Oberseminar `Calculus of Variations and Applications'
Winter Term 2018/2019
Date Speaker Topic Note October 17, 2018 Robert Salzmann Derivation of the 3D energycritical nonlinear SchrÃ¶dinger equation as meanfield limit of the threebody interacting Bose gas October 24, 2018 Nina Gottschling Gamma convergence of the LevyLieb to the ThomasFermi 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 AharonovBohm effect for curved magnetic fields in 3dimensions December 19, 2018 January 9, 2019 Jaroslaw Mederski Timeharmonic 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 CourcyIreland A central limit theorem for integrals of random waves February 6, 2019 February 13, 2019
Abstracts

 Robert Salzmann (LMU): Derivation of the 3D energycritical nonlinear SchrÃ¶dinger equation as meanfield limit of the threebody interacting Bose gas
 In this talk I will derive the 3D quintic NLS as the mean field limit of a Bose gas with threebody interactions. The quintic NLS is energycritical in 3D, leading to several new difficulties in comparison with the cubic NLS which emerges from Bose gases with pairinteractions. 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 Nbody wave function. However, the talk will mostly concetrate on the leading order approximation by the energycritical NLS. The talk is based on joint work with P. T. Nam.

 Nina Gottschling (Cambridge): Gamma convergence of the LevyLieb to the ThomasFermi density functional
 I will present some of the main elements and details of the proof of our work 'Gamma convergence of the LevyLieb to the ThomasFermi density functional' which was supervised by Prof Nam. The main theorem states that the LevyLieb density functional Gammaconverges to the ThomasFermi functional in the semiclassical meanfield limit. This result aides an easy alternative proof of the validity of the atomic ThomasFermi 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 DirichletNeumannBracketing and DirichletNeumannDecoupling 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 Weyltype 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 SchrÃ¶dinger 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.LatosiÅ„ski'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 selfadjointness of the KleinGordon 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 KleinGordon 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 AharonovBohm effect for curved magnetic fields in 3dimensions
 I will review the celebrated AharonovBohm 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 DiracAharonovBohm 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): Timeharmonic Maxwell equations in nonlinear media
 The search for timeharmonic 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 HardyLittlewood maximal operator, for operators of convolution type (associated to elliptic and parabolic PDEâ€™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 CourcyIreland (Zürich):A central limit theorem for integrals of random waves
 We derive a central limit theorem for the meansquare of random waves in the highfrequency 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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