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research [2018/06/07 12:51] ckoehler |
research [2023/11/16 15:06] (current) 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: | ||
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| - | * **A. Analysis of differential and integro-differential equations** \\ Nonlinear dispersive equations, singular integrals, nonlocal generators of jump processes | ||
| - | * **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** \\ Universal limit laws for real and complex eigenvalues, | ||
| - | * **D. Heat semigroups and Dirichlet forms on manifolds and metric spaces** \\ Stochastic differential equations and Sobolev regularity on infinite-dimensional and fractal state spaces, singular drifts, propagation speed on manifolds and metric spaces, heat kernel estimates for operators with singular coefficients and magnetic energy forms | ||
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| - | ===== A. Analysis of differential and integro-differential equations ===== | ||
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| - | **Principal Researchers: | ||
| - | **Scientific Partners:** Kyeong-Hun Kim (Korea University), | ||
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| - | The aim of this project is to study notoriously difficult Cauchy problems. We investigate nonlinear dispersive equations and systems, as well as integro-differential equations. Concerning nonlinear dispersive equations, such as nonlinear Schrödinger and Wave equations, well-posedness problems for initial data in Sobolev spaces of low regularity will be studied using methods from harmonic analysis. With regard to integro-differential operators, research projects will be offered that extend the theory for the fractional Laplace operator to anisotropic and/or nonlinear nonlocal operators. Characteristic properties of the aforementioned problems can often be described in terms of the underlying scaling behavior, e.g., when studying the interplay between dispersion and nonlinearity or the impact of singularities | ||
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| - | ===== B. Dynamics of interacting systems ===== | ||
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| - | **Principal Researchers: | ||
| - | **Scientific Partners:** Seung-Yeal Ha (SNU), Ki-Ahm Lee (SNU), Insuk Seo (SNU)\\ | ||
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| - | In this project, we will study various aspects of the dynamics of interacting systems in deterministic and random media.\\ | ||
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| - | **Markov statistical dynamics** \\ | ||
| - | We will analyze the related kinetic equations. The main focus will be on dynamics given by fractional Fokker–Planck equations (FPEs).\\ | ||
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| - | **Coupled oscillators** \\ | ||
| - | For systems of coupled oscillators, | ||
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| - | * Does noise facilitate synchronization? | ||
| - | * Through which metastable states does the system pass to achieve synchronization? | ||
| - | * How do the answers depend on the coupling structure? | ||
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| - | The following variants of the standard Kuramoto model and related models provide ample mathematical challenges and will be studied subsequently: | ||
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| - | * A stochastic Kuramoto model with inertia and interaction frustration. | ||
| - | * Different types of random couplings. | ||
| - | * Flocking dynamics for the noisy Cucker–Smale model. | ||
| + | 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: | ||
| - | ===== C. Random matrices | + | ==== I. Analysis of differential |
| + | **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 on dispersive and integro-differential equations/ | ||
| + | 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. | ||
| - | **Principal Researchers: | ||
| - | **Scientific Partners:** Nam-Gyu Kang (KIAS), Ji Oon Lee (KAIST), Insuk Seo (SNU)\\ | ||
| - | Random Matrix theory is an extremely active and exciting research area in Mathematics and Mathematical Physics. It connects for example Analysis, Probability Theory | + | ==== II. Stochastic dynamics |
| - | In this project we want to establish new universal limit laws, study rates of convergence as well asymptotic refinements and their applications.\\ | + | |
| - | The methods that we plan to apply include concentration of measure techniques, free probability theory and asymptotic analysis of real and complex orthogonal polynomials. This relates to questions in Coulomb gases, Gaussian free fields, conformal field theory and Quantum many-body systems. | + | |
| - | ===== D. Heat semigroups | + | **Participating researchers in Bielefeld: |
| + | **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 | ||
| + | examples of systems are analyzed: Open quantum systems | ||
| + | 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. | ||
| - | **Principal Researchers:** Alexander Grigor'yan (Bielefeld), Michael Hinz (Bielefeld), Michael | + | ==== III. Heat semigroups and Dirichlet forms on manifolds and metric spaces ==== |
| - | **Scientific Partners:** Sun-Sig Byun (SNU), Panki Kim (SNU), Gerald Trutnau (SNU), Ki-Ahm Lee (SNU)\\ | + | **Participating researchers in Bielefeld:** Alexander Grigor’yan, Michael Hinz, Michael |
| + | **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. | ||
| - | One main aim of the project is to develop criteria and techniques to obtain from Dirichlet form theory indeed a " | ||