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====== Research program ====== | ====== 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 in uence of randomness. Our research program is structured into the following four research areas: | ||
- | * **A. Analysis of differential and integro-differential equations** | + | 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: |
- | * **B. Dynamics of interacting systems** \\ Dynamical systems | + | |
- | * **C. Random matrices and Mathematical Physics** \\ Universal limit laws for real and complex | + | ==== I. Analysis of differential and integro-differential equations |
- | * **D. Heat semigroups and Dirichlet forms on manifolds and metric spaces** \\ Stochastic | + | **Participating researchers in Bielefeld: |
- | < | + | **Participating researchers in Seoul:** Sun-Sig Byun (SNU), Myungjoo Kang (SNU), Panki Kim (SNU), Soonsik Kwon |
+ | (KAIST), Ki-Ahm Lee (SNU), Sanghyuk Lee (SNU)\\ | ||
+ | This area is concerned with the research | ||
+ | research focus is the development of the regularity theory of local solutions to nonlinear dispersive equations. For instance, nonlinear Schrödinger and wave equations | ||
+ | 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:** Sung-Soo Byun (SNU), 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 | ||
+ | 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 | ||
+ | spectral theory, potential theory as well as large-deviations theory. However, a finer analysis is | ||
+ | intended | ||
+ | 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: | ||
+ | **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 | ||
+ | stochastic differential equations. A first compound of projects aims at qualitative features of partial differential equations | ||
+ | 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 | ||
+ | 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. | ||