Bourgain-Brezis discovered Sobolev inequalities that are borderline. We extend these results to the setting of symmetric spaces of non-compact type and to Nilpotent Lie groups. As a first application we obtain estimates for the Poisson equation in borderline cases where Calderon-Zygmund theory fails to give estimates at L¹ .

We also obtain applications to 2D incompressible Navier-Stokes flow and the Maxwell equations of Electromagnetism by virtue of the fact that we obtain improved Strichartz inequalities for the wave and Schrodinger equation. This is joint work with Jean van Schaftingen and Po-lam Yung.

The classical spherical heat kernel is an important object in analysis, probability and physics, among other fields. It is the integral kernel of the spherical heat semigroup and thus provides solutions to the heat equation based on the Laplace-Beltrami operator on the sphere. It is also a transition probability density of the spherical Brownian motion. In this talk we prove sharp two-sided global estimates for the heat kernel associated with a Euclidean sphere of arbitrary dimension. If time permits, we will present a generalization of this result to the compact rank-one symmetric spaces. The talk is based on joint papers with Adam Nowak and Peter Sjögren.

The John-Nirenberg theorem states that functions of bounded mean oscillation are actually exponentially integrable. We will present a few extensions of the classical theorem, concerning maximal functions and weights.

We will also discuss some generalizations to other contexts besides the usual euclidean space, such as spaces of homogeneous type or the anisotropic world of rectangles.

Riemann's non-differentiable function was proposed in the 1860s to give an answer to a long-lived question: do continuous functions always have a derivative? That is why one can consider it to be purely analytic in its origins, that go deep into the foundations of modern mathematical analysis. However, it has been recently discovered that it also has a hidden physical face!

In this talk, I will first relate the historical process that yielded such a fruitful function. Then I will explain how it has been found to play a physical role in the context of vortex filaments, to finally show my results connecting it with turbulence and intermittency obtained in collaboration with Alexandre Boritchev and Victor Vilaça da Rocha.

This work concerns the Cauchy problem associated to a higher dimensional version of the Benjamin-Ono equation. In two dimensions, this model describes the dynamics of three-dimensional slightly nonlinear disturbances in boundary-layer shear flows. For the initial value problem associated, our purpose is to establish local well-posedness in weighted Sobolev spaces as well as some unique continuation principles. In consequence, the optimal spatial decay rate for this model is determined. A key ingredient is the deduction of a commutator estimate involving Riesz transforms.

It is a fundamental fact that neutron stars collapse when their masses are bigger than the Chandrasekhar limit mass. We will study the detail of the collapse phenomenon in the Hartree-Fock-Bogoliubov theory. We prove that in the mass critical limit the ground states develop a universal blow-up profile which solves the Lane-Emden equation. Similar result holds for boson stars, in which the blow-up profile is given by the optimizer of a Gagliardo-Nirenberg-type interpolation inequality.

I show that the 2D Euler equations, when linearized around self-similar states near Coutte flow, contain the full nonlinear resonance cascade mechanism, yielding Gevrey 2 as a critical regularity class. Moreover, there exists a Gevrey regular class of initial data, for which the velocity asymptotically converges in $L^2$ but for which the vorticity diverges to infinity in Sobolev regularity. Thus, on the one hand, the physical phenomenon of inviscid damping holds. On the other hand, "strong damping" cannot hold due to blow-up.

In a lot of industrial and biological applications a thin flat film wrinkles due to thermal enlargement or internal growth. In this talk I consider a mathematical model based on Foppl-von Karman equations describing this phenomena. I establish relation with Monge-Ampere equation, consider a simple case of transformation of flat circular film into a cone, and perform asymptotic analysis for singular region near its tip. I also prove existence of solution for the case of small growth and present a use this model to simulate growth of bacterial biofilms bonded to agar substratum using finite element method.

In this talk, I will explain how the Kubo-Martin-Schwinger (KMS) condition which generally characterise equilibrium in quantum statistical mechanics lead to the relation introduced by G. Gallavotti and E. Verboven in the seventies for classical mechanical systems, in the limiting regime of high temperature and for the simple case of the Bose-Hubbard dynamical system on a finite graph. The main ingredients of the proof are Golden-Thompson inequality, Bogoliubov inequality and semiclassical analysis. This talk is based on a joint work with Antsa Ratsimanetrimanana.

Linear response theory (LRT) is a tool which allows the study of the response of systems that are driven out of equilibrium by external perturbations. In this talk I present a systematic approach to LRT by combining analytic and algebraic ideas. The theory is robust and provides a tool to implement LRT for a wide class of systems like periodic and random systems in the discrete and the continuum. The mathematical framework of the theory is outlined firstly: the relevant von Neumann algebras, non-commutative $L^p$- and Sobolev spaces are introduced; the notion of isospectral perturbations and the associated dynamics and commutators are studied; their construction is then made explicit for various physical systems (quantum systems, classical waves). The final part is dedicated to the presentation of some open problems (e.g. the thermal transport).

Joint work with M. Lein.

We prove global existence for the defocusing, cubic, nonlinear Schr\”odinger equation on two dimensional hyperbolic space in H^s for s>5/6, using a high-low frequency decomposition method.This is a joint work with Gigliola Staffilani.

We will discuss the continuum limit of discrete NLS-type equations and some issues with dispersion and smoothing effect in the discrete setting. We will first study the approach of Kirkpatrick, Lenzmann and Staffilani (2013), as well as some new ideas of Hong and Yang (2019) and explain how to extend these methods to tackle more general dispersive equations.

Abstract

We prove that the Gaertner-Ellis generating functions associated with KMS states of weakly interacting fermions on the lattice can be written as the limit of logarithms of Gaussian Berezin integrals. The covariances of the Gaussian integrals are shown to have a uniform Pfaffian bound and to be summable in general cases of interest (including systems that are not translation invariant). The Berezin integral representation can thus be used to obtain convergent expansions of the generating function in terms of powers of its parameter. Because the results are uniform w.r.t. the free part of the interaction, they are relevant for the study of equilibrium correlations of weakly interacting fermions in random media (background potentials). In this context, a recent application on the accuracy of the macroscopic electric conductivity at microscopic scales will be discussed.

In this talk, I will try to review mathematical results and interests in quantum many-body theory, focussing on the relationship between the Hilbert space formulation and the algebraic one.

In this talk some results in relation with the self-improving properties of generalised Poincaré-Sobolev type inequalities will be discussed. On the one hand, some previous results on the general theory of self-improvement of these inequalities will be introduced to later present some new results in this direction. On the other hand, some new results concerning weighted fractional Poincaré-Sobolev type inequalities will be stated.

In this talk we will study the motion of an interface separating an irrotational fluid from vacuum. In addition we will present several asymptotic models in the regime of small steepness and some of their mathematical properties as the existence and uniqueness of solutions.

Let $\Omega$ be a domain in $\Re$, $1\le p_0,p\le q<\infty$, $a\ge 0,b,\gamma\in \R$. Suppose $\sigma,\mu,w$ are weights. Combining some previous results, we study embedding and compact embedding theorems of sets $\CS\subset L^1_{\sigma,loc}(\Omega)\times L^p_w(\Omega)$ to $L^q_\mu(\Omega)$ (projection to the first component) where $\CS$ (abstract Sobolev space) satisfies a Poincaré type inequality, $\sigma$ satisfies certain doubling property. In particular, when $w,\mu,\rho$ are weights so that $\rho$ is essentially constant on each ball deep inside in $\Omega\setminus F$, $F$ is a finite collection of points and hyperplanes, with the help of a simple observation, we apply our result to the studies of embedding and compact embedding of $L^{p_0}_{\rho^\gamma}(\Omega)\cap E^p_{w\rho^b}(\Omega)$ and weighted fractional Sobolev spaces to $L^q_{\mu\rho^a}(\Omega)$ where $E^p_{w\rho^b}(\Omega)$ is the space of locally integrable functions in $\Omega$ such that their weak derivatives are in $L^p_{w\rho^b}(\Omega)$.

Sobolev and Poincaré inequalities play a central rdylia ole in the regularity theory of elliptic operators. Moreover, the classical Moser-DeGiorgi theory can be adapted to degenerate operators, if one can prove the appropriate versions of Sobolev and Poincaré inequalities on a certain metric measure space. It turns out that Sobolev inequalities themselves imply some interesting properties of the underlying metric measure space --- such as the doubling condition. I will discuss some of these implications, and how they fit with the regularity theory for degenerate elliptic operators.

We present some lower bounds for regular solutions of Schrödinger equations with bounded and time dependent complex potentials. Assuming that the solution has some positive mass at time zero within a ball of certain radius, we prove that this mass can be observed if one looks at the solution and its gradient in space-time regions outsdie of that ball.

There are many problems arising from geometry than can be treat from harmonic analysis. For example, problems about the size of sets containing certain geometric configurations are related to the boundedness of maximal operators. In this talk, we will present a discretized maximal operator associated to averaging over (neighborhoods of) squares in the plane and, more generally, k-skeletons in \mathbb{R}^n . These results are motivated by, and partially extend, recent results on sets that contain a scaled k-skeleton of the unit cube with center in every point of \mathbb{R}^n.

Joint work with Z. Ammari, and F. Nier.

In this work, we study how multiscale analysis and quantum mean field asymptotics can be brought together. In particular we study when a sequence of one-particle density matrices has a limit with two components: one classical and one quantum. The introduction of "separating quantization for a family'' provides a simple criterion to check when those two types of limit are well separated.

We give examples of explicit computations of such limits, and how to check that the separating assumption is satisfied.

C_p weights were introduced by B. Muckenhoupt and later considered by E. Sawyer as a step to characterize the weights for which the Hilbert transform is bounded by the Hardy-Littlewood maximal operator in the weighted norm. In this talk, we introduce a quantitative constant to control the “size” of the weight, which we use to quantify known bounds for Calderón-Zygmund operators and give a quantitative reverse Hölder inequality. Finally, we obtain new bounds for rough homogeneous singular integral operators.

Joint work with Kangwei Li, Luz Roncal and Olli Tapiola.

The talk will be devoted to present some maximum principles for integro-differential operators acting on odd functions, as well as their applications regarding a nonlocal version of the Allen-Cahn equation.

Our main interest is the so-called saddle-shaped solution to the Allen-Cahn equation in R^{2m}, a doubly radial solution which is odd with respect to the Simons cone {(x', x'') \in R^m x R^m: |x'| = |x''|} and that vanishes only in this set. The importance of studying this type of solution is due to its relation with the fractional version of a conjecture by De Giorgi and the theory of local and nonlocal minimal surfaces. Saddle-shaped solutions are the simplest non 1D candidates to be global minimizers in high dimensions, a property not yet established in any dimension.

The key tool in order to develop a theory on existence and uniqueness of the saddle solution is a maximum principle for odd functions with respect to the Simons cone. The main part of the talk will consist on describing it and its proof. After this, we will discuss its applications to the theory of existence and uniqueness of the saddle-shaped solution.

This is a joint work with Juan Carlos Felipe-Navarro (UPC-BGSMath).

The Calderón's problem is to decide whether the conductivity of a body can be uniquely recovered from measurements of potential and current at the boundary. In this talk I will introduce the problem and the main ideas behind the method of Complex Geometrical Optics (CGO) solution. Finally, I will show how bilinear estimates come into the method, and what is the extension of Tao's bilinear theorem we need.

We give an informal survey of the series of papers by Viazovska, Cohn, Kumar, Miller and Radchenko on the problem of sphere packing, and more generally on optimal configurations of points in Euclidean space. We focus especially on the most recent paper, arXiv:1902.05438, concerning the universal optimality of some highly symmetrical lattice configurations in dimension 8 and 24.

In this talk we introduce some regularity properties of solutions for fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian. We extend the results to the case where we move the boundary conditions in an appropriate way.

This talk concerns with boundedness of the resolvent of the Laplacian, which has application to various related problems such as, uniform Sobolev inequality, unique continuation, and spectral analysis. Uniform resolvent estimate which is independent of spectral parameter are now well understood by the work of Kenig-Ruiz-Sogge and Gutierrez. However, the sharp bounds depending on the spectral parameter have not been considered in general framework. In this talk we present a rather complete picture for boundedness of the resolvent outside of uniform boundedness range and also discuss boundedness of related multiplier operators.