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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: TBA
  
  Abstract: TBA
  
• 15.05.2024: Diane Saint Aubin (Universität Zürich).
  
  Title: TBA
  
  Abstract: TBA
  
• 22.05.2024: Yuri Suhov (Mathematics department at Penn State University).
  
  Title: TBA
  
  Abstract: TBA
  
• 29.05.2024: David Gontier (CEREMADE, Université Paris-Dauphine - PSL)
  
  Title: TBA
  
  Abstract: TBA
  
• 19.06.2024: Israel Michael Sigal (University of Toronto)
  
  Title: TBA
  
  Abstract: TBA
  
• 26.06.2024: Benjamin Hinrichs (University of Paderborn)
  
  Title: TBA
  
  Abstract: TBA