Oberseminar Analysis Winter 2023/24

Organizers: S. Conti, H. Koch, S. Müller, B. Niethammer, A. Rüland, M. Rumpf, C. Thiele, J.J.L. Velázquez, E. Demattè

Upcoming Talks:

Previous Talks:

• Thursday, November 23 at 14:15 in Lipschitz Saal
  
  Laurent Desvillettes (Paris Cité)
  About the Landau equation of plasma physics
  

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  We propose a description of recent results for the Landau equation of plasma physics, which describes the binary collisions between charged particles. We explain what are the analogies between this equation and (3D, incompressible) Navier-Stokes system.

  
  

• Thursday, January 25 at 14:15 in Lipschitz Saal
  
  Maria Gordina (University of Connecticut and Bonn Research Chair)
  Ergodicity for Langevin dynamics with singular potentials
  

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  We discuss Langevin dynamics of $N$ particles on $\mathbb{R}^d$ interacting through a singular repulsive potential, such as the Lennard-Jones potential, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted total variation distance. The proof relies on an explicit construction of a Lyapunov function using a modified Gamma calculus (Bakry-Emery). In contrast to previous results for such systems, our results imply geometric convergence to equilibrium starting from an essentially optimal family of initial distributions. This is based on joint work with F. Baudoin and D. Herzog, E. Camrud and G. Stoltz.

  
  

• Thursday, February 1 at 14:15 in Lipschitz Saal
  
  Jürg Fröhlich (ETH Zürich)
  A quantum Poisson process
  

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  The time evolution of a physical system consisting of charged matter interacting with the quantized electromagnetic field is stochastic featuring a sequence of random events. In an idealized mathematical description of such systems a new type of stochastic process is encountered, which I call "quantum Poisson process". I introduce this process, derive some of its mathematical properties and sketch some applications to quantum theory.

  
  

• Wednesday, February 21 at 15:15 in SemR 0.011
  
  Heiko Gimperlein (University of Innsbruck)
  Regularity theory for fractional Laplacians: a geometric approach and applications
  

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  We consider a new perspective on the sharp regularity of solutions to the Dirichlet problem for the fractional Laplacian. Following Caffarelli and Silvestre, the fractional Laplacian is defined as a Dirichlet-to-Neumann operator for a degenerate elliptic problem in a half-space of one higher dimension, and we briefly outline its relevance in macroscopic models of biological or robotic processes. We then use techniques from geometric microlocal analysis to analyse the regularity of solutions for the Dirichlet problem on a smoothly bounded domain in Euclidean space, with particular emphasis on asymptotic expansions and Hölder continuous data. Detailed and sharp results about this problem have been obtained by Gerd Grubb. Our complementary approach provides a framework to apply standard tools from singular and pseudodifferential analysis to fractional boundary problems, and we discuss some applications. This is joint work with Rafe Mazzeo and Nikoletta Louca.

  
  

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