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Oberseminar: Calculus of Variations and Applications

Organizers: Phan Thành Nam, Arnaud Triay

Summer Semester 2023

• 19.04.2023: Long Meng (Cermics, École des ponts Paritech).
  
  Title: A Rigorous Justification Of Mittleman’s Approach To The Dirac–Fock Model
  
  Abstract: In this talk, we study the relationship between the Dirac–Fock model and the electron-positron Hartree–Fock model. We justify the Dirac–Fock model as a variational approximation of QED when the vacuum polarization is neglected and when the fine structure constant α is small and the velocity of light c is large. As a byproduct, we also prove, when α is small or c is large, the no-unfilled shells theory in the Dirac–Fock theory for atoms and molecules. The proof is based on some new properties of the Dirac–Fock model.
  
• 26.04.2023: Prof. Chulkwang Kwak (Ewha university).
  
  Title: FPU to KDV
  
  Abstract: In this talk, we are going to consider the Fermi-Pasta-Ulam (FPU) system with infinitely many oscillators. We particularly see that Harmonic analysis approaches allow us to observe dispersive properties of solutions to a reformulated FPU system, and with this observation, solutions to the FPU system can be approximated by counter-propagating waves governed by the Korteweg de-Vries (KdV) equation as the lattice spacing approaches zero. Additionally, we see different phenomena detected in the periodic FPU system.
  
• 04.05.2023: Mathieu Lewin (Ceremade, Université Paris-Dauphine PSL).
  
  Note: the talk will be given on Thursday 04.05 in the Mathematical Colloquium which takes place at 4:30 in A027.
  
  Title: Chemists and physicists have found how to approximate Schrödinger's equation; here is how mathematicians can contribute
  
  Abstract: Schrödinger's equation is a beautiful piece of mathematics. It fits on just one line and is supposed to accurately describe the behavior of most atoms and molecules of our world. But it is essentially impossible to simulate accurately, due to its very high dimensionality. In this talk I will explain how physicists and chemists have overcome this problem in an impressive way, within a framework called "Density Functional Theory". I will discuss the role that mathematical results have historically played in this revolution and then present more recent results.
  
• 10.05.2023: Peter Madsen (LMU Munich).
  
  Title: Representability and universal bounds in classical density functional theory.
  
  Abstract: We consider a system of indistinguishable classical particles in Euclidean space interacting through a short-range pair potential. Fixing the one-particle density profile of the system, we minimize the free energy over the set of states with this exact density. This can be useful e.g. to model interfaces between two different equilibrium phases of a system. In this talk, I will discuss the question of representability (which functions can arise as densities of many-body states?), which is a non-trivial problem when the interaction potential has a hard core. I will also explain how to obtain bounds on the free energy in terms of the density of the system. A main issue is contructing trial states with a fixed prescribed density. This is joint work with Mathieu Lewin and Michal Jex.
  
• 24.05.2023: Laurent Lafleche (ICJ, Université Claude Bernard Lyon 1).
  
  Title: Mean-field and semiclassical analysis: Quantum Wasserstein and Sobolev distances.
  
  Abstract: In the context of combined mean-field and semiclassical limits, such as the limit from the $N$-body Schrödinger equation to the Hartree--Fock and Vlasov equations, it is useful to obtain inequalities uniform in the Planck constant and the number of particles. It is therefore important to obtain analogous tools and inequalities in the context of quantum mechanics, such as operator versions of Wasserstein, Lebesgue and Sobolev distances, and the corresponding classical inequalities.
  The stability estimates for the Vlasov equation then yields different quantitative versions of the mean-field and semiclassical limits that do not have the same advantages, leading to different initial data, types of potentials, rates of convergence and time of validity of the estimates.
  
• 31.05.2023: Andreas Deuchert (University of Zurich).
  
  Title: A novel upper bound for the grand canonical free energy of the homogeneous Bose gas in the Gross–Pitaevskii limit
  
  Abstract: We consider a homogeneous Bose gas in the Gross–Pitaevskii limit at temperatures that are comparable to the critical temperature for Bose–Einstein condensation in the ideal gas. Our main result is an upper bound for the grand canonical free energy in terms of two new contributions: (a) the free energy of the interacting condensate is given in terms of a φ4 theory describing its particle number fluctuations, (b) the free energy of the thermally excited particles equals that of a temperature-dependent Bogoliubov Hamiltonian. This is joint work with Chiara Boccato, David Stocker.
  
• 01.06.2023: Sascha Lill (Università di Milano).
  
  Title: Friedrichs Diagrams—Bosonic and Fermionic
  
  Abstract: In Many-Body physics and QFT one often encounters tedious computations of commutators involving creation and annihilation operators. A diagrammatic language introduced by Friedrichs in 1965 allows for cutting down these computations tremendously, while representing the occurring operators in a particularly convenient visual form. We revisit a formula for bosonic commutators in terms of Friedrichs diagrams and prove its fermionic analogue. The talk is based on joint work with Morris Brooks from IST Vienna.
  
• 14.06.2023: Giao Ky Duong (LMU Munich).
  
  Title: Finite time blowup solutions for Complex Ginzburg-Landau equations
  
  Abstract: This talk is based on the joint work with Nejla Nouaili (Paris Dauphine University) and Hatem Zaag (Sorbonne Paris Nord University). We show the construction of singular blowup solutions which blow up in finite time T<0. In particular, we describe the asymptotic behaviors of the solutions when they get near their singular domain.
  
• 28.06.2023: Markus Holzmann (Univ. Grenoble Alpes).
  
  Title: Ground state phase diagram of jellium: what do we think we know?
  
  Abstract: The homogeneous electron gas (jellium) where electrons interact with each other and with a positive background charge is one of the simplest model system in condensed matter physics. Still, the precise determination of the zero temperature phase diagram remains challenging. In the talk I will review some recent progress from a computational perspective concerning the ground state phase diagram and Fermi Liquid properties
  
• 12.07.2023: Hoai-Minh Nguyen (Sorbonne Université).
  
  Title: On Hardy's and Caffarelli, Kohn, Nirenberg's inequalities
  
  Abstract: I discuss the full range of Caffarelli, Kohn, Nirenberg's inequalities for fractional Sobolev spaces and several improvements for standard integer order Sobolev spaces. The full range extension for radial functions are also mentioned. Their proofs, which are quite elementary and not involved the integration by parts, are discussed. The talk is based on joint work with Marco Squassina and Arka Mallick.
  
• 19.07.2023: Nathan Metraud (University of the Basque Country (UPV/EHU) - Basque Center for Applied Mathematics (BCAM)).
  
  Title: Quadratic Fermionic Hamiltonians
  
  Abstract: Quadratic Hamiltonians are important in many-body quantum fields theory. Their general studies, which go back to the sixties, are relatively incomplete for the fermionic case. Following Berezin, they are quadratic in the fermionic field and in this way well-defined as self-adjoint operators acting on the fermionic Fock space. In 1994 Bach, Lieb and Solovej defined them to be generators of strongly continuous unitary groups of Bogoliubov transformations. This is shown to be an equivalent definition, under some conditions, and it is demonstrated to be reminiscent of the celebrated Shale-Stinespring condition on Bogoliubov transformations. Moreover, we show that we can implement Bogoliubov transformations through a novel elliptic operator-valued non-linear differential equations. This allows for their (N-) diagonalization under much weaker assumptions than before. Joint work with Jean-Bernard BRU