Department Mathematik
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Oberseminar `Calculus of Variations and Applications'


The seminar takes place on Wednesday, 4:15 - 6:00 pm in B132, unless indicated otherwise.

Winter Term 2018/2019

 

Date Speaker Topic Note
October 17, 2018 Robert Salzmann Derivation of the 3D energy-critical nonlinear Schrödinger equation as mean-field limit of the three-body interacting Bose gas
October 24, 2018 Nina Gottschling Gamma convergence of the Levy-Lieb to the Thomas-Fermi density functional
October 31, 2018 Chokri Manai Weyl's law on the asymptotic distrubution of the eigenvalues of the Dirichlet and Neumann Laplacian and elliptic divergence operators
November 7, 2018 Ari Laptev Spectral inequalities and the Darboux transform
November 14, 2018 Marcin Napiorkowski A mathematical physics perspective on spin wave theory
November 21, 2018 Belén Paredes Boson Lattice Construction for Anyon Models  
November 27, 2018 Jan Derezinski Balanced geometric Weyl quantization with applications to QFT on curved spacetimes Room B349
December 5, 2018 Felipe Gonçalves Sharpened Restriction Estimates on the Paraboloid
December 12, 2018 Jan Philip Solovej On the Aharonov-Bohm effect for curved magnetic fields in 3-dimensions  
December 19, 2018  
January 9, 2019 Jaroslaw Mederski Time-harmonic Maxwell equations in nonlinear media
January 16, 2019  
January 23, 2019 Emanuel Carneiro Regularity theory for maximal operators: an overview  
January 30, 2019 Matthew de Courcy-Ireland A central limit theorem for integrals of random waves
February 6, 2019
February 13, 2019

Abstracts

  • Robert Salzmann (LMU): Derivation of the 3D energy-critical nonlinear Schrödinger equation as mean-field limit of the three-body interacting Bose gas
    In this talk I will derive the 3D quintic NLS as the mean field limit of a Bose gas with three-body interactions. The quintic NLS is energy-critical in 3D, leading to several new difficulties in comparison with the cubic NLS which emerges from Bose gases with pair-interactions. The used method is based on Bogoliubov's approximation, which also provides the information on the fluctuations around the condensate in terms of a norm approximation for the N-body wave function. However, the talk will mostly concetrate on the leading order approximation by the energy-critical NLS. The talk is based on joint work with P. T. Nam.

  • Nina Gottschling (Cambridge): Gamma convergence of the Levy-Lieb to the Thomas-Fermi density functional
    I will present some of the main elements and details of the proof of our work 'Gamma convergence of the Levy-Lieb to the Thomas-Fermi density functional' which was supervised by Prof Nam. The main theorem states that the Levy-Lieb density functional Gamma-converges to the Thomas-Fermi functional in the semiclassical mean-field limit. This result aides an easy alternative proof of the validity of the atomic Thomas-Fermi theory which was first established by Lieb and Simon.

  • Chokri Manai (TU): Weyl's law on the asymptotic distrubution of the eigenvalues of the Dirichlet and Neumann Laplacian and elliptic divergence operators
    I am going to present the main ideas of the classical proof of Weyl's law for the case of Dirichlet boundary conditions which is based on the Dirichlet-Neumann-Bracketing and Dirichlet-Neumann-Decoupling and which can be found in several mathematical textbooks. Weyl's law still holds for the Neumann Laplacian if we assume that the domain has a smooth boundary. However, this result is usually proven by fairly advanced methods. I want to show the sketch of a proof which only requires "elementary" ideas and the usage of Weyl's law for the Dirichlet Laplacian. Finally, I want to discuss the Weyl-type asymptotic distributions of eigenvalues of elliptic divergence operators.

  • Ari Laptev (London): Spectral inequalities and the Darboux transform
    We shall discuss the application of the commutator method (Darboux transform) that allows us to obtain inequalities for the 3/2 moments of negative eigenvalues of a number of classes of Schrödinger operators.

  • Marcin Napiorkowski (LMU): A mathematical physics perspective on spin wave theory
    Spin wave theory suggests that low temperature properties of the Heisenberg model can be described in terms of noninteracting quasiparticles called magnons. In my talk I will review the basic concepts and predictions of spin wave approximation and report on recent rigorous results in that direction.

  • Belén Paredes (LMU): Boson Lattice Construction for Anyon Models
    This work makes a shift in our physical understanding of anyons. It establishes a duality between the complex mathematical properties of anyons and the intuitive physical properties of systems of bosons in a lattice. Moreover, it establishes a duality between anyons and curved space geometries, between anyons and gravity.

  • Jan Derezinski (Warsaw): Balanced geometric Weyl quantization with applications to QFT on curved spacetimes
    First I will describe a new pseudodifferential calculus for (pseudo-)Riemannian spaces, which in our opinion (my, D.Siemssen's and A.Latosiński's) is the most appropriate way to study operators on such a manifold. I will briefly describe its applications to computations of the asymptotics the heat kernel and Green's operator on RIemannian manifolds. Then I will discuss analogous applications to Lorentzian manifolds, relevant for QFT on curved spaces. I will mention an intriguing question of the self-adjointness of the Klein-Gordon operator. I will describe the construction of the (distinguished) Feynman propagator on asymptotically static spacetimes. I will show how our pseudodifferential calculus can be used to compute the full semiclassical asymptotics around the diagonal of various inverses and bisolutions of the Klein-Gordon operator.

  • Felipe Gonçalves (Bonn): Sharpened Restriction Estimates on the Paraboloid
    In this talk I will discuss how to produce sharp restriction estimates for the paraboloid in small dimensions and how to improve them with a second term that measures the distance of the inital data to the set of extremizers.

  • Jan Philip Solovej (Copenhagen): On the Aharonov-Bohm effect for curved magnetic fields in 3-dimensions
    I will review the celebrated Aharonov-Bohm effect and in particular discuss its generalization to curved solenoids in 3 dimensions. I will discuss the effect for massless Dirac fermions where there is a natural notion of spectral flow of the corresponding families of Dirac operators parametrized by the magnetic flux. I will define spectral flow and discuss how it can be computed explicitly in the Dirac-Aharonov-Bohm case for a large class of field line geometries given as knots or links. The spectral flow will, indeed, depend on the geometry and not only the topology of the links.

  • Jaroslaw Mederski (Institute of Mathematics, Polish Academy of Sciences): Time-harmonic Maxwell equations in nonlinear media
    The search for time-harmonic solutions of nonlinear Maxwell equations in the absence of charges and currents leads to a semilinear equation that under certain conditions has a variational structure. Our goal is to find ground state and bound state solutions for a general class of such equations.

  • Emanuel Carneiro (Trieste): Regularity theory for maximal operators: an overview
    This talk will be a brief survey of past and recent results on the regularity of maximal operators acting on Sobolev and BV functions. The problems in this area can be posed either in a continuous or in a discrete setting - each format bearing their own challenges. We will present some results for the classical Hardy-Littlewood maximal operator, for operators of convolution type (associated to elliptic and parabolic PDE’s) and also for fractional maximal operators - and discuss at length some of the related open problems. I expect the talk will be fully accessible to a broad audience of analysts.

  • Matthew de Courcy-Ireland (Zürich):A central limit theorem for integrals of random waves
    We derive a central limit theorem for the mean-square of random waves in the high-frequency limit over shrinking sets. The proof applies to any compact Riemannian manifold of arbitrary dimension, thanks to the universality of the local Weyl law. The key technical step is an estimate on a triple integral of Bessel functions which we achieve using Gegenbauer's addition formula. This is joint work in progress with Marius Lemm.




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Last modified: February 4, 2018