BCAM, Thursday, June 29th, 2023, 17:00 - 18:00
Title: A singular variant of the Falconer distance problem
Tainara Borges (she/her) - Brown University
In this talk we will discuss the following variant of the Falconer distance problem. Let $E$ be a compact subset of ${\mathbb{R}}^d$, $d \geq 1$, and define $$ \Box(E)=\{|(y,z)-(x,x)|: (y,z)\in E\times E,x \in E,\, y\neq z \}\subseteq \mathbb{R}.$$ This is the set of distances between points of $E\times E $ and the diagonal $\mathcal{D}_{E\times E}=\{(x,x)\colon x\in E\}$ with the additional non-degeneracy condition $y\neq z$.
We showed using a variety of methods that if the Hausdorff dimension of $E$ is greater than $\frac{d}{2}+\frac{1}{4}$, then the Lebesgue measure of $\Box(E)$ is positive. This problem can be viewed as a singular variant of the classical Falconer distance problem because considering the diagonal $(x,x)$ in the definition of $\Box(E)$ poses interesting complications stemming from the fact that the set $\{(x,x): x \in E\}\subseteq \mathbb{R}^{2d}$ is much smaller than the sets for which the Falconer type results are typically established.
This talk is based on joint work with Alex Iosevich and Yumeng Ou.
BCAM, Thursday, June 15th, 2023, 17:00-18:00
Title: Scattering for nonlinear waves outside two strictly convex obstacles
David Lafontaine (he/him) - Université de Toulouse
We will be interested in the long time asymptotic of the defocusing energy-critical nonlinear wave equation posed outside an obstacle. For the problem in the free space, it is known that the solutions behave linearly in large time: we say that they scatter. A natural question is therefore to try to understand under which geometrical conditions on the obstacle scattering occurs. I will explain why we expect scattering outside any non-trapping obstacle (for which all the rays of geometrical optics go to infinity), but that such a result seems out of reach for now; and I will present a result of scattering in a trapping geometry: the exterior of two strictly convex obstacles, for which there is one, unstable, trapped trajectory. Joint work with Camille Laurent.
UPV/EHU, Thursday, June 8th, 2023, 12:00-13:00
Title: La ecuación de Ostrovsky en espacios de Sobolev con peso
Jose Manuel Jimenez Urrea (he/him) - Universidad Nacional de Colombia - Medellin
En este seminario consideramos el problema de valor inicial (PVI) asociado a las ecuaciones de Ostrovsky,
\begin{align} \left. \begin{array}{rl}
u_t+\partial_x^3 u\pm \partial_x^{-1}u +u \partial_x u &\hspace{-2mm}=0,\qquad\qquad x\in\mathbb R,\; t\in\mathbb R,\\
u(x,0)&\hspace{-2mm}=u_0(x).
\end{array} \right\}\label{O} \end{align}
UPV/EHU, Thursday, June 1st, 2023, 12:00-13:00
Title: Current Topics in the Theory of Differentiation of Integrals
Paul Hagelstein - Baylor University
In this talk we will highlight recent developments in the theory of differentiation of integrals. In particular, we will discuss the Halo Conjecture and a recent result with Alexander Stokolos that any homothecy invariant basis in $\mathbb{R}^2$ consisting of convex sets is a density basis if and only if it differentiates $L^p(\mathbb{R}^2)$ for \emph{every} $1 < p \leq \infty$.
BCAM, Thursday, May 25th, 2023, 17:00 - 18:00
Title: A variational approach to nonlocal interactions: discrete-to-continuum analysis, ground states and geometric evolutions.
Andrea Kubin (he/him) - Technische Universität München
We consider a core-radius approach to nonlocal perimeters governed by isotropic kernels having critical and supercritical exponents, extending the nowadays classical notion of s-fractional perimeter, defined for 0 < s < 1, to the case s>1.
In the second part of the talk we introduce a model for hard spheres interacting through attractive Riesz type potentials, and we study its thermodynamic limit.
In the third part of the talk we prove that s -fractional heat flows converge to the standard heat flow as s go to 1, and to a degenerate ODE type flow as s go to 0.
BCAM, Tuesday, May 23rd, 2023, 15:00 - 16:00
Title: Torus-like solutions for the Landau de Gennes model
Adriano Pisante (he/him) - University of Rome "La Sapienza"
We report on some recent progress (in collaboration with F. Dipasquale and V.Millot) about the study of global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals in three-dimensional domains. First, we discuss absence of singularities for minimizing configurations under norm constraint, as well as absence of the isotropic phase for the unconstrained minimizers, together with the related biaxial escape phenomenon. Then, under suitable assumptions on the topology of the domain and on the Dirichlet boundary condition, we show that smoothness of energy minimizing configurations yields the emergence of nontrivial topological structure in their biaxiality level sets. Then, we discuss the previous properties under both the norm constraint and an axial symmetry constraint, showing that in this case only partial regularity is available, away from a finite set located on the symmetry axis. In addition, we show that singularities may appear due to energy efficiency and we describe precisely the asymptotic profile around singular points. Finally, in an appropriate class of domains and boundary data we obtain qualitative properties of the biaxial surfaces, showing that smooth minimizers exhibit torus structure, as predicted in numerical simulations.
UPV/EHU, Thursday, May 18th, 2023, 12:00 - 13:00
Title: Weighted Gagliardo-Nirenberg interpolation inequalities
Rodrigo Duarte (he/him)- Instituto Superior Técnico
In this talk, we explore Gagliardo-Nirenberg inequalities with the Lebesgue measure modified by a weight. The classical Gagliardo-Nirenberg inequality generalizes the Sobolev inequality and is an absolutely central tool in the study of PDEs. The first results using weighted norms were given by Caffarelli, Kohn and Nirenberg, and they were subsequently improved by Chang-Shou Lin, who settled the inequality with homogeneous radial weights and integer derivatives. Here we will explain how to make use of the modern methods of weighted inequalities in Harmonic Analysis to extend Lin's inequality to the fractional derivatives setting. Moreover, these methods are flexible enough to give rise to similar inequalities with other weights, including non-homogeneous weights. This talk is based on joint work with Jorge Drumond Silva (IST).
BCAM, Thursday, May 11th, 2023, 17:00 - 18:00
Title: L^2-stability for the 4-waves kinetic equation around the Rayleigh-Jeans equilibrium
Angeliki Menegaki (she/her) - Institut des Hautes Études Scientifiques, Université Paris-Saclay
We consider the four-waves spatial homogeneous kinetic equation arising in weak wave turbulence theory. In this talk I will present some new results on the existence and long-time behaviour of solutions around the Rayleigh-Jeans thermodynamic equilibrium solutions. In particular, introducing a cut-off on the frequencies, I will present an $L^2$ stability of mild solutions for initial data close to Rayleigh-Jeans, when the dispersion relation is weakly perturbed around the quadratic one.
UPV/EHU, Thursday, May 4th, 2023, 12:00-13:00
Title: Varopoulos' extensions in domains with Ahlfors-regular boundaries
Thanasis Zacharopoulos (he/him) - UPV/EHU
In this talk we shall describe the construction of Varopoulos' type extensions of L^p and BMO boundary functions in rough domains. That is, smooth extensions of functions such that the L^p-norms of their non-tangential maximal function and the Carleson functional of their gradients can be controlled by the norm of the boundary data. After giving the geometric motivation and a brief survey of known results, we will proceed to present a new and more general approach of constructing Varopoulos' extensions in domains with minor geometrical assumptions for the boundaries. This talk is based on joint work with Mihalis Mourgoglou.
BCAM, Thursday, April 27th, 2023, 17:00-18:00
Title: On Fourier uncertainty and extremal problems
Emily Quesada-Herrera (she/her) - TU Graz
The uncertainty principle states, broadly, that one cannot have an unrestricted control of a function and its Fourier transform, simultaneously. This paradigm is related to different sorts of Fourier optimization problems, where one imposes conditions on a function and its transform and seeks to optimize a certain quantity of interest. We will discuss work (joint with E. Carneiro) on an uncertainty principle involving the sign of a function; some Fourier extremal problems (joint with A. Chirre and D. Dimitrov, and with E. Carneiro, M. Milinovich, and A. Ramos), and some connections to problems on other fields, such as the sphere packing problem and Linnik’s problem on the least prime in an arithmetic progression.
UPV/EHU, Thursday, April 20th, 2023, 12:00 - 13:00
Title: Threshold resonances induced by spin
Hynek Kovarik (he/him) - Università di Brescia
It is well-known that the interaction between the spin and a magnetic field gives rise to zero energy eigenfunctions of the two-dimensional Pauli operator, the so called Aharonov-Casher states. In this talk we will consider an additional multiplicative perturbation with a small coupling parameter. We will show that if the perturbation is positive in certain weak sense, then the Aharonov-Casher states turn into resonances. We will also provide the leading terms of their lifetimes in the weak coupling expansion. The talk is based on a joint work with Jonathan Breuer (Hebrew University, Jerusalem).
BCAM, Thursday, March 30th, 2023, 17:00-18:00
Title: Polynomials with nice properties and equidistributed points on the sphere.
Ujué Etayo (she/her) - Universidad de Cantabria
In this talk we will present two mathematical results about polynomials: an inequality that compares the norm of a polynomial with the product of the norms of its factors and a bound for the condition number (a quantity that measures the stability of the roots of a polynomial). Both problems can be expressed as a problem of equidistributing points on a sphere of dimension 2.
UPV/EHU, Thursday, March 23rd, 2023, 12:00-13:00
Title: A sharp Fourier extension inequality on the circle under arithmetic constraints
Valentina Ciccone (she/her) - University of Bonn
In this talk, we discuss a sharp Fourier extension inequality on the circle in the Stein-Tomas endpoint case for functions whose spectrum satisfies a certain arithmetic constraint. Such arithmetic constraint corresponds to a generalization of the notion of B_3-set. This talk is based on a joint work with Felipe Gonçalves.
BCAM, Thursday, March 16th, 2023, 17:00 - 18:00
Title: Strichartz estimates for Schrödinger equations with slowly decaying potentials
Haruya Mizutani (he/him) - Osaka University
The Strichartz estimate is one of fundamental tools in the study of nonlinear dispersive equations, especially scattering theory. This talk deals with (global-in-time) Strichartz estimates Schrödinger equations with potentials decaying at infinity. The case when the potential decays sufficiently fast has been extensively studied in the last three decades. However, it has remained mostly unknown for slowly decaying potentials in which case the standard argument, based on the local smoothing effects for the perturbed equation and the free Strichartz estimate, does not work. We instead employ several techniques from long-range scattering theory and microlocal/semiclassical analysis, and prove Strichartz estimates for a class of positive potentials decaying arbitrarily slowly. A typical example is the positive Coulomb potential in three space dimensions. As an application, we also obtain a modified scattering type result for the final state problem of the nonlinear Schrödinger equations with long-range nonlinearity and potential. This is partly joint work with Masaki Kawamoto (Ehime University).
UPV/EHU, Thursday, March 9th 2023, 12:00 - 13:00
Title: Regularity of envelopes swept by rigid bodies
Felipe Ponce-Vanegas (he/him) - BCAM
Turbine blades and other aircraft components are usually manufactured by 5-axis flank CNC machining, in which a cutting tool removes material from the workpiece leaving behind an envelope swept by a rotational symmetric rigid body. In the talk I will recall the classical theory of envelopes, and I will present two results about the regularity of envelopes depending on the smoothness of motion and cutting tool profile. In general, we need two derivatives to get some regularity, but we will see that even after dropping derivatives at some points, the envelope can still retain some smoothness.
Joint work with Michael Bartoň and Michal Bizzarri.
BCAM, Thursday, March 2nd, 2023, 17:00 - 18:00
Title: Wavelet representation and Calderón-Zygmund theory on domains
Francesco di Plinio (he/him) - Università di Napoli "Federico II"
This work devises a new smooth representation formula for CZ operators on domains. As a first order consequence of this representation, we obtain a weighted, sharply quantified T(1)-type theorem on Sobolev spaces. Previous results of Prats and Prats-Tolsa are limited to unweighted bounds for convolution-type operators. Our weighted Sobolev inequalities are subsequently applied to obtain quantitative regularity results for solutions to the Beltrami equation with symbol in the critical class W^{k,2}(Omega). All past results, due to Prats among others, based on the Iwaniec scheme are of qualitative nature.
Talk is based on current and ongoing joint work with Green and Wick.
BCAM, Thursday, February 23rd, 2023, 17:00 - 18:00
Title: Extrapolation in quasi-Banach function spaces
Zoe Nieraeth (she/her) - BCAM
Rubio de Francia’s extrapolation theorem allows one to show that an operator that is bounded on weighted Lebesgue spaces for a single exponent and with respect to all weights in the associated Muckenhoupt class has to also be bounded for every exponent. As a matter of fact, in the previous years it has been shown that the operator has to be bounded on a much larger class of spaces, including Lorentz, variable Lebesgue, and Morrey spaces, and further weighted Banach function spaces.
In this talk I will discuss a recently obtained unification and extension of some of these results by presenting an extrapolation theorem for general Banach function spaces. I will also discuss limited range and off-diagonal variants in the setting of quasi-Banach function spaces.
UPV/EHU, Monday, February 13th, 2023, 12:00 - 13:00
Title: Smoothing properties of averages over curves
Sanghyuk Lee (he/him) - Seoul National University
The regularity property of integral transforms is a fundamental subject in classical harmonic analysis. In this talk, we are concerned with the smoothing properties of the averaging operator defined by convolution with a measure on a smooth non-degenerate curve. Despite the simple geometric structure of the curve, the sharp smoothing estimates have remained largely unknown except for those in low dimensions until recently. We prove the sharp Sobolev regularity estimates in every dimension bigger than 4. Besides, we obtain the sharp local smoothing estimates, which consequently establish a nontrivial Lp bound on the maximal function associated with the nondegenerate curves in dimensions bigger than 3.
UPV/EHU, Thursday, February 9th, 2023, 12:00 - 13:00
Title: Nonequilibrium Neural Computation: Stochastic thermodynamics of the asymmetric Sherrington-Kirkpatrick model
Most systems in nature operate far from equilibrium, exhibiting time-asymmetric, irreversible dynamics; giving rise to entropy production as they exchange energy and matter with their environment. In neuroscience, effective information processing entails flexible architectures integrating multiple sensory streams that vary in time with internal and external events. Physically, neural computation is, in a thermodynamic sense, an out-of-equilibrium, non-stationary process that changes dynamically. Cognitively, nonequilibrium neural activity results in dynamic changes in sensory streams and internal states. In contrast, classical neuroscience theory focuses on stationary, equilibrium information paradigms (e.g., efficient coding theory), which often fail to describe the role of nonequilibrium fluctuations in neural processes.
Inspired by the success of the equilibrium Ising model in investigating disordered systems and related associative-memory neural networks, we study the nonequilibrium thermodynamics of the asymmetric Sherrington-Kirkpatrick system as a prototypical model of large-scale nonequilibrium networks. We employ a path integral method to calculate a generating functional over the trajectories to derive exact solutions of the order parameters, conditional entropy of trajectories, and steady-state entropy production of infinitely large networks. We find that entropy production peaks at a critical order-disorder phase transition but is more prominent in a regime with quasi-deterministic disordered dynamics. While entropy production is becoming popular to characterize various complex systems as well as neural activity, our results reveal that increased entropy production is linked with radically different scenarios, and combining multiple thermodynamic quantities yields a more precise picture of the system. These results contribute to an exact analytical theory for studying the thermodynamic properties of large-scale nonequilibrium systems and their phase transitions.
BCAM, Thursday, February 2nd, 2023, 17:00 - 18:00
Title: On the square of Laplacian with inverse square potential in higher dimensions
Vladimir Georgiev (University of Pisa)
The talk treats the domain of Laplace operator A with inverse square potential and its square. In the case when this operator is essentially self adjoint, its domain has an explicit representation in terms of classical Sobolev spaces and the value of the function at the origin. If the coefficient of the inverse square potential is sufficiently large, then similar characterization is obtained for A^2.
UPV/EHU, Thursday, January 26th, 2023, 12:00 - 13:00
Reduced order models of the Tropical Atmosphere
Scott Hottovy (United States Naval Academy)
The Madden-Julian Oscillation is the major contributor to rainfall in tropical regions and influences the climate in Europe regularly. Unlike Hurricanes and El Niño, the MJO is still not well understood. In an effort to understand the mechanisms of the MJO, I will describe a reduced order model of the MJO. This model is built starting from rotating hydrostatic Boussinesq fluid equations for the atmosphere. These equations are simplified through linearizations and projects which aim to capture large scale phenomena typical of the MJO. I will explain these approximations in detail. Then these equations are coupled to moisture in a simple way to give a linear PDE model of the MJO.
BCAM, Wednesday, January 25th, 2023, 17:00 - 18:00
Best subspace data fitting
Úrsula Molter (Departamento de Matemática, FCEyN, UBA e IMAS, UBA-CONICET)
In this talk, we will show how to approximate given data with appropriate 'small' subspaces.
We apply this to functions in a Hilbert space to approximate them by Shift Invariant Spaces. We further consider the case when we include spaces that are invariant under the action of different groups, such as crystal groups.
Finally, we address the question of whether we can obtain the approximation with 'good' functions.
BCAM, Wednesday, January 25th, 2023, 16:00 - 17:00
Frames by Orbits of Operators and Model Spaces
Carlos Cabrelli (UBA-IMAS-CONICET)
We will characterize frames generated by orbits of operators and their relationship with certain subspaces of the Hardy space on the unit disk. This work is motivated by the recent problem of Dynamical Sampling.
Online, Thursday, January 19th, 2023, 17:00 - 18:00
Global Existence and Long Time Behavior in the 1+1 dimensional Principal Chiral Model with Applications to Solitons
Jessica Trespalacios (Universidad de Chile)
We consider the 1+1 dimensional vector valued Principal Chiral Field model (PCF) obtained as a simplification of the Vacuum Einstein Field equations under the Belinski-Zakharov symmetry. PCF is an integrable model, but a rigorous description of its evolution is far from complete. Here we provide the existence of local solutions in a suitable chosen energy space, as well as small global smooth solutions under a certain non degeneracy condition. We also construct virial functionals which provide a clear description of decay of smooth global solutions inside the light cone. Finally, some applications are presented in the case of PCF solitons, a first step towards the study of its nonlinear stability.
UPV/EHU, Thursday, January 12th, 2023, 12:00 - 13:00
Seminorm inequalities for ergodic averages along primes
Wojciech Słomian (Wroclaw University of Science and Technology)
Various kinds of seminorms have been intensively studied since Bourgain's seminal papers at the end of 1980s, where oscillation and variation seminorms were used to prove the pointwise convergence of the ergodic averages along polynomials. Since then those seminorms became an indispensable tool in studying the pointwise convergence in ergodic theory and harmonic analysis.
In this talk we present recent results concerning oscillation and jump seminorms for the polynomial ergodic averages modelled over multi-dimensional subset of primes.
Joint work with Nathan Melhhop (Rutgers University).
UPV/EHU, Monday, December 19th, 2022
Examples of explicit solutions to the cubic wave equation
Giuseppe Negro (Instituto Superior Técnico, Lisboa)
We construct a two-parameter family of solutions to the focusing cubic wave equation in $\mathbb{R}^{1+3}$. Depending on the values of the parameters, these solutions either scatter to linear ones, blow-up in finite time, or exhibit a new type of unstable behaviour that acts as a threshold between the other two. We further prove that the blow-up behaviour is stable and we characterize the threshold behaviour precisely, both pointwise and in Sobolev sense.
Joint work with Thomas Duyckaerts (Sorbonne Paris Nord)
BCAM, Thursday, December 15th, 2022
A priori estimates for dispersive equations via frequency-restricted estimates
The derivation of nonlinear dispersive estimates in low regularity has been the focus of active research in the past fifty years. Many tools have been developed in order to capture precisely the underlying oscillatory structure: Strichartz estimates, normal form reductions, space-time resonances, multilinear Bourgain estimates, and so on. In this talk, I will introduce frequency-restricted estimates, which are sublevel estimates for certain multipliers in the spatial frequency. The goal is twofold: first, I'll show how these estimates relate to the aforementioned techniques. Second, I'll present the general methodology to prove them, illustrating its wide-range applicablity with several examples. This is partially joint work with R. Côte, F. Oliveira and J. Silva.
BCAM, Thursday, December 15th, 2022, 12:00 - 14:00
Inverse problems: a new approach to PDE
Alberto Ruiz (Universidad Autónoma de Madrid)
We give some properties that characterize a problem as inverse, as well as several historical examples.
UPV/EHU, Thursday, December 1st, 2022
Solvability of the Poisson problem with interior data in L^p Carleson spaces and its applications to the regularity problem
Bruno Poggi (Universitat Autònoma de Barcelona)
On Corkscrew domains with Ahlfors-regular boundary, we prove the equivalence of the classically considered Lp-solvability of the (homogeneous) Dirichlet problem with a new concept which we introduce, of the solvability of the inhomogeneous Poisson problem with interior data in an Lp-Carleson space (with a natural bound on the Lp norm of the non-tangential maximal function of the solution), and we study several applications. Our main application is towards the Lq Dirichlet-regularity problem for second-order elliptic operators satisfying the Dahlberg-Kenig-Pipher condition (this is, roughly speaking, a Carleson measure condition on the square of the gradient of the coefficients), in the geometric generality of bounded Corkscrew domains with uniformly rectifiable boundaries. This solves an open problem from 2001. Other applications include: several new characterizations of the Lp-solvability of the Dirichlet problem, new non-tangential maximal function estimates for the Green's function, a new local T1-type theorem for the Lp solvability of the Dirichlet problem, new estimates for eigenfunctions, and a bridge to the theory of the landscape function (also known as torsion function). This is joint work with Mihalis Mourgoglou and Xavier Tolsa.
BCAM, Thursday, November 17th, 2022
Simply connected translating solitons of low entropy
Francisco Martín (Universidad de Granada)
We consider self-translating solitons for the mean curvature flow of complete embedded 2-surfaces of finite genus and finite entropy. Under a collapsedness condition - amounting to the confinement into slabs of 3-space - we first define the concepts of "wing types" and "wing numbers". In terms of these, a simple formula then computes the entropy, which is in particular quantized into integer values. Asking further which examples of solitons exist in the low entropy range, we combine the wing number ideas with Morse theory for minimal surfaces a la Radó to prove a uniqueness theorem: The unique simply connected complete embedded translating solitons contained in slabs and of entropy $\lambda(\Sigma) = 3$ are the Hoffman-Martín-White pitchforks. This is joint work with E.S. Gama and N.M. Moller.
BCAM, Thursday, November 3rd, 2022
The Kac Limit for Fermionic Lattice Systems
The Kac limit is a way of obtaining the thermodynamic behavior of certain mean-field models as the limiting case of a system subject only to short-range interactions. We prove new results for fermionic systems on a lattice, stating that the convergence holds not only for the thermodynamic pressure but also for equilibrium states (i.e., for all correlation functions). When both an attractive and a repulsive Kac interaction are present, we also show that the pressure of the system and the corresponding correlation functions do not necessarily converge to those of the conventional mean-field model, and can possibly lead to very unconventional (infinite volume) mean-field models.
UPV/EHU, Thursday, October 27th
BMO with respect to Banach function spaces
Sheldy Ombrosi (Universidad Nacional del Sur)
In the 1960s John and Niremberg introduced the space of bounded mean oscillation functions $BMO$ in connection with differential equations. Since that time, and because of the diverse and direct relationship with other relevant objects in Harmonic Analysis, such as duality of Hardy spaces, upper endpoint estimates of Calderón-Zygmund operators, and the $L^p$ estimates of Commutators of those operators, $BMO$ spaces have been objective of much study. In this talk, we will discuss necessary and sufficient (geometric) conditions in a Banach function space $X$ in such a way that $BMO$ and $BMO_{X}$ are equivalent spaces.
The new results that we will present in this talk are based on joint works with E. Lorist and A. Lerner.
UPV/EHU, Thursday, October 20th, 2022
Local existence and scattering for models with internal structure
In this talk we will consider the nonlinear matrix Schrödinger equation on the half-line with general self-adjoint boundary condition. This model corresponds to a star graph describing the behavior of n connected very thin quantum wires that form a graph with only one vertex and a finite number of edges of infinite length. We will discuss the existence of local solutions for this model in the energy space. Moreover, we consider the scattering problem for this model. Under certain assumptions on the input data, we construct the wave, inverse wave and scattering operators. In particular, thanks to the general boundary conditions, we are able to obtain a scattering result for the nonlinear matrix Schrödinger equation on the line with a potential and point interactions.
BCAM, Thursday, October 13th, 2022
Nikodym sets and maximal functions associated with spheres
Giorgios Dosidis (Charles University)
We study Lp bounds on Nikodym maximal functions associated to spheres. In contrast to the spherical maximal function introduced by Stein, our maximal functions are uncentered: for each point in Rn, we take the supremum over families of spheres containing that point.
BCAM, Thursday, October 6th, 2022
Extrapolation and compact operators
One of the main results in Harmonic Analysis is the extrapolation theorem of Rubio de Francia for Muckenhoupt weights. Over the years, it has become a fundamental tool to deal with many problems in harmonic analysis, since, for example, one can obtain general $L^p$ estimates for a linear operator just from an appropriate case $p=p_0$. In this talk, we present some results about extrapolation of compactness for multilinear operators.
The results are based on a joint work with M. Cao and K. Yabuta.
Thursday, September 29th, 2022
Fourier decay of self-similar measures on the complex plane.
Carolina Mosquera (Universidad de Buenos Aires and IMAS-CONICET)
We prove that the Fourier transform of self-similar measures on the complex plane has fast decay outside of a very sparse set of frequencies, with quantitative estimates, extending the results obtained in the real line by R. Kaufman. Also we derive several applications concerning correlation dimension and Frostman exponent of complex Bernoulli convolutions. Furthermore, we present a generalisation for a particular case on Rd, with d ≥ 3.
The results are based on a joint work with Andrea Olivo.
Thursday, September 22th, 2022
On solutions to Interaction equations for short and long dispersive waves
In this lecture we will be concerned with properties of solutions to two nonlinear dispersive models called the Schrödinger–Korteweg–de Vries and Schrödinger-Benjamin-Ono systems.
First we will describe the decay of long-time solutions of the initial value problem (IVP) associated with the Schrödinger–Korteweg–de Vries system. We use recent techniques in order to show that solutions of this system decay to zero in the energy space. Our result is independent of the integrability of the equations involved and it does not require any size assumptions.
In the second part of the talk we will discuss the local well-posedness of the IVP associated with the Schrödinger-Benjamin-Ono system.
Thursday, September 15th, 2022
Higher-order Transversality in Fourier Analysis
Jennifer Duncan (University of Edinburgh)
In modern Fourier analysis, there is a ubiquity of operators whose functional-analytic properties depend on the geometric properties of an underlying submanifold, such as Fourier extension operators or Radon-like transforms, for example. In the analysis of the boundedness of such operators, it is often useful to employ a linear to multilinear reduction so that one may appeal to a multilinearised version of the linear estimate one would like to obtain. These multilinear estimates usually require that the submanifolds involved are uniformly transversal in some suitable sense, however, standard linear algebra tools are sometimes insufficient to capture an appropriately general notion of transversality for the estimates in which we are interested. Brascamp–Lieb inequalities offer a robust framework for understanding higher-order transversality in a manner that is well-suited to applications in Fourier analysis. In my talk, I shall introduce the notion of a Brascamp–Lieb inequality, describe the broader role they play in the subject, and discuss the topic of nonlinear Brascamp–Lieb inequalities, a recent variant that generalises this framework to the manifold setting.
Thursday, September 8th, 2022
Motion of a Rigid body in a Compressible Fluid with Navier-slip boundary condition
In this talk, we present a well posedness result of fluid-structure interaction model regarding the motion of a rigid body in a bounded domain which is filled with a compressible isentropic fluid. We consider the Navier-slip boundary condition at the interface as well as at the boundary of the domain. We prove the existence of a weak solution of the fluid-structure system up to collision.
Thursday, September 1st, 2022
Loop measures and isomorphism theorems for space-time random walks
Stefan Adams (University of Warwick)
Loop measures have become important in the analysis of random walks and their applications in mathematical physics and are currently an active field of research. The concepts of measures go back to Symanzik in the late 1960s in the context of Euclidean field theory. We introduce a novel class of such measures - namely loop measures on graphs with countable infinite different time horizons. The loops are generated by space-time random walks where we add dimension - called 'time' - to the integer lattice in $ d $ dimensions, $ d\ge 3$. These measures are connected to the cycle representation of partition functions in quantum systems (Boson systems). We derive corresponding Dynkin isomorphism theorems for space-time random walks which are complex Gaussian measures, and we show the onset of the so-called Bose-Einstein condensation for some examples.