Talks

You can find the schedule HERE.

Gernot Akemann (Bielefeld): "Asymptotic analysis of orthogonal polynomials in random matrix theory: overview and recent developments"
Random matrices enjoy a large variety of interesting applications in physics, mathematics and other sciences. Many questions about the correlations of eigenvalues of such matrices can be expressed in terms of the kernel of classical orthogonal polynomials, such as Laguerre or Hermite polynomials. Taking the limit of large matrices requires an analysis of the asymptotic behaviour of these polynomials of large degree. I briefly review in which situations the standard limiting Bessel, Sine and Airy behaviour occurs, which is in fact shared by a much larger class of polynomials. This phenomenon is called universality. I will try to cover recent developments about the distribution of the smallest eigenvalue of a random matrix as well as generalisations to kernels of bi-orthogonal functions of a more general kind.

Stefan Bauer (Bielefeld): "Monopoles on Bounded Four-Manifolds"
The Seiberg-Witten monopole map is being studied on bounded 4-manifolds. Connections with equivariant stable homotopy and the geometry of 4-manifolds are discussed.

Sun-Sig Byun (SNU): "Weighted estimates for nonlinear parabolic problems with Orlicz data"
We discuss weighted estimates for nonlinear parabolic problems in divergence form with Orlicz data

Zhen-Qing Chen (Seattle): "Brownian Motion on Spaces with Varying Dimension"
Brownian motion is a building block of modern probability theory. It has important and intrinsic connections to analysis and partial differential equations as the infinitesimal generator of Brownian motion is the Laplace operator. In real world, there are many examples of spaces with varying dimensions. For example, image an insect moves randomly in a plane with an infinite pole installed on it. In this talk, I will introduce and discuss Brownian motion (or equivalently, "Laplace operator") on a state space with varying dimension. I will present sharp two-sided estimates on its transition density function (also called heat kernel). The two-sided estimates is of Guassian type but the parabolic Harnack inequality fails for such process and the measure on the underlying state space does not satisfy volume doubling property.

Barbara Gentz (Bielefeld): "Synchronization, noise-induced phase slips and log-periodic oscillations"
We consider a system of two coupled oscillators. It is well-known that in the absence of noise, for sufficiently strong coupling, synchronization is observed (the oscillators are "in phase"). In the presence of small noise, the system will still spend most of the time in metastable equilibrium, i.e., the difference in phase will show only small fluctuations. Occasionally however, transitions to other metastable states are observed. The different metastable states are characterized by the corresponding phase difference -- each (integer) multiple of $2\pi$ corresponds to a metastable state -- and a transition to another metastable state can by viewed as a "phase slip". We will discuss the distribution of phase slips and the required tools for a rigorous description. These include a novel approach to noise-induced transitions through unstable periodic orbits. Joint work with Nils Berglund (Orleans).

Seung-Yeal Ha (SNU): "Synchronization of classical and quantum oscillator systems"
In this talk, I will report recent progress on the complete synchronization problem of the classical and quantum oscillator systems appearing in biological and physical complex systems.
This is a joint work with Dr. Sun-Ho Choi (NUS).

Sebastian Herr (Bielefeld): "Endpoint Strichartz estimates and the cubic Dirac equation"
We will review classical Strichartz estimates for the wave equation in dimension three and related tools from harmonic analysis. For Dirac and Klein-Gordon equations we will present a new endpoint estimate. As an application, we discuss a critical well-posedness result for the cubic Dirac equation with small initial data (joint work with Ioan Bejenaru).

Biung-Ghi Ju (SNU): "Hierarchical outcomes and collusion neutrality on networks"
We investigate TU-game solutions that are neutral to collusive agreements among players. A collusive agreement binds collusion members to act as a single player and are feasible when they are connected on a network. Collusion neutrality requires that no feasible collusive agreement can change the total payoff of collusion members. We show that on the domain of network games, there is a solution satisfying collusion neutrality, efficiency and null-player property if and only if the network is a tree. Considering a tree network, we show that affine combinations of hierarchical outcomes (Demange 2004, van den Brink 2012) are the only solutions satisfying the three axioms together with linearity. As corollaries, we establish characterizations of the average tree solution (equally weighted average of hierarchical outcomes); one established earlier in the literature and the others new.

Moritz Kaßmann (Bielefeld): "Differential operators of arbitrary order between zero and two"
We discuss a subclass of (integro-)differential operators of fractional order. These operators are related to semi-groups and stochastic processes in a natural way. In the talk we present definitions, basic results as well as recent developments for linear and nonlinear equations. The main emphasis is on new intrinsic scaling properties.

Panki Kim (SNU): "Parabolic Littlewood-Paley inequality for $\phi(-\Delta)$-type operators and applications to Stochastic integro-differential equations"
In this talk we introduce a parabolic version of the Littlewood-Paley inequality for the operators of the type $\phi(-\Delta)$, where $\phi$ is a Bernstein function. As an application, we construct an $L_p$-theory for the stochastic partial integro-differential equations of the type $du=(-\phi(-\Delta)u+f)dt +gdW_t$. This is a joint work with Ildoo Kim and Kyeonghun Kim.

Yuri Kondratiev (Bielefeld): "Nonlinear Markov processes from statistical dynamics"
We describe a general concept of the statistical dynamics for Markov stochastic evolutions in the continuum. A mesoscopic scaling limit algorithm for statistical dynamics leads to the kinetic description of considered systems. The latter is formulated in terms of nonlocal evolution equations for the densities. These equations define nonlinear Markov propagators and related nonlinear Markov processes. We apply this approach to a number of models of interacting particle systems. In particular, certain models of spatial ecology describing notions of growth, expansion and aggregation of populations will be discussed.

Soonsik Kwon (KAIST): "Normal form reduction for unconditional well-posedness of canonical dispersive equations"
Normal form method is a classical ODE technique begun by H. Poincare. Via a suitable transformation one reduce a differential equation to a simpler form, where most of nonresonant terms are cancelled. In this talk, I begin to explain the notion of resonance and the normal form method in ODE setting and Hamiltonian systems. Afterward, I will present how we apply the method to nonlinear dispersive equations such as KdV, NLS to obtain unconditional well-posedness for low regularity data.

Sanghyuk Lee (SNU): "Sharp bounds for Stein's square functions"
In this talk we consider sharp bounds for Stein's square functions in Lebesgue spaces. The problem is regarded as an extension of Bochner-Riesz conjecture, and thanks to square average the bounds have various applications. We obtain an improved range of boundedness and discuss related problems.

Jongil Park (SNU): "How to construct 4-manifolds with $b^+_2 = 1$"
One of the fundamental problems in the topology of 4-manifolds is to classify all smooth/symplectic 4-manifolds and complex surfaces homeomorphic to the same underlying topological 4-manifold.Even though gauge theory has been very successful in this question, the complete answer for the case of 4-manifolds with $b^+_2 = 1$ is still far from reach. The aim of this talk is to briefly review known techniques producing simply connected 4-manifolds with $b^+_2 = 1$ (equivalently, $p_g = 0$ in complex category) in three levels - smooth category, symplectic category and complex category.

Mikko Parviainen (Jyväskylä): "Regularity results for game values and nonlinear PDEs"
We consider a two-player zero-sum stochastic game called a tug-of-war with noise. Our main objective is to describe regularity results for the value functions of the game, and in particular, a local Lipschitz estimate. As an application, the result provides a new technique to prove local Lipschitz continuity and Harnack’s inequality for solutions to certain nonlinear PDEs. The proof is based on a careful choice of strategies and is thus quite different from the original techniques of De Giorgi, Moser, or Nash.

Frank Riedel (Bielefeld): "Knightian Uncertainty in Finance and Economics"
In many economic situations, the probabilities of uncertain outcomes are not known. In such a situation, we speak of Knightian or model uncertainty. A large decision-theoretic literature has been developed in the last two decades. We discuss several applications of the new models to typical optimization and game-theoretic problems.

Michael Röckner (Bielefeld): "An operatorial approach to stochastic partial differential equations driven by multiplicative noise"
In this talk, we develop a new general approach to the existence and uniqueness theory of infinite dimensional stochastic equations of the form $dX+A(t)X dt=X dW$ in $(0,T)\times H$, where $A(t)$ is a nonlinear monotone and demicontinuous operator from $V$ to $V'$, coercive and with polynomial growth.
Here, $V$ is a reflexive Banach space continuously and densely embedded in a Hilbert space $H$ of (generalized) functions on a domain $\mathcal{O}\subset\mathbb{R}^d$ and $V'$ is the dual of $V$ in the duality induced by $H$ as pivot space.
Furthermore, $W$ is a Wiener process in $H$.
The new approach is based on an operatorial reformulation of the stochastic equation which is quite robust under perturbation of $A(t)$. This leads to new existence and uniqueness results of a larger class of equations with linear multiplicative noise than the one treatable by the known approaches.
In addition, we obtain regularity results for the solutions with respect to both the time and spatial variable which are sharper than the classical ones. New applications include stochastic partial differential equations, as e.g. stochastic transport equations.

Sönke Rollenske (Bielefeld): "Explicit constructions of (very singular, stable) algebraic surfaces with $K_X^2 = 1$"
We will present work in progress with M. Franciosi & R. Pardini on the classification of Gorenstein stable surfaces with $K_X^2 = 1$. There will be many pictures as well.

Christoph Thiele (Bonn): "The Hilbert transform along vector fields."
An old conjecture by A. Zygmund proposes a Lebesgue Differentiation theorem along a Lipschitz vector field in the plane. E. Stein formulated a corresponding conjecture about the Hilbert transform along the vector field. If the vector field is constant along vertical lines, the Hilbert transform along the vector field is closely related to Carleson's operator. We discuss some progress in the area by and with Michael Bateman and by my student Shaoming Guo.

Gerald Trutnau (SNU): "On countably skewed Brownian motion with accumulation point Countably skewed Brownian motion (CSBM) is a special case of distorted Brownian motion in dimension one"
Existence and pathwise uniqueness of CSBM was presented by LeGall in 1984 in an abstract frame for some special cases and then explicitly presented by Takanobu in 1986 assuming a uniform, strictly positive distance between the skew reflection points. In this case CSBM is a semimartingale and conservative, i.e. without explosion in finite time. This is not the case when the sequence of skew reflection points has an accumulation point. In this case we shall discuss conditions for existence, pathwise uniqueness, non-explosion, recurrence and positive recurrence, and conditions for CSBM to be a semimartingale. We shall also consider applications. (This is joint work with Youssef Ouknine (Cadi Ayyad University Marrakech) and Francesco Russo (ENSTA ParisTech))

Rudolf Zentel (Mainz University): "International graduate colleges between Germany and South Korea. Case study: Mainz-Seoul (SNU)."
Prof. Rudolf Zentel is the coordinator of the international research and training group "Self-Organized Materials for Optoelectronics" between Mainz University and Seoul National University. This joint program between Germany and Korea was funded in 2006. Further information can be found here. R. Zentel will be present in a video-chat and share with us some of his experiences.

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Talks

You can find the schedule HERE.