Tobias‘ research is broadly centered around the study of partial differential equations with links to the calculus of variations and probability. The topics he finds most exciting have applications in kinetic theory, statistical physics, materials science and quantum mechanics.
Typically, the problems he works on are described by few and simple basic mechanisms that can lead to a rich class of complex phenomena, whose description involves interesting mathematics!

Optimal Transport and Applications

The optimal transport problem can be stated quite easily: given two piles of mass and a cost function that tells you how expensive it is to move a particle between them, what is the cheapest way of transporting the initial to the target distribution. The solution – the optimal transport plan – has some very interesting properties, with an extremely delicate regularity theory. Tobias’ research is concerned with a relatively novel variational approach to the regularity theory of optimal transport, in particular trying to understand the regularity of a generalization of optimal transport to more than two measures. This multi-marginal optimal transport problem has applications in the theory of Wasserstein barycenters (an important concept in data science, statistics, and image processing) and electronic density functional theory in quantum chemistry.

Microstructures and Universality in Materials Science

The magnetization in ferromagnetic samples can exhibit a wealth of complex patterns. The attempt of understanding the their emergence has led to the discovery and development of effective theories that are nowadays of crucial importance in engineering applications. From a mathematical point of view, proving the existence and formation of complex patterns in such effective models is an extremely challenging question with only a handful of examples where this has been achieved. In Tobias’ research he is particularly interested in branched micro-structures, which occur in so-called strongly uniaxial ferromagnets, but also give rise to the complex patterns at the surface of certain type-I superconductors. Mathematically, his goal is to describe these branched patterns in a statistical sense through local energy estimates, which makes a nice connection to elliptic regularity theory.

Mathematical Quantum Mechanics

A central question in quantum mechanics is how many electrons an atom can bind. In mathematical terms, this translates into a bound on the number of negative eigenvalues of a Schrödinger operator. An estimate with the correct semi-classical behavior has independently been proved by Cwikel, Lieb, and Rozenblum in the 1970s. The optimal constant in this inequality has been a long-standing open problem. Together with colleagues from Karlsruhe Tobias recently made some progress by a so far overlooked connection of this problem to estimates that are well- known in harmonic analysis. However, their estimate on the optimal constant is still far away from the conjectured optimal one — so completely new ideas are needed to resolve this question!