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research [2018/05/29 14:02] ckoehler |
research [2023/11/16 15:06] seckert [II. Stochastic dynamics and mathematical physics] add Sung-Soo as PR |
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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** \\ Nonlinear dispersive equations, singular integrals, nonlocal generators of jump processes | + | 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 on configuration spaces, fractional Fokker-Planck equations, nonlocal evolution equations, stochastic Kuramoto model, synchronization, | + | |
- | * **C. Random matrices and Mathematical Physics** | + | ==== I. Analysis of differential and integro-differential equations |
- | * **D. Heat semigroups and Dirichlet forms on manifolds | + | **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 | ||
+ | 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 | ||
+ | and nonlinear boundary value problems including appropriate function | ||
+ | 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. | ||
- | ===== A. Analysis of differential | + | ==== II. Stochastic dynamics |
- | **Principal Researchers:** Sebastian Herr (Bielefeld), Moritz Kassmann (Bielefeld)\\ | + | **Participating researchers in Bielefeld:** Gernot Akemann, Barbara Gentz, Benjamin Gess\\ |
- | **Scientific Partners:** Kyeong-Hun Kim (Korea University), Panki Kim (SNU), | + | **Participating researchers in Seoul:** Sung-Soo Byun (SNU), Seung-Yeal Ha (SNU), |
+ | 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, | ||
+ | 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. | ||
- | The aim of this project is to study notoriously difficult Cauchy problems. We investigate nonlinear dispersive equations | + | ==== III. Heat semigroups |
+ | **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 | ||
+ | are governed by the geometry | ||
+ | Dirichlet forms to study first order equations and stochastic | ||
+ | Continuing former | ||
+ | coefficients | ||
+ | Dirichlet forms to McKean-Vlasov type equations. The research program develops further | ||