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 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

**Principal Researchers:** Sebastian Herr (Bielefeld), Moritz Kassmann (Bielefeld)

**Scientific Partners:** Panki Kim (SNU), Soonsik Kwon (KAIST), Ki-Ahm Lee (SNU), Sanghyuk Lee (SNU), Insuk Seo (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

**Principal Researchers:** Barbara Gentz (Bielefeld), Oleksandr Kutovyi (Bielefeld), Yuri Kondratiev (Bielefeld)

**Scientific Partners:** Seung-Yeal Ha (SNU), Ki-Ahm Lee (SNU), Insuk Seo (SNU)

In this project, we will study various aspects of the dynamics of interacting systems in deterministic and random media.

**Markov statistical dynamics**

We will analyze the related kinetic equations. The main focus will be on dynamics given by fractional Fokker–Planck equations (FPEs).

**Coupled oscillators**

For systems of coupled oscillators, we will study the effect of noise on synchronization and the subtle interplay between coupling strength, coupling structure and noise. A number of important questions regarding the onset of synchronization in the Kuramoto model will be investigated:

- Does noise facilitate synchronization?
- Through which metastable states does the system pass to achieve synchronization?
- How do the answers depend on the coupling structure?

The following variants of the standard Kuramoto model and related models provide ample mathematical challenges and will be studied subsequently:

- A stochastic Kuramoto model with inertia and interaction frustration.
- Different types of random couplings.
- Flocking dynamics for the noisy Cucker–Smale model.

**Principal Researchers:** Gernot Akemann (Bielefeld), Friedrich Götze (Bielefeld)

**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 and Combinatorics in various ways. The area is characterized by emerging universal laws for spectral statistics, which apply to surprisingly many phenomena ranging from Number Theory to Quantum Field Theory.

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.

**Principal Researchers:** Alexander Grigor'yan (Bielefeld), Michael Hinz (Bielefeld), Michael Röckner (Bielefeld)

**Scientific Partners:** Sun-Sig Byun (SNU), Panki Kim (SNU), Gerald Trutnau (SNU), Ki-Ahm Lee (SNU)

One main aim of the project is to develop criteria and techniques to obtain from Dirichlet form theory indeed a “pointwise” analysis, i.e. develop a regularity theory for Dirichlet forms. It is well-known that to get to a pointwise analysis, the study of the heat kernel is fundamental. In the case of singular coefficients there are many open questions in this respect.