 - 13:00
Title: A Rough Nikodym Maximal Function Estimate
We introduce a generalisation of the local smoothing problem for the wave equation, where time averaging is carried out with respect to an Ahlfors—David-regular measure $\mu$ supported on $[1,2]$. An incidence geometry based proof of the $L^p$ boundedness of the associated Nikodym maximal function is outlined, which is novel even in the classical setting where $\mu$ is Lebesgue measure restricted to $[1,2]$.
UPV/EHU, Thursday, May 23rd, 2024, 12:00 - 13:00
Title: Some extremal problems in de Branges spaces
Louis de Branges introduced a class of Hilbert spaces of entire functions which now bear his name. They are generalizations of the classical Paley–Wiener space of square integrable functions whose Fourier transform is compactly supported. Evaluation functionals are continuous in these spaces, so they enjoy a reproducing kernel structure and there are corresponding interpolation formulas with nodes at zeros of certain auxiliary functions. We explore some extremal problems in these spaces and discuss their applications towards existence of zeros in families of L-functions and sharp Poincaré inequalities.
BCAM, Thursday, May 16th, 2024, 17:00 - 18:00
Title: Convolution Inequalities and Applications
In this talk we will discuss a collection of convolution inequalities for real valued functions on the hypercube, motivated by combinatorial applications.
UPV/EHU, Thursday, May 9th, 2024, 12:00 - 13:00
Title: Catalan operators in Operator Theory
Let $c=(C_n)_{n\ge 0}$ be the Catalan sequence and $T$ a linear and bounded operator on a Banach space $X$ such $4T$ is a power-bounded operator. The Catalan generating function is defined by the following Taylor series,
C(T) := \sum_{n=0}^\infty C_nT^n.
Note that the operator $C(T)$ is a solution of the quadratic equation $TY^2-Y+I=0.$ In this talk we study this algebraic equation in the case that $T$ is the infinitesimal generator of a C_0-semigroup. We express $C(T)$ by means of an integral representations which involves the resolvent operator $(\lambda-T)^{-1}$ or the C_0-semigroup. In the case that $T$ is a bounded operator, we define powers of the Catalan generating function $C(T)$ in terms of the Catalan triangle numbers. Finally, we give some particular examples to illustrate our results and some ideas to continue this research in the future. This is a research proyect with Alejandro Mahillo (Universidad de Zaragoza) and Natalia Romero (Universidad de La Rioja).
Online, Tuesday, May 7th, 2024, 11:00 - 12:00
Title: Minimality of the vortex solution for Ginzburg-Landau systems
We consider the standard Ginzburg-Landau system for N-dimensional maps defined in the unit ball for some parameter eps>0. For a boundary data corresponding to a vortex of topological degree one, the aim is to prove the (radial) symmetry of the ground state of the system. We show this conjecture in any dimension N≥7 and for every eps>0, and then, we also prove it in dimension N=4,5,6 provided that the admissible maps are curl-free. This is part of joint works with L. Nguyen, M. Rus, V. Slastikov and A. Zarnescu.
UPV/EHU, Thursday, May 2nd, 2024, 12:00 - 13:00
Title: Singular integrals: Multi-parameter framework and Zygmund dilation
Over a half-century ago, Alberto Calderón and Antoni Zygmund made groundbreaking studies that initiated the study of modern harmonic analysis that focuses on singular integrals. These operators still lie at the core of today's harmonic analysis. Recently, studies of singular integrals have been fast-paced due to new techniques that are based on similar underlying methods as in the classical studies by Calderón and Zygmund. These new techniques, namely the sparse method and the representation theory, have enabled us to find, for example, sharp weighted estimates and handle more complex operators such as multi-parameter singular integrals. Multi-parameter framework refers to the study of singular integrals whose singularity is more delicate to work with as it is expanded over the underlying space.
In this talk, I will be giving an introduction to the study of singular integrals and extensions to the multi-parameter framework, where we will be focusing, especially on the kernels that are dilation invariant under a specific "entangled" dilation - the so-called Zygmund dilation.
BCAM, Thursday, April 25th, 2024, 17:00 - 18:00
Title: Uncertainty principles for bandlimited functions
In this talk I plan to have a broad discussion on certain types of uncertainty principles for bandlimited functions (i.e. functions that have Fourier transforms compactly supported), and its connections to multiplication operators in certain Hilbert spaces of entire functions. Some of these problems are related to applications in analysis and number theory, and I also plan to describe a few of these.
The talk should be accessible to a broad audience.
BCAM, Tuesday, April 23rd, 2024, 17:00 - 18:00
Title: Limiting embedding of Fractional Sobolev Spaces
The Bourgain-Brezis-Mironescu(BBM) formula gives us a way to express the Sobolev seminorms in terms of the fractional Sobolev seminorms and thus gives an alternative characterization of Sobolev spaces without the notion of weak derivatives. In this talk, I shall discuss the origin and some recent developments on the extensions of the BBM formula. In the first part of the talk, I shall discuss the case of Sobolev extension domains, and in the second part, I shall talk about BBM formula for spaces related to Triebel-Lizorkin spaces.
Part of the results in this talk are based on a joint work with Kaushik Bal and Prosenjit Roy.
UPV/EHU, Thursday, April 18th, 2024, 12:00 - 13:00
Title: Continuum limit for a discrete Hodge-Dirac operator on square lattices
In this talk, we examine the limit of Schrödinger and Dirac operators defined on the n-dimensional square lattice hℤn as h approaches 0. We review the positive results obtained for Schrödinger operators and the negative ones for the Dirac case, which is related with the so-called "fermion doubling". Then we introduce a discrete Hodge-Dirac operator, which is different from the usual discretization of the Dirac operator. To be able to define this operator, we begin by establishing an alternative framework for a higher-dimensional discrete differential calculus. We express our operator as a differential operator acting on discrete forms and obtain the limit to the continuous standard Hodge-Dirac operator. In the final part, we discuss the relation of the discrete Hodge-Dirac operator with other alternatives aimed at overcoming the "fermion doubling".
BCAM, Thursday, April 11th, 2024, 15:00 - 16:00
Title: Some problems related to the Fourier decay of homogeneous self-similar measures
In this seminar, we are going to talk about the behavior of homogeneous self-similar measures and their Fourier transform. It is known that, in some cases, the Fourier transform of a self-similar measure does not go to zero as the frequencies go to infinity; nevertheless, it can have power decay outside of a sparse set of frequencies. We will go over this result and show some consequences of it. Additionally, we will discuss about the $L^p$ improvement property of such measures as part of an ongoing project.
Online, Tuesday, April 2nd, 2023, 11:00 - 12:00
Title: Microlocal analysis of strong magnetic fields, from magnetic bottles to edge states
I will talk about recent work with Rayan Fahs, Loïc Le Treust, Léo Morin, and Nicolas Raymond. It concerns the spectral study of purely magnetic Schrödinger operators in dimension 2, in the limit of large fields, which is transformed into a semiclassical limit. A precise geometric and microlocal analysis (of "normal forms" ) gives a very useful heuristic to reduce the problem to an effective 1D operator. I will present the case of the confinement of classical and quantum particles by a variable magnetic field, as well as more recent work on the appearance of edge states on bounded domains in the plane, with constant magnetic field. In both cases we obtain spectral asymptotics with 2 or more terms, for Weyl formulas but also for the precise individual descriptions of a large number of eigenvalues, and their relation with the Landau levels.
UPV/EHU, Tuesday, March 26th, 2024, 12:00 - 13:00
Title: Statistical stationary solutions to the compressible Rayleigh-Benard convection problem
Stationary statistical solutions (invariant measures) represent a standard tool for describing turbulent phenomena in fluid mechanics. We consider the complete Navier-Stokes-Fourier system describing the motion of a compressible, viscous and heat conducting fluid driven by boundary temperature fluctuations (Rayleigh-Benard problem). We show that any omega-limit set associated to a global in time (weak) solution supports an invariant measure - stationary statistical solution. The proof is based on careful analysis of propagation of density oscillations.
Joint work with A. Swierczewska-Gwiazda (Warsaw)
BCAM, Thursday, March 21th, 2024, 17:00 - 18:00
Title: Degenerate Boundary Value Problems
In Boundary Value Problems (BVPs) one aims to understand solutions to a differential equation (the problem) under some constraint (value at the boundary).
On euclidean space, BVPs can be attacked with Fourier methods: the Fourier transform provides us with a representation of the solution and a characterisation of the trace space of the solutions. What happens when the boundary of our domain becomes rough and symmetries are lost? Can we still find a way to describe solutions and trace space when Fourier methods break down? Ultimately: how do solutions (and these methods) depend on small perturbations of the boundary?
In this talk I will introduce the "first order approach" for divergence form equations -div A ∇u = 0, which relates harmonic extensions from the real line and holomorphic functions. This relation works in higher dimensions as well, and allows us to rewrite our problem in a suitable way so that holomorphic functional calculus can be applied.
We will take a closer look at degenerate BVPs: when the coefficient A(x) of our divergence form equation lacks uniform boundedness and accretivity, and can exhibit singularities. Current state-of-the-art results can handle singularities characterised by scalar Muckenhoupt weights. These results have been recently extended on manifolds satisfying some curvature assumption. But even on flat euclidean space, anisotropic degenerate coefficients have been out of reach, due to the lack of off-diagonal estimates. Is there another way to handle more general matrix-degenerate coefficients? How far can we push the new methods before the theory falls apart?
New results are from a joint work with Andreas Rosén.
UPV/EHU, Tuesday, March 19th, 2024, 12:00 - 13:00
Title: Hardy inequalities in the Heisenberg group
In this talk, we revisit Hardy inequalities for the Heisenberg sub-Laplacian, first proved by Garofalo and Lanconelli in the 90s.
We discuss what happens when the Hardy weight is not of Garofalo-Lanconelli-type, and is instead associated with the sub-Riemannian distance, i.e., the control distance induced by the Heisenberg distribution. Surprising phenomena concerning the optimal constant emerge, raising several unanswered questions.
Time permitting, we delve into Hardy inequalities for Folland-Stein operators, which generalize the Heisenberg sub-Laplacian, and their connection with a newly introduced concept of magnetic sub-Laplacian.
Based on joint works with Dario Prandi and Biagio Cassano, David Krejčiřík and Dario Prandi.
UPV/EHU, Thursday, March 14th, 2024, 12:00 - 13:00
Title: Asympotic of fast rotating incompressible fluids
In this talk, we are going to present the problem of the asympotic of a family of global weak solutions of the incompressible Navier-Stokes in the case of fast rotation. This problem is the toy problem related to the study of large scale oceanic motion. After recalling the classical result when there is no topography, we are going to present the problem in the case of topography. The result turns out to be very different and compactness method plays an important role.
BCAM, Thursday, March 7th, 2024, 17:00 - 18:00
Title: Test function method in KdV hierarchy
One can extend KdV equation from 1 dimension to higher dimension finding many equations that appears in mathematical physics. For example one has KP, BLMP, YTSF, CBS equations. They inherits from KdV a quasilinear evolution structure: L(∂t, ∇)u + Q(u, ∇u) = 0 with linear L and Q in divergence form. We imagine to perturb one of these equations with a semilinear forcing term. We gain Lu + Q(u, ∇u) = N(u, ∇u) with polynomial growth for N. The interaction between L, Q, N destroy the symmetry of the equations and for suitable N one can prove non-existence of solutions and lifespan estimate. The test function method is the first idea for proving this, but due to the peculiar for of L and Q something new appears in initial data condition.
Online, Tuesday, March 5th, 2024, 11:00 - 12:00
Title: Confinement versus Singularity for incompressible Euler in high dimensions
The three-dimensional incompressible Euler equations under axisymmetry have been widely studied. While the “no-swirl” assumption makes the system very similar to the two-dimensional vorticity equations, it is still possible for solutions to have unbounded vortex stretching. After reviewing classical confinement results in two dimensions, we report some progress on the issue of vortex stretching for Euler equations under rotational symmetries in three and higher dimensions. (Based on joint works with Kyudong Choi and Deokwoo Lim.)
UPV/EHU, Thursday, February 29th, 2024, 12:00 - 13:00
Title: Selfimproving phenomena: Poincaré-Sobolev inequalities and BMO estimates
In this talk I will present some recents results on weighted Poincaré and Poincaré-Sobolev type inequalities with an explicit quantitative analysis on the dependence on the involved weights. This is a consequence of a sort of self-improving property related to a discrete summability property of the functional controlling the oscillation of a given Lipschitz function.
As a consequence of our results (and the method of proof) we obtain further extensions to two weights Poincaré inequalities and to the case of higher order derivatives. We also obtain results in the same spirit valid for the geometry of product spaces.
We also study minimal integrability conditions via Luxemburg-type expressions with respect to generalized oscillations that imply the membership of a given function f to the space BMO. Our method is simple, sharp and flexible enough to be adapted to several different settings, like spaces of homogeneous type, non doubling measures and also BMO spaces defined over more general bases than the basis of cubes.
BCAM, Thursday, February 15th, 2024, 17:00 - 18:00
Title: Heteroclinic solutions of the Allen-Cahn equation and mean curvature flows
The Allen-Cahn equation is a semilinear partial differential equation which models phase transition and separation phenomena and which provides a regularization for the mean curvature flow (MCF), one of the most studied geometric flows.
In this talk, we combine analytic, geometric and topological strategies to obtain existence results for eternal solutions of this parabolic PDE connecting unstable nontrivial stationary solutions, namely heteroclinic solutions, in certain compact manifolds. In the concrete setting of a 3-dimensional round sphere, we describe the space of all low energy eternal solutions and explain how they can be used to construct geometrically interesting MCFs.
This is joint work with Jingwen Chen (University of Pennsylvania).
BCAM, Thursday, February 15th, 2024, 15:30 - 16:30
Title: Large conformal metrics with prescribed gauss and geodesic curvatures
In this talk, our goal is to discuss the existence of at least two distinct conformal metrics with prescribed gaussian curvature and geodesic curvature respectively, $K_{g}= f + \lambda$ and $k_{g}= h + \mu$, where f and h are nonpositive functions and $\lambda$ and $\mu$ are positive constants. Utilizing Struwe's monotonicity trick, we investigate the blowup behavior of the solutions and establish a non-existence result for the limiting PDE, eliminating one of the potential blow-up profiles.
UPV/EHU, Thursday, February 8th, 2024, 12:00 - 13:00
Title: Transport structures in incompressible fluid mechanics
In this talk, we are interested in the well-posedness theory for a system of PDEs describing the dynamics of an incompressible fluid which presents non-dissipative viscosity effects. At the level of the mathematical model, the non-dissipative nature of the viscosity is encoded by a skew-symmetric term, dubbed odd viscosity tensor. Differently from classical viscosity, the odd viscosity term does not provide any gain of regularity; on the contrary, it is responsible for a loss of derivatives in the a priori estimates.
In this talk, we show how to circumvent such a loss of derivatives and establish a well-posedness result in the framework of Sobolev (or, more generally, Besov) spaces of high enough regularity. The key is the identification of a suitable effective velocity in the model, which allows to recast the system as a system of transport equations.
The talk is based on joint works with Rafael Granero-Belinchón (Universidad de Cantabria), Stefano Scrobogna (Università degli Studi di Trieste) and Alexis Vasseur (University of Texas Austin).
BCAM, Thursday, January 25th, 2024, 17:00 - 18:00
Title: On quantum Wasserstein distance
A classical Wasserstein distance is a metric between two probability distributions, induced by the problem of optimal mass transportation. It reflects the minimal effort that is required in order to morph the mass of the first probability distribution into the mass of the other one.
Optimal transport is a central problem in mathematics and engineering, which has been generalized to the quantum setting. The quantum
Wasserstein distance has recently been defined based on a minimization of a cost operator over bipartite states with given marginals, such that it is also related to quantum channel formalism. It has been found that in this case the self-distance of the state is nonzero and equals
the Wigner-Yanase skew information. If we restrict the optimization to separable states then, surprisingly, the self-distance is related to the quantum Fisher information, a quantity central to quantum metrology.
The talk is based on the common work with Géza Tóth, Dániel Virosztek and Tamás Titkos.
BCAM, Thursday, January 18th, 2024, 17:00 - 18:00
Title: Endpoint bounds on the Hermite spectral projection
In this talk we are concerned with L2-Lq bounds on the Hermite spectral projection operator Πλ in ℝd. For d≥2, the optimal L2-Lq bound on Πλ has been known except for the endpoint case q=2(d+3)/(d+1).
However, the endpoint L2-L2(d+3)/(d+1) bound has been left unsettled for a long time. We prove this missing endpoint case for every d≥3. Our result is based on a new phenomenon: improvement of bounds due to asymmetric localization near the sphere.
UPV/EHU, Thursday, December 14th, 2023, 12:00 - 13:00
Title: Almost sure local well-posedness of the nonlinear Schrödinger equation using directional estimates
The nonlinear Schrödinger equation (NLS) on ℝd is a prototypical dispersive equation, i.e. it is characterized by different frequencies traveling at different velocities and by the lack of a smoothing effect over time.
Furthermore, NLS is a prototypical infinite-dimensional Hamiltonian system. Constructing an invariant measure for the NLS flow is a natural, albeit very difficult problem. It requires showing local well-posedness in low regularity spaces, in an appropriate probabilistic sense.
Deterministic local well-posedness for the NLS is well-understood: it holds only for initial data with regularity above a certain scaling-critical threshold.
We show how directional behavior of solutions can be used to obtain better interaction estimates to control the non-linearity. Combined with multilinear tree expansions for the solutions, this provides the framework to deal with randomized initial data in any positive regularity for the cubic power nonlinearity in dimension 3. This approach improves our understanding of the structure of the solutions and sheds light on NLS in dimensions d ≥ 3 and potentially with other power nonlinearities.
BCAM, Friday, December 1st, 2023, 17:00 - 18:00
Title: Decay estimates for 2D wave in electromagnetic fields
In this talk, we will talk about the dispersive estimates for wave with critical electromagnetic potentials such as Aharonov-Bohm and inverse-square potentials. The construction of the wave propagator and heat kernel associated with the potentials will be presented. As applications, we finally show the dispersive estimates and Strichartz estimates. This is based on a joint work with L. Fanelli(BCAM), J. Zheng(IAPCM).
BCAM, Thursday, November 23rd, 2023, 16:00 - 17:00
Title: Some new and some not so new Fourier interpolation formulas
In this talk we will discuss some problems about Fourier interpolation, mainly focusing on interpolation formulas on the space PWπ of L2 functions whose Fourier transform is supported on [-1/2,1/2]. In particular we will talk about a method that allows us to obtain new interpolation formulas from already known ones.
UPV/EHU, Thursday, November 16th, 2023, 12:00 - 13:00
Title: Scattering theory for cubic inhomogeneous NLS with inverse square potential
In this talk, we will discuss the scattering theory for the cubic inhomogeneous Schr\"odinger equations with inverse square potential iut+Δ u-a/|x|2 u=λ |x|-b|u|2u with a>-1/4 and 0<b<1 in dimension three. In the defocusing case (i.e. λ=1), the global well-posedness and scattering for any initial data in the energy space H1a(ℝ3) can be established. While for the focusing case(i.e. λ=-1), we can obtain the scattering for the initial data below the threshold of the ground state, by making use of the virial/Morawetz argument as in Dodson-Murphy and Campos-Cardoso that avoids the use of interaction Morawetz estimate.
BCAM, Thursday, November 9th, 2023, 17:00 - 18:00
Title: Sharp embeddings between weighted Paley–Wiener spaces
In this talk we will discuss some extremal problems related to embeddings between weighted Paley–Wiener spaces. We will present some asymptotic results for sharp constants in terms of the parameters involved, deduce existence results for extremal functions as well as radial symmetry of those, and talk about some numerical results. For certain cases, these extremal problems can be reformulated in terms of sharp Poincaré inequalities, and for those cases we will present a characterisation of extremizers and sharp constants that recover several classical results.
UPV/EHU, Thursday, November 2nd, 2023, 12:00 - 13:00
Title: Keller estimates of the Eigenvalues in the gap of Dirac operators
This talk aims to present estimates on the lowest eigenvalue in the gap of a Dirac operator in terms of a Lebesgue norm of the potential. Domain, self-adjointness, optimality and critical values of the norms are addressed, while the optimal potential is given by a Dirac equation with a Kerr nonlinearity. A new critical bound appears, which is the smallest value of the norm of the potential for which eigenvalues may reach the bottom of the gap in the essential spectrum. Most of our result are established in the Birman-Schwinger reformulation of the problem.
This is a collaboration work with Jean Dolbeault and David Gontier (University Paris-Dauphine), and Hanne Van Den Bosch (University of Chile).
BCAM, Thursday, October 26th, 2023, 17:00 - 18:00
Title: Pointwise localization and sharp weighted bounds for Rubio de Francia square functions
The Rubio de Francia square function is the square function formed by frequency projections over a collection of disjoint intervals of the real line. J. L. Rubio de Francia proved in 1985 that this operator is bounded in L^p for p\ge 2 and in L^p(w) for p > 2 with weights w in the Muckenhoupt class A_{p/2}. What happens in the endpoint L^2(w) for w\in A_1 was left open, and Rubio de Francia conjectured the validity of the boundedness.
In this talk we will show a new pointwise sparse bound for the Rubio de Francia square function. This sparse bound leads to quantified weighted norm inequalities. We will also show that the weighted L^2-conjecture holds for radially decreasing even weights and in full generality for the Walsh group analogue of the Rubio de Francia square function; in general the weighted L^2 inequality is at this point still an open problem.
In the first part of the talk, we will give the background of the problem while in the second part we will explain the new results mentioned above.
This talk is based on a joint work with F. Di Plinio, I. Parissis and L. Roncal.
UPV/EHU, Thursday, October 19th, 2023, 12:00-13:00
Title: Poincaré-Sobolev inequalities in domains
Hanne van den Bosch - Universidad de Chile
We study a Poincaré-Sobolev inequality with Neumann boundary conditions in bounded domains. We find an explicit threshold value that allows to show that minimizers exist in smooth domains and do not exist in some corner domains. This is joint work with Rafael Benguria and Cristóbal Vallejos
BCAM, Wednesday, October 11th, 2023, 17:00-18:00
Title: Applications of Catastrophe Theory to the Study of Phase Transitions in Quantum Statistical Mechanics.
Kauê Rodrigues (he/him) - BCAM
In both classical and quantum statistical mechanics, mean-field interactions are generally used to model phase transitions, by obtaining a variational problem over the pressure where the phase diagram of the system can be computed. The aim of this talk is to demonstrate that it is possible to analyze this variational problem —for certain fermionic systems with mean-field interactions— by means of catastrophe theory, and to show that catastrophe theory can be a useful tool for obtaining a qualitative description of the phase diagram of a quantum system.
UPV/EHU, Friday, October 8th, 2023, 12:30 - 13:30
Title: Curvature contribution to the essential spectrum of Dirac operators with critical shell interactions
Badreddine Benhellal
This talk is devoted to the characterization of the essential spectrum of three-dimensional Dirac operators with critical combinations of electrostatic and Lorentz scalar shell interactions supported on a smooth compact surface.
After giving the rigorous definition and basic spectral properties of the perturbed operator, we show that its essential spectrum within the gap of the free Dirac operator is nonempty and depends on the surface geometry. More precisely, we show that the criticality of the interaction leads to a new interval of the essential spectrum whose position and length are explicitly controlled by the coupling constants and the principal curvatures of the surface, reducing to a single point only in the case of a sphere.
This work was carried out jointly with Konstantin Pankrashkin (Carl von Ossietzky Universität Oldenburg).
Online, Tuesday, October 3rd, 2023, 11:00-12:00
https://u-bordeaux-fr.zoom.us/j/84229893689
Title: A nonlinear forward-backward problem
Frédéric Marbach (Bordeaux)
In this presentation, we will construct regular solutions for a nonlinear elliptic-parabolic equation in which the natural direction of parabolicity reverses along a critical line. To prevent the emergence of singularities, we will impose orthogonality conditions on the source terms, and follow them during the execution of the nonlinear scheme.
This is a joint work with Anne-Laure Dalibard and Jean Rax, motivated by recirculation problems in boundary layer theory for fluid mechanics, and based on the preprint https://arxiv.org/abs/2203.11067
BCAM, Thursday, September 28th, 2023, 17:00 - 18:00
Title: Spectral stability via the method of multipliers
Lucrezia Cossetti (she/her) - BCAM
Originally arisen to understand characterising properties connected with dispersive phenomena, in the last decades the method of multipliers has been recognized as a useful tool in Spectral Theory, in particular in connection with proof of absence of point spectrum for self-adjoint and non self-adjoint operators.
In this seminar we will see the developments of the method reviewing some recent results concerning self-adjoint and non self-adjoint Schrödinger operators in different settings and relativistic Pauli and Dirac operators. Moreover we will show how this technique can be fruitful developed to obtain similar results for higher order operators. In particular special emphasis will be given to the case of perturbed bilaplacian.
The talk is based on joint works with L. Fanelli and D. Krejcirik.
UPV/EHU, Thursday, September 21st, 2023, 11:30- 12:30
Title: The time-frequency analysis of the uniform bounds for Hilbert forms
Marco Fraccaroli (he/him) - BCAM
Forms associated with the superposition of bilinear Hilbert transforms appear in many contexts in analysis. For example, developing calculus for pseudo differential operators and studying Cauchy intergrals on Lipschitz curves.
In these contexts it arose the question of uniform bounds for such bilinear Hilbert transforms. We will explore this problem with a special focus on the multidimensional case. In particular, we will describe the main tool in the time-frequency analysis of such operators, the phase plane projection. This projection concerns the appropriate simultaneous localization of both a function and its Fourier transform to specific regions of the time-frequency support.
This talk is based on joint work with Olli Saari, Christoph Thiele, and Gennady Uraltsev.
UPV/EHU, room 0.22, Thursday, September 14th, 2023, 12:00 - 13:00
Title: Gabor orthonormal bases, tiling and periodicity
Alberto Debernardi Pinos (he/him) - Universitat Autònoma de Barcelona
Given a Gabor orthonormal basis of L2(ℝ)
𝒢(g,T,S):={ g(x-t) e2π is x: g∈ L2(ℝ), t∈ T, s∈ S},
we study periodicity properties of the translation and modulation sets T and S. In particular, we show that if the window function g is compactly supported, then T and S must be periodic sets, i.e., of the form
T = aℤ+ {t1,…,tn}, S = bℤ + {s1,…,sm}.
To achieve this, we first obtain a result of independent interest: if the system 𝒢(g,T,S) is an orthonormal basis of L2(ℝ), then both |g|2 and |ĝ|2 tile ℝ by translations (when translated along the sets T and S, respectively), and moreover,
∑t∈ T |g(x-t)|2=D(T), ∑s∈ S |ĝ(x-s)|2=D(S), a.e. x∈ ℝ,
where D(Λ) denotes the uniform density of a set Λ⊂ ℝ.
Partial results towards the Liu-Wang conjecture are also obtained.