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Oberseminar: Calculus of Variations and Applications

Organizers: Phan Thành Nam, Arnaud Triay

Past semesters

• Summer semester 24
• Winter semester 23
• Summer semester 23
• Winter semester 22

Winter Semester 2024/2025

• 30.10.2024: Long Meng (LMU Munich).
  
  Title: Mathematical insight into incommensurate systems
  
  Abstract: In this talk, I am trying to discuss some mathematical results on the incommensurate systems. The incommensurate systems refer to electrons in a potential with two or more competing incommensurate periods, leading to the absence of overall translational symmetry. A typical model is the twisted bilayer graphene model which has received widespread attention from physical community due to its controllability and unconventional superconductivity. As a multi-scale problem, it is extremely hard to be calculated numerically. From semiclassical analysis, we are trying to give some insight into this problem, and discuss some known and unknown results from theoretical and numerical point of view.
• 06.11.2024: Martin Peev (Imperial College London).
  
  Title: Renormalising Non-Commutative Singular PDEs
  
  Abstract: When attempting to construct QFTs that include Fermions using the methods of Stochastic Quantisation, one is naturally forced to consider noncommutative stochastic PDEs. I shall show how to formulate SPDEs driven by noncommutative noises in terms of algebra-valued singular PDEs. Furthermore, I will describe how one can renormalise the singular products appearing in such equations for a set of algebras interpolating between Fermions and Bosons by appropriately modifying their topologies. This talk will be based on joint work with Ajay Chandra and Martin Hairer.
  
• 13.11.2024: Jiasheng Lin (Sorbonne).
  
  Title: Around Segal Axioms for QFT: construction and interpretations
  
  Abstract: This talk will introduce G. Segal's framework for describing in an axiomatic way Quantum Field Theories defined on curved Riemannian surfaces with or without boundaries, particularly how composition of evolution operators corresponds to gluing surfaces along boundaries. Based on a recent work which fits the classical P(phi)_2 QFT into this framework using probabilistic tools, we will first describe precisely the analytic problems behind such a task, then sketch the ideas for the construction. As an application we point out how the Segal framework results naturally in asymptotic properties of the partition function of a QFT on a periodic surface with high genus, as the genus goes to infinity. If time permits we will discuss how in the slightly more general setting of branched covers the Segal framework could also be related to the ''entanglement entropy'' of 1d quantum systems and possible future works in this direction.
  
• 27.11.2024: Phuoc-Tai Nguyen (Masaryk University).
  
  Title: Quantitative bounds for bounded solutions to the Navier-Stokes equations in critical besov spaces
  
  Abstract: In this talk, I will discuss the quantitative regularity and blow-up criteria for classical solutions to the three-dimensional incompressible Navier-Stokes equations. By adapting Tao's breakthrough strategy [in Nine Mathematical Challenges: An Elucidation, American Mathematical Society, Providence, RI, 2021, pp. 149-193], I will show a pointwise estimate for solutions in terms of their critical Besov norm. Consequently, I will give a blow-up rate for solutions blowing up at a finite time in term of their critical Besov norm. This talk is based on joint work with R. Hu, Q.-H. Nguyen and P. Zhang
• 04.12.2024: Francois Visconti (LMU Munich).
  
  Title: Derivation of Hartree theory for two-dimensional trapped Bose gases in almost Gross-Pitaevskii regime
  
  Abstract: We study the ground state energy of trapped two-dimensional Bose gases with mean-field type interactions. We prove the stability of second kind of the many-body system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger (NLS) energy functional. Notably, we can take the scaling of the interaction to be arbitrarily close to the Gross-Pitaevskii scaling. As a consequence of the stability of second kind we also obtain Bose-Einstein condensation for the many-body ground states and dynamics for a much improved range of the diluteness parameter. Based on joint work with Lukas Junge.
• 11.12.2024: Cornelia Vogel (Universität Tübingen).
  
  Title: Macroscopic Thermalization for Highly Degenerate Hamiltonians
  
  Abstract: An isolated macroscopic quantum system thermalizes if its initial state eventually reaches a suitable thermal equilibrium subspace and stays there for most of the time. For non-degenerate Hamiltonians, a sufficient condition for the thermalization of every initial state is an appropriate version of the eigenstate thermalization hypothesis (ETH). Shiraishi and Tasaki recently proved the ETH for a perturbation of the Hamiltonian of a large number of free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of the Hamiltonian. We point out that also for degenerate Hamlitonians, all initial states thermalize if the ETH holds for every eigenbasis, and we show that this is the case for free fermions in 1d. Additionally, we develop another strategy of proving thermalization by adding small generic perturbations to Hamiltonians for which it can be shown that one eigenbasis (but not necessarily all) fufills the ETH. This strategy applies to arbitrarily small generic perturbations of the Hamiltonian of free fermions in arbitrary spatial dimensions. This is joint work with Barbara Roos, Stefan Teufel, and Roderich Tumulka.
• 12.12.2024: Toan Nguyen (Pennsylvania State University).
  
  Title: Landau damping
  
  Abstract: Of great interest in plasma physics is to determine whether excited charged particles in a non-equilibrium state will relax to neutrality or transition to a nontrivial coherent state. Due to the long range interaction between particles, the self-consistent generating electric field oscillates in time and disperses in space like a Klein-Gordon wave, known in the physical literature as plasma oscillations or Langmuir’s oscillatory waves. Landau in his original work addresses the decay of such an electric field, namely the energy exchange between the oscillatory electric field and charged particles, in a linearized setting. This talk will provide an overview of the recent mathematical advances in the nonlinear setting. The talk should be accessible to graduate students and the general audience.