Probability and variational methods in PDEs

Head:
Tobias Ried

Phone:
+49 (0) 341 - 9959 - 959

Fax:
+49 (0) 341 - 9959 - 658

Address:
Inselstr. 22
04103 Leipzig

Probability and variational methods in PDEs

The interplay between analysis and probability has always been very fruitful and has led to numerous advancements in both fields and applications. In this group, we aim to use probabilistic and variational techniques to better understand nonlinear partial differential equations, with applications in kinetic theory, statistical physics, materials science and quantum mechanics.

The projects we are working on are related to a series of current and highly active lines of research: optimal transportation, singular stochastic PDEs, effective equations, pattern formation problems, and density functional theory.

These diverse projects, upon closer inspection, share some common features:

  • ideas from regularity theory play a crucial role;
  • nonlocal interactions are driving mechanisms;
  • the analysis of continuum models is combined with probabilistic tools and thinking.

Quite often, very few and simple basic mechanisms can lead to a rich class of complex phenomena, whose description involves interesting mathematics.

Selected research projects

  • Regularity theory of optimal transport
  • Multi-marginal optimal transport and Wasserstein barycentres
  • Branched transport and domain branching in thin ferromagnetic films and type-I superconductors
  • Variational Methods for singular stochastic PDEs and variational problems with quenched randomness
  • The Local Density Approximation in Density Functional Theory
  • Regularisation properties of the Boltzmann equations
  • Convergence of many-body dynamics to the Vlasov-Poisson equation

Group Members

Surname, first nameEmailPhoneOfficeHomepage
Group Leader
Ried, TobiasTobias.Ried840G2 05external

Former Group Members and Visitors

  • Prodhomme, Maxime (until 31.12.2021)
    • Next affiliation: Institut de Mathématiques de Toulouse, France

Tobias Ried

I obtained my PhD from the Karlsruhe Institute of Technology (KIT) in October 2017. The topics of my doctoral research, supervised by Dirk Hundertmark, were smoothing properties of the Boltzmann equation, the study of convergence to equilibrium in the Kac model, as well as the existence of solitons in optical fiber cables with periodically varying dispersion.

Before, I studied Theoretical and Mathematical Physics at TU and LMU Munich, where my main research interests were in many-body quantum mechanics. In my Master’s thesis, supervised by Simone Warzel (TUM), I studied quantum spin systems in random energy landscapes.

After my PhD, I stayed one more year at KIT as a postdoc within the CRC 1173 “Wave Phenomena: Analysis and Numerics”. Since October 2018, I have been at the MPI in Leipzig, first as a postdoc and since 2021 as a group leader. In the academic year 2021/22, I had a substitute professor position at LMU Munich, and in the summer of 2022, I was a substitute professor at Leipzig University.

04.10.2022, 01:40