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Oberseminar: Calculus of Variations and Applications

Organizers: Phan Thành Nam, Arnaud Triay

Past semesters

• Winter semester 23
• Summer semester 23
• Winter semester 22

Summer Semester 2023

• 18.04.2024: Nguyen Viet Dang (Institut de Mathématiques de Jussieu).
  
  Title: Aspects of constructive quantum field theory
  
  Abstract: In this talk, I will try to motivate the subject of constructive quantum field theory which was born in the 70's as an attempt to give rigorous constructions of quantum field theory models on Minkowski space and also describe scaling limits of spin systems. We will focus on some examples which give a taste of the theory and then discuss recent advances and open problems.
  
• 24.04.2024: Salma Lahbabi ( Ecole Nationale Supérieure d’Electricité et de Mécanique (ENSEM), Université Hassan II de Casablanca (UHIIC), université Paris-Dauphine).
  
  Title: Density functional theory for two dimensional homogeneous materials
  
  Abstract: We study Density Functional Theory models for 2D materials in the 3D space. Our interest comes from the recent developments of two-dimensional materials, such as graphene and phosphorene, in the physics community [2]. In this work, we focus on homogeneous systems. We first show that a homogeneous material can be seen as a limit of periodic systems [1]. Next, we derive reduced models in the remaining orthogonal direction, for DFT models with and without magnetic fields [3, 4]. We show how the different terms of the energy are modified and we derive reduced equations in the remaining direction. We prove some properties of the ground state, such as perfect screening and precise decay estimates in the Thomas-Fermi model, and in Kohn-Sham models, we prove that the Pauli principle is replaced by a penalization term in the energy.
  
  [1] S. Benjelloun, S. Lahbabi, and A. Moussa, Homogenization of 2D materials in the Thomas-Fermi-von Weizsacker theory, arXiv preprint arXiv:2312.08067, (2023).
  [2] A. Geim and I. Grigorieva, Van der Waals heterostructures, Nature, 499 (2013), pp. 419 425.
  [3] D. Gontier, S. Lahbabi, and M. A., Density functional theory for ho- mogeneous two dimensional materials, Comm. Math. Phys., 388 (2021), pp. 1475 1505.
  [4] D. Gontier, S. Lahbabi, and A. Maichine, Density functional theory for two-dimensional homogeneous materials with magnetic fields, Journal of Functional Analysis, 285 (2023), p. 110100.
  
  
• 25.04.2024: Jinyeop Lee (LMU Munich).
  
  Title: Derivation of the Vlasov Equation from Fermionic Many-Body Schrödinger Systems via Husimi Measure
  
  Abstract: This work offers a derivation of the Vlasov equation from fermionic many-body Schrödinger systems, utilizing the Husimi measure as a connecting tool between classical mechanics and quantum mechanics. We start with an intuitive overview of the Vlasov equation, followed by a concise investigation of the many-body Schrödinger equation. The core of our discussion is about the usage of the Husimi measures to bridge these two equations. Participants will be introduced to the underlying formalism and techniques for the derivation process.
  
• 08.05.2024: Quoc Hung Nguyen ( Academy of Mathematics and Systems Science, Chinese Academy of Sciences).
  
  Title: Boundary regularity for the non cut-off Boltzmann and Landau equations on general bounded domains
  
  Abstract: In this talk, I will discuss some optimal boundary regularities for the non cut-off Boltzmann and Landau equations on general bounded domains with any of the usual physical boundary conditions: in-flow, bounce-back, specular-reflection and diffuse-reflection. Then, I state some open problems in this field. If time allows, I will present few recent results on Nonlinear landau damping for the 2d/3d Vlasov-Poisson system and the 3d Vlasov-Poisson- Landau/ non-cutoff Boltzmann system with the weakly collisional regime in the whole domain.
  
• 15.05.2024: Diane Saint Aubin (Universität Zürich).
  
  Title: Third order corrections to the ground state energy of a Bose gas in the Gross-Pitaevskii regime
  
  Abstract: In recent decades, substantial progress has been made in understanding Bose gases, from both experimental and theoretical realms. In this talk, we consider a system of N bosons in the Gross-Pitaevskii regime, moving on the unit torus and interacting through a repulsive potential with scattering length 1/N. We present the derivation of the ground state energy of such a system. While the leading order (of order N) and second order (of order 1) of the ground state energy have been established in recent years, this approach resolves the next order term of order logN/N. The correction to the energy is consistent with predictions of the ground state energy per particle in the thermodynamic limit. This talk is based on joint work with Cristina Caraci, Alessandro Olgiati and Benjamin Schlein.
  
• 22.05.2024: Yuri Suhov (Mathematics department at Penn State University).
  
  Title: Hard-ball packings on lattices
  
  Abstract: I will discuss some recent results on packings of identical hard spheres of diameter $D$ on 2D- and 3D- lattices and graphs (unit triangular lattice, unit square lattice, unit cubic lattice). Our results identify dense-packing configurations and their random`perturbations' which describe high-density `pure phases' of the lattice hard-sphere model of statistical mechanics. No preliminary knowledge of a special material will be assumed.
  
• 05.06.2024: David Gontier (CEREMADE, Université Paris-Dauphine - PSL)
  
  Title: Spectral properties of half-periodic systems
  
  Abstract: The spectral properties of Schrödinger operators for periodic systems (crystals) are well-understood, thanks to Bloch-Floquet theory. When such periodic systems are cut in half, an "edge spectrum" can appear. This spectrum describes physical properties appearing at the boundary of the system. In this talk, we discuss tools to study this edge spectrum, in the continuum and tight-binding setting. We prove in particular that, for any material cut with an incommensurate angle, the edge spectrum fills all the bulk gaps. joint work with Hanne Van Den Bosch and Camilo Gómez Araya
  
• 12.06.2024: Avy Soffer (Rutgers University,)
  
  Title: Scattering Theory of Linear and Nonlinear Waves: A Unified New Paradigm
  
  Abstract: I will present a new approach to Mathematical Scattering of multichannel Dispersive and Hyperbolic Equations. In this approach we identify the targe time behavior of such equations, both linear and non-linear, for general (large) data, and interactions terms which can be space-time dependent. In particular, for the NLS equations with spherically symmetric data and Interaction terms, we prove that all global solutions in H^1 converge to a smooth and localized function plus a free wave, in 5 or more dimensions. Similar result holds for 3,4 dimensions, though the argument proving localization is different. We also show similar results in any dimension for localized type of interactions, provided they decay fast enough. We show breakdown of the standard Asymptotic Completeness conjecture if the interaction is time dependent and decays like r^{-2} at infinity. Many of these results extend to the non-radial case, for NLS, NLKG and Bi-harmonic NLS in three or more dimensions. Furthermore, we prove Local-Decay Estimates for Time dependent potentials in 5 or more dimensions. Finally, we apply this approach to N-body scattering, and prove AC for three quasi-particle scattering. This is based on joint works with Baoping Liu and Xiaoxu Wu.
  
• 19.06.2024: Israel Michael Sigal (University of Toronto)
  
  Title: Ginzburg-Landau equation on non-compact Riemann surfaces
  
  Abstract: In this talk I will consider Ginzburg-Landau equations (GLE) on non-compact surfaces, more precisely on line bundles over such surfaces modelling superconductivity on thin membranes. I will report on recent results on existence of energy minimizing solutions for such equations. I will introduce all necessary definitions and structure the presentation so that no preliminary knowledge of line bundles over Riemann surfaces is not required. The talk is based on the joint work with Nick Ercolani and Jingxuan Zhang.
  
• 26.06.2024: Benjamin Hinrichs (University of Paderborn)
  
  Title: Feynman-Kac Formulas for Polaron Models & the Ultraviolet Problem
  
  Abstract: We present Feynman-Kac formulas for the semigroup generated by the Hamiltonian of a single particle linearly coupled to a bosonic quantum field, such as the spin-boson model, the Fröhlich polaron and the non- and semi-relativistic Nelson models. Based on concrete examples, we discuss how to obtain similar descriptions for the full semigroup of ultraviolet renormalized Hamiltonians. As an application, it is shown how these formulas yield the ergodicity of the described semigroup. This talk is based on joint work with Fumio Hiroshima and Oliver Matte.
  
• 10.07.2024: Dinh-Thi Nguyen (Uppsala University)
  
  Title: 2D focusing "almost-bosonic'' anyons gases.
  
  Abstract: In the two-dimensional space, we consider a trapped system of N anyons interacts via an attractive two-body interaction. The maximum value of the interaction strength is defined by the “magnetic” Gagliardo-Nirenberg inequality. In the stable regime, we derive the average-field-Pauli functional (also know as Chern-Simon-Schroedinger) as the mean-field limit of many-body quantum mechanics. Furthermore, we investigate the collapse phenomenon in the collapse regime where the strength of attractions tends to a critical value (defined by the “non-magnetic" cubic NLS equation) while simultaneously considering the weak field regime where the strength of the self-generated magnetic field tends to zero.
  
• 17.07.2024: Lukas Junge (University of Copenhagen)
  
  Title: Stability of the two dimensional dilute Bose gas.
  
  Abstract: We consider a 2D quantum system of N bosons in a trapping potential $v(x)=\vert x \vert^s$, interacting via a pair potential of the form $N^{2\beta-1}w(N^\beta x)$. We show that for any $\beta\geq 0$ the system exhibits stability of second kind when $w$ is not too negative. This constitutes an extension of the known result for $\beta < 1$ and is a step towards understanding Bosons with attractive potentials in harder scaling regimes. The proof builds on a strong quantitative version of quantum de-Finetti by "Brandao and Harrow".