Thursday, July 28th, 2022, 17:00 - 18:00
Dimension-free Lp estimates for odd order maximal Riesz transforms in terms of the Riesz transforms
Błażej Wróbel (University of Wrocław)
Thursday, July 14th, 2022, 12:00 - 13:00
On the topology of the magnetic lines of solutions of MHD
Renato Lucà (Ikerbasque and BCAM)
Thursday, July 7th, 2022, 17:00 - 18:00
The Vlasov-Poisson-Boltzmann/Landau equation with polynomial perturbation near Maxwellian
We consider the Vlasov-Poisson-Boltzmann system without angular cutoff and the Vlasov-Poisson-Landau system with Coulomb potential near a global Maxwellian µ. We establish the global existence, uniqueness and large time behavior for solutions in a polynomial-weighted Sobolev space $H^2_{x,v}(⟨v⟩k)$ for some constant k > 0. For the domain union of cubes, we will consider the specular-reflection boundary condition and its high-order compatible specular boundary condition. The proof is based on extra dissipation generated from the semigroup method and energy estimates on electrostatic fields. It is a joint work with Chuqi Cao and Dingqun Deng (Tsinghua University).
Thursday, June 30th, 2022, 12:00 - 13:00
Nodal sets of monochromatic waves from a deterministic and random point of view
In this talk we present recent results on the nodal set (i.e., the zero level set) of monochromatic waves (i.e., solutions of the Helmholtz equation) on the Euclidean space. Following the breakthrough work of F. Nazarov and M. Sodin, a growing literature gives us powerful probabilistic results for the number of connected components of the nodal set of random monochromatic waves. The aim of this talk is to explore the properties of these standard random monochromatic waves and, consequently, define a more general class of random monochromatic waves depending on a real parameter, which includes the standard definition as a particular case. This parameter controls some regularity (of the Fourier transform) and decay properties of these waves. Given that, we study the structure of the nodal set depending on that parameter from a deterministic and from a random point of view. Finally, we show how to construct deterministic realizations or examples of monochromatic waves satisfying the probabilistic Nazarov-Sodin volumetric growth for the number of connected components of the nodal set and similarly for the volume of the nodal set. This is a joint work with A. Enciso, D. Peralta-Salas and A. Sartori.
Thursday, June 23rd, 2022, 17:00 - 18:00
Sparse bounds for the bilinear spherical maximal function
Tainara Borges (Brown University)
Monday, June 20th, 2022, 12:00 - 13:00
Elementary geometric measure theory ideas in data science
Alex Iosevich (University of Rochester)
A frequently arising problem in applied data science is determining the dimensionality, in a suitable sense, of a large multi-dimensional data set. For example, if a million points are contained in a 1000-dimensional space, it would be useful to know whether %90 of them live on or near a 100-dimensional affine plane. The classical and highly effective method to study such problems is called PCA (Principal Component Analysis). However, PCA is not applicable if the lower dimensionality is due to "fractal" phenomena that do arise in practice. We are going to see how a discretized variant of the energy integral from geometric measure theory and related analytic techniques can be used to study the dimensionality of large data sets. We will briefly discuss further prospects for applying advanced mathematical techniques to some of the key questions in Big Data.
Thursday, June 16th, 2022, 17:00 - 18:00
Riesz basis of exponentials for convex polytopes with symmetric faces
Alberto Debernardi (CIDMA, Universidade de Aveiro)
We will discuss a joint result with Nir Lev, which states that for any convex and centrally symmetric polytope Ω ⊂ R^d , whose faces of all dimensions are also centrally symmetric, there exists a Riesz basis of exponential functions for L^2(Ω). This result extends previously known statements in this direction due to Lyubarskii and Rashkovskii, and also due to Walnut (d = 2), and by Grepstad and Lev (in arbitrary dimensions), where the same conclusion is obtained under the additional assumption that all the vertices of Ω lie in the lattice Z^d .
Thursday, June 9th, 2022, 17:00 - 18:00
Gibbs measures as unique KMS equilibrium states of nonlinear Hamiltonian PDEs
Vedran Sohinger (University of Warwick)
Gibbs measures for nonlinear dispersive PDEs have been used as a fundamental tool in the study of low-regularity almost sure well-posedness of the associated Cauchy problem following the pioneering work of Bourgain in the 1990s. In this talk, we will discuss the connection of Gibbs measures with the Kubo-Martin-Schwinger (KMS) condition. The latter is a property characterizing equilibrium measures of the Liouville equation. In particular, we show that Gibbs measures are the unique KMS equilibrium states for a wide class of nonlinear Hamiltonian PDEs. Our proof is based on Malliavin calculus and Gross-Sobolev spaces. This is joint work with Zied Ammari.
Thursday, June 2nd, 2022, 12:00 - 13:00
Vitushkin’s conjecture and sets with plenty of big projections
Damian Dabrowski (University of Jyväskylä)
In this talk I am going to describe recent progress made on Vitushkin’s conjecture: if a compact set E in the plane has plenty of big projections, then E is non-removable for bounded analytic functions. Based on joint work with Michele Villa.
Tuersday, May 31st, 2022, 12:00 - 13:00
Towards a converse to Schur's criterion
Anthony Carbery (University of Edinburgh)
We discuss progress towards reversing the implication of Schur's famous criterion for the L² boundedness of positive integral operators. This is work in progress with Timo Hanninen.
Thursday, May 26th, 2022, 12:00 - 13:00
Principal Eigenvalue and Landscape Function of the Anderson Model on a Large Box
Daniel Sánchez Mendoza (Université de Strasbourg)
We state a precise formulation of a conjecture concerning the product of the principal eigenvalue and the sup-norm of the landscape function of the Anderson model restricted to a large box. We first provide the asymptotic of the principal eigenvalue as the size of the box grows and then use it to give a partial proof of the conjecture. For a special case in one dimension we give a complete proof.
Thursday, May 26th, 2022, 12:00 - 13:00
Principal Eigenvalue and Landscape Function of the Anderson Model on a Large Box
Daniel Sánchez Mendoza (Université de Strasbourg)
We state a precise formulation of a conjecture concerning the product of the principal eigenvalue and the sup-norm of the landscape function of the Anderson model restricted to a large box. We first provide the asymptotic of the principal eigenvalue as the size of the box grows and then use it to give a partial proof of the conjecture. For a special case in one dimension we give a complete proof.
Thursday, May 19th, 2022, 12:00 - 13:00
Non-classical shocks in a non-local generalised Korteweg-de Vries-Burgers equation
Xuban Diez (UPV/EHU)
Hyperbolic conservation laws are ill-posed in general and a common way to derive uniqueness of weak solutions are the so called regularisations. In this talk we study a non-local diffusive and dispersive regularisations of a hyperbolic conservation law given by a fractional derivative. We will analyse the travelling wave solutions in relation with shock formation and show the existence of solutions that in the limit of vanishing diffusion and dispersion lead to non-classical shocks.
Tuesday, May 17th, 2022, 14:00 - 15:00
On a derivative nonlinear Schrödinger equation on the Hardy space of the line
Patrick Gérard (Université Paris-Saclay)
We introduce a nonlinear Schrödinger equation on the line, with a mass critical nonlocal cubic nonlinearity of DNLS type, which conserves the Hardy property of a Fourier transform supported in the positive half line.
We identity a Lax pair for this equation, and we use this structure for studying multisoliton solutions. This a joint work with Enno Lenzmann (Basel).
Thursday, May 12th, 2022, 17:00 - 18:00
Kakeya-type sets for Geometric Maximal Operators
Anthony Gauvan (Laboratoire Mathématiques d`Orsay)
Thursday, May 5th, 2022, 17:00 - 18:00
Strong convergence of the vorticity for the 2D Euler equations in the inviscid limit
The goal of this talk is to study the inviscid limit of a family of solutions of the 2D Navier-Stokes equations towards a renormalized/Lagrangian solution of the Euler equations. First I will prove the uniform-in-time L^p convergence in the setting of unbounded vorticities. Then I will show that it is also possible to obtain a rate in the class of solutions with bounded vorticity. The proofs are based on the stochastic Lagrangian formulation of the incompressible Navier-Stokes equations. In particular, the results are achieved by studying the zero-noise limit from stochastic towards deterministic flows of irregular vector fields. Finally, I will show that solutions of the Euler equations with L^p vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy.
This is joint work with G. Crippa (Universität Basel) and S. Spirito (Università degli Studi dell’Aquila).
Thursday, April 28th, 2022, 12:00 - 13:00
Nikodym-type spherical maximal functions
Alan Chang (Princeton University)
We study L^p bounds on Nikodym maximal functions associated to spheres. In contrast to the spherical maximal functions studied by Stein and Bourgain, our maximal functions are uncentered: for each point in R^n, we take the supremum over a family of spheres containing that point.
Thursday, April 21st, 2022, 17:00 - 18:00
Unique continuation for the fractional discrete Laplacian
Luz Roncal (Ikerbasque and BCAM)
We present several qualitative and quantitative unique continuation properties for the fractional discrete Laplacian. Unlike the fractional continuous Laplacian, which enjoys striking rigidity properties in the form of global unique continuation, we will show that these properties fail to hold in general for the fractional discrete Laplacian.
We also discuss quantitative versions of unique continuation which illustrate how the properties in the continuous setting are recovered up to exponentially small (in terms of the lattice size) correction factors. Such quantitative versions will allow us to deduce stability properties for a linear inverse problem involving the fractional discrete Laplacian.
Joint work with Aingeru Fernández-Bertolin and Angkana Rüland.
Tuesday, April 5th, 2022, 11:00 - 12:00
Renormalisation group and SPDEs
Antti Kupiainen (University of Helsinki)
Non-linear diffusive PDEs driven by space-time white noise require infinite renormalisations to be well posed. I will discuss why this is the case and how the renormalisations can be found by using an idea from quantum field theory, the renormalisation group.
Link to the recorded seminar:
https://drive.google.com/file/d/1U21NgJFt6vZ3Umgf6DOTKK7g6dyuvz1E/view?usp=sharing
Monday, April 4th, 2022, 12:00 - 13:00
An inverse problem for a fragmentation equation modeling the breakage of amyloid fibrils
Thursday, March 31st, 2022, 12:00 - 13:00
Large Deviation Principle for the Cubic NLS Equation
Ricardo Grande (University of Michigan – Ann Arbor)
In this talk we will explore the weakly nonlinear cubic Schrodinger equation with random initial data as a model for the formation of large waves in deep sea. First we will prove a large deviation principle for the solution of the equation, i.e. we derive the top order asymptotics for the probability of seeing a large wave at a certain time as the height of the wave grows. Then we will study a related problem: if we do see a large wave, what is the most likely initial datum that produced it? We answer this question in the weakly nonlinear regime by giving a probabilistic characterization of the set of rogue waves. This is joint work with M. Garrido, K. Kurianski and G. Staffilani.
Link to the recorded seminar:
https://drive.google.com/file/d/16q70pbKNj3Y13hTMyA5jSq4Yf7WJy01t/view?usp=sharing
Thursday, March 24th, 2022, 17:00 - 18:00
Local well-posedness for the gKdV equation on the background of a bounded function
José Manuel Palacios (Université de Tours)
In this talk we will prove the LWP for the gKdV equation in Hs(ℝ), s>1/2, under general assumptions on the nonlinearity f(x), on the background of a bounded function Ψ(t,x), with Ψ(t,x) satisfying some suitable conditions. As a consequence of our estimates, we also obtain the unconditional uniqueness of the solution in Hs(ℝ). This result not only gives us a framework to solve the gKdV equation around a Kink, for example, but also around a periodic solution, that is, to consider localized non-periodic perturbations of a periodic solution.
Thursday, March 17th, 2022, 12:00 - 13:00
The regularity problem for the Laplace equation in rough domains
Mihalis Mourgoglou (Ikerbasque and UPV/EHU)
Tuesday, March 15th, 2022, 11:00 - 12:00
Mean field limits and singular kernels: some recent advances
In this talk, I will present mathematical justifications for mean field limits with singular nuclei based on the control of appropriate weights. These weights must be dynamic and fully relevant to the problem under consideration. We will explain some recent results obtained with Pierre-Emmanuel Jabin (Penn-State) and initially with Z. Wang (Peking Univ) then in a second time with J. Soler (Granada Univ.) respectively around systems of order 1 and then around systems of order 2. This idea of well-adapted dynamical weights finds for us its origin in a joint work with P.-E. Jabin on compressible Navier-Stokes.
Thursday, March 10th, 2022, 17:00 - 18:00
Decoupling, Cantor sets, and additive combinatorics
Decoupling and discrete restriction inequalities have been very fruitful in recent years to solve problems in additive combinatorics and analytic number theory. In this talk I will present some work in decoupling for Cantor sets, including Cantor sets on a parabola, decoupling for product sets, and give applications of these results to additive combinatorics. Time permitting, I will present some open problems.
Contains joint work with Alan Chang (Princeton), Rachel Greenfeld (UCLA), Asgar Jamneshan (Koç University), Zane Li (IU Bloomington), José Ramón Madrid Padilla (UCLA).
Link to the recorded seminar:
https://drive.google.com/file/d/1nx16PQnoLg6vDRjPK9QUeYSUOj3azZ6f/view?usp=sharing
Thursday, March 3rd, 2022, 12:00 - 13:00
Positron Position Operators
Roderich Tumulka (University of Tübingen)
I address the question of what kind of position operators should accompany the free standard quantized Dirac field. I propose a POVM on a configuration space of particle positions of two kinds of point particles, positrons and electrons, acting on the Hilbert space of the quantized Dirac field. It is supposed to serve in several roles of position operators: (i) characterizing what an ideal detector sees, (ii) clarifying how probabilities of macroscopic configurations are provided by the quantum state, and (iii) defining the distribution of Bohmian particles. The existence of this POVM, which I call P_natural, depends on a mathematical conjecture which at present I can neither prove nor disprove; in the talk I explore consequences of the conjecture. P_natural is different from the obvious POVM that one would guess from the wave function representation (as in Thaller’s book) of Fock space vectors; several considerations make P_natural seem more physically plausible than P_obvious. I will also explain what the corresponding Bohmian trajectories look like, how P_natural avoids Malament’s no-go theorem, how it arises naturally as a limiting object from taking the Dirac sea literally, and how it is expected to extend to the cases with external fields, curved space-time, and interaction.
Link to the recorded seminar:
https://drive.google.com/file/d/1NRyZJvW7vp3UrDXdCFadQgqUSJdxYC5j/view?usp=sharing
Thursday, February 24th, 2022, 17:00 - 18:00
On global solutions of the obstacle problem and the behaviour of the regular part of the free boundary close to singularities
In this talk we investigate global solutions of the classical obstacle problem. I give a partial result (under some dimension bound) towards a conjecture by H. Shahgholian ('92) saying that the coincidence sets of global solutions of the obstacle problem are in the closure of ellipsoids, i.e. ellipsoids, paraboloids, cylinders with one of the two as basis, or half-spaces. I will briefly sketch the proof and then present - as a corollary - a new characterization of the behaviour of the regular part of the free boundary in the obstacle problem close to singularities of certain types.
Link to the recorded seminar:
https://drive.google.com/file/d/1EN9qcuJty88bhKfDrTR50EzrPYiirh5N/view?usp=sharing
Thursday, February 17th, 2022, 12:00 - 13:00
Observability inequalities for elliptic equations with potentials in 2D and applications to control theory
Kévin le Balc'h (INRIA and Sorbonne Université)
Link to the recorded seminar:
https://drive.google.com/file/d/1ycyOiKix3iShD5IJLuCvFS4SWIlZTA1Q/view?usp=sharing
Thursday, February 10th, 2022, 12:00 - 13:00
Positivity and convexity properties for harmonic functions on the n-sphere
Racheli Yovel (The Hebrew University)
Consider a harmonic function defined on a spherical disc (an open ball contained in the sphere). In this talk I will discuss the nonnegativity of the iterated Laplace-Beltrami operator applied on a square of such a harmonic function. I will give an outline for the proof in 2 dimensions and of a generalization for any dimension. Furthermore, using spherical means, I will show how this property implies a strong convexity property for the radial L² -growth function. The latter gives an inequality between the radial 2-norm of a harmonic function over three spherical circles (lattitudes), and thus it is a three-circles-type theorem. The talk is based on a joint work with Gabbor Lippner, Dan Mangoubi and Zachary McGuirk.
Link to the recorded seminar:
https://drive.google.com/file/d/1DFqs_eHrGbnjb4q8w3RNLW6Ebijk7O_-/view?usp=sharing
Thursday, February 3rd, 2022, 17:00 - 18:00
The Boundary of Bounded Motions in the Restricted Three Body Problem
One of the most classical mechanisms generating chaos in Hamiltonian systems is the transverse intersection of the stable and unstable manifolds of a hyperbolic fixed point. In the planar circular restricted three body problem (PCR3BP), the intersection of stable and unstable manifolds associated to fixed points "at infinity" lead to chaotic "oscillatory motions", which leave every bounded region but return infinitely often to some bounded region. Conversely, we can bound motions by searching for invariant tori of the system guaranteed by the K.A.M. theorem.
In this talk we discuss methods to estimate the location of the last invariant torus before the onset of such chaotic motions. Due to the delicate nature of the problem - namely issues coming from the parabolic nature of the fixed point and exponentially small nature of the splitting, careful control of the errors of the associated Hamilton-Jacobi equation are required. The control of such errors are achieved by geometric means.
Joint work with Amadeu Delshams.
Link to the recorded seminar:
https://drive.google.com/file/d/1vvJm1x_45owK66fpPiKbrX5DTBlTIQSm/view?usp=sharing
Thursday, January 27th, 2022, 17:00 - 18:00
Operator-free sparse domination
Emiel Lorist (University of Helsinki)
Sparse domination is a recent technique, allowing to estimate (in norm, pointwise or dually) many operators in harmonic analysis by simple, positive expressions. This technique has led to many new results in harmonic analysis over the past decade. In this talk I will discuss a sparse domination principle for an arbitrary family of functions f(x,Q), where x ∈ Rn and Q is a cube in Rn. When applied to operators, this result recovers various recent sparse domination results. In contrast to preceding results, our sparse domination principle can also be applied to non-operator objects, which allows one to use sparse techniques in new areas. I will discuss one of these new applications in detail: weighted Poincare inequalities. This talk is based on joint work with Andrei Lerner and Sheldy Ombrosi.
Link to the recorded seminar:
https://drive.google.com/file/d/11z_7pzYgS6i6In62fKXcDcetUbPjrcMu/view?usp=sharing
Thursday, January 20th, 2022, 17:00 - 18:00
Implementing Bogoliubov transformations beyond the Shale-Stinespring condition
Sascha Lill (BCAM and Universität Tubingen)
Link to the recorded seminar:
https://drive.google.com/file/d/17KsjLe6qc7kxVw9D2ICjijgfIoXUWboH/view?usp=sharing
Thursday, January 13th, 2022, 17:00 -18:00
The role of the Algebraic Multiplicity in Topological Degree Theory
Juan Carlos Sampedro (Universidad Complutense de Madrid)
Thursday, December 16th, 2021, 17:00 -18:00
Maximal operators on the infinite-dimensional torus
Link to the recorded seminar:
https://drive.google.com/file/d/1KyfGnvD0r_njpRahXxjGOk1xSpVWN-pp/view?usp=sharing
Wednesday, December 15th, 2021, 12:00 - 13:30
What is Harmonic Analysis?
María Jesús Carro (Universidad Complutense de Madrid)
Sala Adela Moyua, Facultad de Ciencia y Tecnología of the UPV/EHU
The purpose of this master class is to give an introduction to the area of Harmonic Analysis that every graduate student can easily follow. It will start with a very basic introduction to Mathematical Analysis in order to better understand what part of this big area corresponds to Harmonic Analysis. We shall be both concerned with theoretical results and applications to real problems that can be described using Fourier Analysis techniques. No theorems will be proved, but some of them will be stated and motivated. We shall also present several open problems in the area.
Link to the recorded Master Class:
https://drive.google.com/file/d/1zOZO_oPeq2AwVUhyom2T2ui4khXaYzNq/view?usp=sharing
Thursday, December 9th, 2021, 12:00 -13:00
Vertical square functions and other operators associated with an elliptic operator
Cruz Prisuelos (Universidad Alcalá)
Thursday, December 2nd, 2021, 12:00 -13:00
Quantitative differentiability on uniformly rectifiable sets
Michele Villa (University of Oulu and University of Jyväskylä)
A basic fact of Lipschitz functions is that they are differentiable almost everywhere. This is Rademacher’s theorem. It says a lot about the asymptotic behaviour of Lipschitz functions at small scales (where they look affine) but not much at any definite scale. How long do we need to wait for the the smoothness of f to kick in and make it look like an affine map Dorronsoro’s theorem is a quantification of Rademacher’s which tells us: not long. Indeed, a Lipschitz function looks approximately affine at most scales (in some precise sense). Dorronsoro’s theorem has plenty of applications. It is, for example, a cornerstone of the theory of uniform rectifiability. In this talk, I will discuss how to extend it to functions defined on non-necessarily-smooth subsets of Euclidean space.
Based on a joint work with Jonas Azzam and Mihalis Mourgoglou.
Link to the recorded seminar:
https://drive.google.com/file/d/1LnRl0wfcOgTDOnmd_tySObiFz_MTFdAr/view?usp=sharing
Thursday, November 25th, 2021, 18:00 -19:00
The Calderón problem for nonlocal operators
María Ángeles García-Ferrero (BCAM)
The classical Calderón problem is the inverse problem which the electrical impedance tomography is based on. Its fractional counterpart can be studied by exploiting Runge approximation results for the fractional Laplacian, which are based on unique continuation or antilocal properties. In this talk we will consider other nonlocal operators which “see” conical domains and are generators of stable processes. We will see the implications of directional antilocality for the approximation theorems and for the associated Calderón problem and we will discuss the new phenomena which arise.
This is a joint work with Giovanni Covi and Angkana Rüland.
Link to the recorded seminar:
https://drive.google.com/file/d/1QHgleGLkqcUGWnSjnWVsLRHOdVQUtin_/view?usp=sharing
Thursday, November 18th, 2021, 17:00 -18:00
Caloric measure and regular Lip(1,1/2) graphs
Steve Hofmann (University of Missouri)
Link to the recorded seminar:
https://drive.google.com/file/d/1d2rqRHgUbvYweL2XUtKRkOUQ6G6L2xxK/view?usp=sharing
Thursday, November 4th, 2021, 12:00 -13:00
Layer potentials and Boundary Value Problems in unbounded domains
José María Martell (ICMAT CSIC-UAM-UC3M-UCM)
S. Hofmann, M. Mitrea, and M. Taylor considered boundary value problems in bounded Semmes-Kenig-Toro using the method of layer potentials. This method allows one to solve boundary value problems for the Laplacian and other elliptic operators once it is shown that a certain singular integral operator is invertible. The previous authors established the desired invertibility using the Fredholm theory and exploiting the compactness of the boundary-to-boundary double layer, fact that follows from the extra cancellation of its kernel based on the good oscillation properties of the outer unit normal. In this talk we will study the case of unbounded domains, where the Fredholm theory is not expected to work. We assume that the outer unit normal has sufficiently small oscillation and we establish the desired invertibility by using a Neumann series. Our theory works for the Laplacian and, more generally, for other elliptic systems with constant complex coefficients such as the complex version of the Lamé system of ellipticity.
Joint work with J.J. Marín, D. Mitrea, I. Mitrea, and M. Mitrea.
Link to the recorded seminar:
https://drive.google.com/file/d/1e0ySo380NiriiA2mTzbIcPbTXgh4fQWG/view?usp=sharing
Thursday, October 28th, 2021, 12:00 -13:00
Inhomogeneous cancellation conditions and boundedness of Calderón-Zygmund type operators on local Hardy spaces
Claudio Vasconcelos (Universidade Federal de São Carlos)
Thursday, October 21st, 2021, 12:00 -13:00
Duality for outer L^p spaces and relation to tent spaces
Marco Fraccaroli (Universität Bonn)
Link to the recorded seminar:
https://drive.google.com/file/d/1K1dP7fzePwsqE5SyOq9lOhL1jSE145ox/view?usp=sharing
Thursday, October 14th, 2021, 12:00 -13:00
Endpoint Fourier restriction and unrectifiability
Andrea Merlo (Université Paris-Saclay)
Thursday, October 7th, 2021, 12:00 -13:00
The dilute Fermi gas via Bogoliubov theory
Emanuela Giacomelli (LMU München)
We consider N spin 1/2 fermions interacting with a positive and regular enough potential in three dimensions. We compute the ground state energy of the system in the dilute regime making use of the almost-bosonic nature of the low-energy excitations of the systems.
Link to the recorded seminar:
https://drive.google.com/file/d/1D0U2xhuzQXqOzFON-PIjzD4iy7eJ9p4O/view?usp=sharing
Thursday, September 30th, 2021, 12:00 -13:00
Rotational smoothing
Rotational smoothing is a phenomenon consisting in a gain of regularity by means of averaging over rotations. This phenomenon is present in operators that regularize only in certain directions, in contrast to operators regularizing in all directions. The gain of regularity is the result of rotating the directions where the corresponding operator performs the smoothing effect. In this talk I will present very recent results on the rotational smoothing phenomenon for a specific class of operators. These results were obtained in collaboration with Cristóbal J. Meroño and Ioannis Parissis. Despite the analysis of rotational smoothing is motivated by the resolution of some inverse problems under low-regularity assumptions, we will focus on the theoretical framework of this phenomenon, and put applications aside.
Link to the recorded seminar:
https://drive.google.com/file/d/1_6NReykVE13JoFOOK_dxz1Hu0sjeIF50/view?usp=sharing
Thursday, September 23rd, 2021, 12:00 -13:00
An analogue of Beurling's theorem for the Heisenberg group
Sundaram Thangavelu (Indian Institute of Sciences)
Link to the recorded seminar:
https://drive.google.com/file/d/1B9fx8P9-MFSJk6WQo_47eXjqB5497GmM/view?usp=sharing
Thursday, September 16th, 2021, 17:00 -18:00
Generic Smoothness for the nodal sets of solutions to the Dirichlet problem for Elliptic PDE
Max Engelstein (University of Minnesota-Twin Cities)
We prove that for a broad class of second order elliptic PDEs, including the Laplacian, the zero sets of solutions to the Dirichlet problem are smooth for “generic” data. Additionally, we can ensure the perturbation is “mean zero” for which there are additional technical difficulties to ensure that we do not introduce new singularities in the process of eliminating the original ones. Of independent interest, in order to prove the main theorem, we establish an effective version of the Lojasiewicz gradient inequality with uniform constants in the class of solutions with bounded frequency. This is joint work with M. Badger (UConn) and T. Toro (U. Washington/MSRI).
Link to the recorded seminar:
https://drive.google.com/file/d/1eqK0x2pn9FNapAgEEbSR-z8WFdWqpQZj/view?usp=sharing
Thursday, September 9th, 2021, 12:00 -13:00
Asymptotic behavior of solutions for some local and nonlocal diffusion problems on metric graphs
Liviu Ignat (University of Bucharest)
Link to the recorded seminar:
https://drive.google.com/file/d/1z5ylQUruweDPm1ZmM6UCVfGn2_B0n6xR/view?usp=sharing
Thursday, September 2nd, 2021, 17:00 -18:00
The Fourier Extension problem through a new perspective
Itamar Oliveira (Cornell University)
Link to the recorded seminar:
https://drive.google.com/file/d/1Xm-EIDNavJKIwjYWdDt_Jn9b_4X3v8h-/view?usp=sharing