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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
  • B. Dynamics of interacting systems
    Dynamical systems on configuration spaces, fractional Fokker-Planck equations, nonlocal evolution equations, stochastic Kuramoto model, synchronization, metastability
  • C. Random matrices and Mathematical Physics
    Universal limit laws for real and complex eigenvalues, asymptotic analysis using free probability and orthogonal polynomials
  • 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

A. Analysis of differential and integro-differential equations

Principal Researchers: Sebastian Herr (Bielefeld), Moritz Kassmann (Bielefeld)
Scientific Partners: Kyeong-Hun Kim (Korea University), Panki Kim (SNU), Soonsik Kwon (KAIST), Ki-Ahm Lee (SNU), Sanghyuk Lee (SNU)

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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