Workshop "Fourth Bielefeld-SNU Joint Workshop in Mathematics"

# Fourth Bielefeld-SNU Joint Workshop in Mathematics

The Department of Mathematics will host the

Fourth Bielefeld-SNU Joint Workshop in Mathematics

February 23-24, 2016

ZiF Bielefeld

## Participants

• Gernot Akemann (Bielefeld)
• Sun-Sig Byun (SNU)
• Timothy Candy (Bielefeld)
• Lars Diening (Osnabrück University)
• Barbara Gentz (Bielefeld)
• Friedrich Götze (Bielefeld)
• Alexander Grigoryan (Bielefeld)
• Seung-Yeal Ha (SNU)
• Sebastian Herr (Bielefeld)
• Michael Hinz (Bielefeld)
• Nam-Gyu Kang (KIAS)
• Moritz Kassmann (Bielefeld)
• Mario Kieburg (Bielefeld)
• Panki Kim (SNU)
• Yuri Kondratiev (Bielefeld)
• Soonsik Kwon (KAIST)
• Ji Oon Lee (KAIST)
• Ki-Ahm Lee (SNU)
• Sanghyuk Lee (SNU)
• Ionut Munteanu (Iasi)
• Barbara Niethammer (Bonn)
• Michael Röckner (Bielefeld)
• Gerald Trutnau (SNU)

## Talks

All talks will be held in ZiF Plenarsaal.

• We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + \mathcal{L} F(u)=0$ posed in a bounded domain, $x\in\Omega\subset \mathbb{R}^N$ with appropriate homogeneous Dirichlet boundary conditions. As $\mathcal{L}$ we can use a quite general class of linear operators that includes the two most common versions of the fractional Laplacian $(-\Delta)^s$, $0 < s < 1$, in a bounded domain with zero Dirichlet boundary conditions, but it also includes many other examples, since our theory only needs some basic properties that are typical of linear heat semigroups''. The nonlinearity $F$ is assumed to be increasing and is allowed to be degenerate, the prototype being $F(u)=|u|^{m-1}u$, with $m>1$. We will present some recent results about existence, uniqueness and a priori estimates for a quite large class of very weak solutions, that we call weak dual solutions. We then show sharper lower and upper estimates up to the boundary, which fairly combine into various forms of Harnack type inequalities. The standard Laplacian case $s=1$ is included and the linear case $m=1$ can be recovered in the limit in most of the results. When the nonlinearity is of the form $F(u)=|u|^{m-1}u$, with $m>1$, global Harnack estimates are the key tool to understand the sharp asymptotic behaviour of the solutions. We finally show that solutions are classical, and even $C^{\infty}$ in space in the interior of the domain, when the operator $\mathcal{L}$ is the (restricted) fractional Laplacian. The above results are contained on a series of recent papers with J. L. Vazquez, and also with A. Figalli, Y. Sire and X. Ros-Oton.
• We discuss on recent regularity results for double phase problems with non-standard growth over bounded domains with irregular boundary. This is a joint work with Jehan Oh.
• We give an overview of recent work on the problem of small data global well-posedness for the Cubic Dirac equation in 2+1 and 3+1 dimensions. As with other dispersive PDE, the main obstruction to studying the global dynamics for the nonlinear Dirac equation, is the lack of available Strichartz estimates in small dimensions. To get around this difficulty, there are two key ideas. The first is the observation, originally due to Tataru, that it is possible to recover certain endpoint Strichartz estimates by using coordinate frames which are oriented in null directions. The second observation is that the cubic nonlinearity exhibits a subtle cancellation (also known as "null structure"), which removes otherwise dangerous parallel interactions. This is joint work with Nikolaos Bournaveas.
• Lars Diening (Osnabrück)
Nonlinear Calderon-Zygmund theory
For many linear partial differential equations there is a correlation between the data and the solution in terms of a singular integral operator. Due to this fact the data and the solution are in the same function space. For example the standard Laplace equation with $L^p$ data has solutions with second gradients in $L^p$. For non-linear partial differential equations this approach is not possible. However, starting with the result of Iwaniec, many similar quantative statements have been derived for the p-Laplace equation/system. In this talk I present a novel pointwise estimate of the gradients of the solution in terms of maximal functions. This is a joint work with Dominic Breit, Andrea Chiani, Tuomo Kuusi, Sebastian Schwarzacher
• The rate of convergence for the spectral distribution of random matrices from Wigner ensembles with more than four moments to the semicircular law is investigated. An optimal rate is shown for the convergence of the expected spectrum as well as partially optimal (local) rates for the random spectral distributions. Applications to the convergence to the Marcenko-Pastur laws are discussed. This is joint work with A. Naumov, D. Timushev and A. Tikhomirov.
• We consider time-dependent perturbations of symmetric diffusions on locally compact spaces. Given a strongly local regular symmetric Dirichlet form, the corresponding first order calculus allows to define time-dependent drift and potential perturbations of its infinitesimal generator. Under standard conditions corresponding Cauchy problems admit unique classical solutions. We provide probabilistic representations for these solutions in terms of an associated space-time process and its additive functionals. If the original diffusion is Feller, we can rewrite these representation using this Feller process and its additive functionals.
• I will present an elementary introduction to conformal field theory in the context of complex analysis and probability theory. Introducing Ward functional as an insertion operator under which the correlation functions are transformed into their Lie derivatives, I will explain several formulas in conformal field theory including Ward's equations. This presentation will also include relations between conformal field theory and Schramm-Loewner evolutions. This is joint work with Nikolai Makarov.
• The study of the relation between eigenvalues and singular values of an arbitrary square matrix has a long history. It reaches back to works by Schur and Weyl in the last century. They derived inequalities between both kinds of spectra. Horn reversed the question and asked whether it is possible to find a matrix having a certain set of eigenvalues and singular values. Again those inequalities found by Weyl yielded the answer. Introducing randomness on the set of matrices changes the situation drastically. One has to rephrase the question to whether there is a relation between the eigenvalue and singular value statistics. I will show that for a specific class of random matrix ensembles called bi-unitarily invariant there is indeed a relation between both statistics. This relation is even bijective. The proof makes use of harmonic analysis on matrix spaces.
• In this talk, We consider a large class of symmetric pure jump Markov processes dominated by isotropic unimodal Lévy processes with weak scaling conditions. We first establish sharp two-sided heat kernel estimates for this processes in $C^{1,\rho}$ open sets. As a corollary of our main result, we obtain a sharp two-sided Green function and a scale invariant boundary Harnack inequality with explicit decay rates in $C^{1,\rho}$ open sets.This is a joint work with Tomasz Grzywny and Kyung-Youn Kim.
• We start with Markov dynamics of interacting particle systems in the continuum. A fractional time evolution in such systems corresponds to random time changes. In the Vlasov type scaling, it leads to a fractional mesoscopic hierarchy for correlation functions. Corresponding state evolutions are obtained by means of a subordination of Poisson flows (which describe the kinetic behavior of initial Markov dynamics). We will discuss subordination effects, in particular, the notion of intermittency which appear as a result of fractional evolution and never is possible in the Markov kinetics.
• We will begin with explaining the Poincare-Dulac normal form idea to prove the local well-posedness of nonlinear dispersive equations. Later, we will discuss with a particular example, quadratic derivative NLS. We develop an infinite iteration scheme of normal form reductions for dNLS. By combining this normal form procedure with the (modified) Cole-Hopf transformation, we prove unconditional global well-posedness in $L^2(T)$, and more generally in certain Fourier-Lebesgue spaces $FL^{s,p}(T)$, under the mean-zero and smallness assumptions. With this example, we observe a relation between normal form approach and canonical nonlinear transform.
• We consider the spherical spin glass model, also known as the spherical Sherrington-Kirkpatrick model. With the aid of recent results of random matrix theory, we prove that the fluctuation of the free energy converges to a Gaussian distribution at high temperature and to the Tracy-Widom distribution at low temperature. The orders of the fluctuations are markedly different in these two regimes. This is a joint work with Jinho Baik.
• In this talk, we are going to discuss recent development in the theory of homogenization of nonlinear partial differential equations. First, we will consider the convergence rate of oscillating solutions of nonlinear equations with oscillations to a limit solution satisfying homogenized (or effective) equations. Second, we will also consider the oscillations in lower dimensional surfaces that have different difficulties from those in the full dimension.
• This talk is concerned about multilinear estimates of which usefulness and importance in hormonic analysis have been proven recently. We show that the best constant in the general Brascamp-Lieb inequality is bounded under small perturbation of the underlying linear transformations. As applications we obtain multilinear Fourier restriction, Kakeya-type estimates, and nonlinear variants of the Brascamp-Lieb inequality. This is a joint work with Jon Bennett, Neal Bez, and Taryn Flock.
• We consider stochastic nonlinear differential equations with highly singular diffusitivity term and multiplicative Stratonovich unbounded operator involving the gradient. We study existence and uniqueness of variational solutions in terms of stochastic differential equations. These kind of equations arise in the use for simulations of image restoring.
• In 1916 Smoluchowski derived a mean-field model in order to develop a mathematical theory for coagulation processes. Since Smoluchowski's groundbreaking work his model has been used in a diverse range of applications such as aerosol physics, polymerization, population dynamics, or astrophysics. After reviewing some basic properties of the model I will address the fundamental question of dynamic scaling, that is whether solutions develop a universal self-similar form for large times. This issue is well understood for some exactly solvable cases, but in the general case many questions are still completely open. I will give an overview of the results that have been obtained in the last decade and the remaining main open problems.
• We develop sufficient analytic conditions for recurrence and transience of non-sectorial perturbations of possibly non-symmetric Dirichlet forms on a general state space. These form an important subclass of generalized Dirichlet forms. In case there exists an associated process, we show how the analytic conditions imply recurrence and transience in the classical probabilistic sense. As an application, we consider a generalized Dirichlet form given on a closed or open subset of $\mathbb{R}^d$ which is given as a divergence free first order perturbation of a non-symmetric energy form. Then using volume growth conditions of the sectorial and non-sectorial first order part, we derive an explicit criterion for recurrence. Moreover, we present concrete examples with applications to Muckenhoupt weights and counterexamples. The counterexamples show that the non-sectorial case differs qualitatively from the symmetric or non-symmetric sectorial case. Namely, we make the observation that one of the main criteria for recurrence in these cases fails to be true for generalized Dirichlet forms. This is joint work with Minjung Gim (Seoul National University).

## Contact

In case of questions feel free to contact Rebecca Reischuk.

rreischuk@math.uni-bielefeld.de

## Organization

Moritz Kaßmann (Bielefeld University)

Panki Kim (Seoul National University)

## Support

This workshop is part of the conference program of CRC 701 "Spectral Structures and Topological Methods" funded by the German Science Foundation (DFG).