Research program

The aim of this International Research Training Group is to advance Mathematics in a joint initiative between Bielefeld University and Seoul National University. The focus will be on the mathematical analysis of problems which exhibit singular features or the influence of randomness. Our research program is structured into the following three research areas:

I. Analysis of differential and integro-differential equations

Participating researchers in Bielefeld: Lars Diening, Sebastian Herr, Moritz Kassmann
Participating researchers in Seoul: Sun-Sig Byun (SNU), Panki Kim (SNU), Soonsik Kwon (KAIST), Ki-Ahm Lee (SNU), Sanghyuk Lee (SNU)
This area is concerned with the research on dispersive and integro-differential equations/systems. One research focus is the development of the regularity theory of local solutions to nonlinear dispersive equations. For instance, nonlinear Schrödinger and wave equations as well as systems arising in Mathematical Physics are studied. Harmonic analysis methods are important in this context because solutions are given by oscillatory integrals. Concerning the theory of integro-differential operators, we also offer research projects on systems such as those from peridynamics. Linear and nonlinear boundary value problems including appropriate function spaces are studied. Moreover, questions of numerical analysis in this framework are considered. The research program develops further the one of research area A from the first funding period.

II. Stochastic dynamics and mathematical physics

Participating researchers in Bielefeld: Gernot Akemann, Barbara Gentz, Benjamin Gess
Participating researchers in Seoul: Seung-Yeal Ha (SNU), Nam-Gyu Kang (SNU), Ji Oon Lee (KAIST), Insuk Seo (SNU)
In this research area we consider aspects of dynamical systems such as stability, synchronization and the influence of randomness. Motivated from Physics and machine learning, the following examples of systems are analyzed: Open quantum systems and the coupled statistics of eigenvalues and eigenvectors of the Hamiltonian, modeled by non-Hermitian random matrices; Metastability in non-reversible system exhibiting periodic orbits and oscillations as well as noise- induced phenomena in Filippov systems; Methods from stochastic dynamics in machine learning; Stability in infinite dimensional dynamical systems and the influence of randomness, e.g., the effect of stabilization by noise. Common tools in the analysis of such systems are stochastic analysis, spectral theory, potential theory as well as large-deviations theory. However, a finer analysis is intended and requires to go beyond these standard techniques. This research area merges areas B and C form the first funding period, where synchronization in the stochastic Kuramoto model (area B) and aspects of many-body and Coulomb systems using random matrices (area C) were studied.

III. Heat semigroups and Dirichlet forms on manifolds and metric spaces

Participating researchers in Bielefeld: Alexander Grigor’yan, Michael Hinz, Michael Röckner
Participating researchers in Seoul: Sun-Sig Byun (SNU), Panki Kim (SNU), Gerald Trutnau (SNU), Ki-Ahm Lee (SNU)
This research area targets problems related to semigroups, Dirichlet forms, partial differential equations, Markov processes and stochastic differential equations. A first compound of projects aims at qualitative features of partial differential equations on Riemannian manifolds and metric spaces and the way these features are governed by the geometry of the respective manifold. A second compound of projects uses Dirichlet forms to study first order equations and stochastic differential equations on fractal spaces. Continuing former research projects on questions of pointwise regularity within Dirichlet form theory, a third compound of projects is dedicated to stochastic differential equations with singular coefficients and, as a new addition to the portfolio of possible topics, applications of generalized Dirichlet forms to McKean-Vlasov type equations. The research program develops further the one of research area D from the first funding period.

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