The seminar takes place on selected** Wednesdays 8-9 am German time**.

Zoom: https://uni-bielefeld.zoom-x.de/j/62660499705?pwd=Yy84K3hPZnUzeS9BeXEzNnovd3hydz09

Meeting ID: 626 6049 9705

Passcode: 2235

**Denis Brazke (Heidelberg)**: Asymptotic analysis for a sharp interface model related to pattern formation in biomembranes

We derive a macroscopic limit for a sharp interface version of a model proposed by Komura, Shimokawa and Andelman (2006) to investigate pattern formation due to competition of chemical and mechanical forces. The problem is reformulated as a non-local isoperimetric problem, for which we identify sub- and supercritical parameter regimes in terms of the relative strength of the competing local and non-local interactions. Using the autocorrelation function, we find an asymptotic expansion of the energy in terms of the length scale parameter up to first order and derive the Γ-limit in the subcritical and critical parameter regime.

This is joint work with Hans Knüpfer and Anna Marciniak-Czochra.

**Shiping Cao (Seattle)**: Boundary Harnack principle of non-local operators

In this talk, I will present a scale invariant boundary Harnack principle on any open sets for a large class of jump diffusions on metric measure spaces. These jump diffusions can be non-symmetric. In particular, we establish a scale invariant boundary Harnack principle on any open sets for divergence form as well as non-divergence form non-local operators having measurable mixed stable-like jumping kernels on Euclidean spaces.

This is a joint work with Zhen-Qing Chen.

**Alexandra Carpentier (Potsdam)**: Tight concentration inequalities for weakly dependent fields, and applications to the mixing bandit problem

In this talk we will first consider the mixing bandit problem, namely a sequential learning problem over weakly dependent data. For solving optimally this problem, it is important to understand tightly the concentration of weakly dependent processes. With this motivation in mind, I will then present a tight Azuma-Hoeffding-type inequality for partial sums of discrete processes in dimension 1, satisfying a weak dependency assumption of projective type - namely that the conditional expectation given the past of the process at a distance more than u is bounded by a known decreasing function of u. The proof is based on a smart multi-scale approximation of random sums by martingale difference sequences, which was first introduced in [Peligrad, Utev and Wu, 2007]. Based on this, a natural question is on whether this type of results and proof techniques can be extended to weakly dependent random fields in dimension d. I will then present Azuma-Hoeffding and Burkholder-type inequalities for partial sums over a rectangular grid of a random field satisfying a weak dependency assumption of projective type. The analysis is also based on multi-scale approximation of random sums by martingale difference sequences, but a careful decomposition of the d dimensional rectangular grid is essential here in order to obtain tight results.

This is based on joint works with Oleksandr Zadorozhnyi and Gilles Blanchard.

**Soobin Cho (Illinois)**: Probabilistic methods in the analysis of non-local Dirichlet forms

In this talk, I will discuss some results concerning local and non-local regular Dirichlet forms on metric measure spaces, adopting a probabilistic approach. I will emphasize function inequalities like localized Poincaré, cutoff Sobolev, and Faber-Krahn inequalities, showcasing their inherent stability structures. Then I will revisit findings on subordinate Markov processes, highlighting their contribution to stability results. In the last part, I will explore recent advancements in the analysis of these processes, unveiling robust function inequalities and regularity outcomes for harmonic functions in metric measure spaces.

**Sungsoo Byun (Seoul)**: The Product of $m$ real $N×N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$

The study of products of random matrices was proposed many decades ago by Bellman and by Furstenberg and Kesten, with the motivation to understand the properties of the Lyapunov exponents in this toy model for chaotic dynamical systems. In this talk, I will discuss the real eigenvalues of products of random matrices with i.i.d. Gaussian entries. In the critical regime where the size of matrices and the number of products are proportional in the large system, I will present the mean and variance of the number of real eigenvalues. Furthermore, in the Lyapunov scaling, I will introduce the densities of real eigenvalues, which interpolate Ginibre's circular law with Newman's triangular law. This is based on joint work with Gernot Akemann.

**Claudia Bucur (Milan)**: Functions of least $W^{s,1}$-fractional seminorm

We discuss some properties of minimizers of the $W^{s,1}$-fractional seminorm, which exhibit similar characteristics to their classical counterparts known as *functions of least gradient*. Specifically, we examine the connection between these minimizers and nonlocal minimal sets and use this connection to show existence of *functions of least $W^{s,1}$-seminorm*. We further reason about the existence of minimizers and weak solution by investigating the asymptotics as $p$ approaches $1$ for both the $W^{s,p}$-fractional seminorm and its corresponding Euler-Lagrange equation.
The results presented are obtained in collaboration with S. Dipierro, L. Lombardini, J. Maz ́on and E. Valdinoci

**Seokchang Hong (Bielefeld)**: Cauchy problems for nonlinear Dirac equations in the Minkowski spacetime

This talk is devoted to introducing the Cauchy problems for nonlinear Dirac equations with low regularity initial data. As an introductory step, I shall review dispersive inequality and then Strichartz estimates, which play a crucial role in the study of the initial value problems for dispersive PDE. Then we discuss several technical difficulties which arise in both linear estimates and multilinear estimates, especially in the low dimensional setting such as two or three spatial dimensions. Then I will introduce a recent result of scattering of the cubic Dirac equations with the Hartree-type nonlinearity [1]. When enough time is allowed, I will discuss briefly Cauchy problems for Dirac equations in the Lorentz manifold, whose curvature is in general not zero.

- Y. Cho, S. Hong, T. Ozawa, Charge conjugation approach to scattering for the Hartree type Dirac equations with chirality.
*J. Math. Phys.*1 Feb 2023; 64 (2): 021508.

**Myeongju Kang (Seoul)**: Phase-locking in a stochastic Winfree model with inertia

In this talk, we study the emergence of phase-locking for Winfree oscillators under the effect of inertia and multiplicative noise. It is known that in a large coupling regime, oscillators governed by the deterministic Winfree model with inertia converge to a unique equilibrium. In contrast, in this talk, we observe the asymptotic emergence of non-trivial synchronization in a suitably small coupling regime. Moreover, we study the effect of multiplicative noise and obtain lower estimates in probability for the pathwise emergence of such a synchronizing pattern, provided the noise is sufficiently small.

**Kyeong Song (Seoul)**: Regularity results for double phase problems involving nonlocal operators

Abstract: In this talk, we investigate the De Giorgi-Nash-Moser theory for nonlocal problems and mixed local and nonlocal problems. We first recall known regularity results for fractional p-Laplacian problems in the context of the calculus of variations. We then introduce two kinds of double phase problems; one is a nonlocal double phase problem, and the other is a mixed local and nonlocal double phase problem. We present our recent results about Hölder regularity and Harnack inequality for such problems under sharp assumptions.

**Jörn Kommer (Bielefeld)**: A tensor product approach to non-local differential complexes

Abstract: We start the talk by giving a brief introduction to classical differential forms and the de Rham complex on a manifold.

We then define differential complexes of Alexander-Spanier(-Kolmogoroff) type on metric measure spaces associated with unbounded non-local operators like the fractional Laplacian. These complexes are Hilbert complexes, and they can be seen as non-local counterparts of the de Rham complex. We can obtain self-adjoint non-local analogues of Hodge Laplacians. If time permits, we prove a Mayer-Vietoris principle and a Poincaré lemma, and verify that in the compact Riemannian manifold case the de Rham cohomology can be recovered.

This talk is based on joint work with Michael Hinz.

**Patricia Alonso Ruiz (Texas)**: Sobolev embeddings in Dirichlet spaces

Abstract: A classical Sobolev embedding in $\mathbb{R}^n$ asserts that functions in the Sobolev space $W^{1,p}(\mathbb{R}^n)$ with $1 \leq p < n$ belong to a suitable $L^q$-space with an explicit optimal exponent q that depends on n and p. How does this embedding read in the context of Dirichlet spaces? This talk will provide an answer in the context of metric measure spaces equipped with a regular Dirichlet form. In particular, we will discover how the optimal exponent depends on the Hausdorff dimension, the walk dimension, and a further invariant of the space. The main assumptions on the underlying space are heat kernel estimates and a non-negative curvature type condition that we call weak Bakry-Émery and is satisfied in classical settings as well as in fractals like (infinite) Sierpinski gaskets and carpets. The talk is based on joint work with F. Baudoin.

**Ho-Sik Lee (Seoul)**: Global Maximal Regularity for Equations with Degenerate Weights.

Abstract: Elliptic equations are usually studied under the uniform ellipticity assumption. In this talk, I allow the degeneracy to the uniform ellipticity and consider the elliptic equations with degenerate weights, where the weight is given by a function belonging to a certain Muckenhoupt class. Then I introduce the result of global Calderon-Zygmund type estimates, which are obtained by imposing a small BMO condition on the logarithm of the degenerate weight and a small Lipschitz constant condition on the boundary of the domain. I also discuss the sharpness of our result by an example.

**Patrícia Gonçalves (Lisbon)**: From particle systems to partial differential equations.

Abstract: In this seminar I will describe the derivation of partial differential equations that rule the space-time evolution of the conserved quantity (ies) of stochastic processes. The random dynamics conserves a quantity (as the total mass) that has a non-trivial evolution in space and time. The goal is to describe the connection between the macroscopic (continuous) equations and the microscopic (discrete) system of random particles. The former can be either PDEs or stochastic PDEs depending on whether one is looking at the law of large numbers or the central limit theorem scaling; while the latter is a collection of particles that move randomly according to a transition probability and rings of Poisson processes. I will focus on a particular particle system, namely, the exclusion process and explain that depending on the choice of the jump rates we get to different partial differential equations.

**Jaehun Lee (Seoul)**: General law of iterated logarithm for Markov processes

Abstract : Law of iterated logarithm for a stochastic process describes the magnitude of the fluctuations of its sample path behaviors. In this talk, we discuss the law of iterated logarithm for continuous-time Markov processes, which cover a large class of subordinated diffusions, jump processes with mixed polynomial local growth, jump processes with singular jumping kernels and random conductance models with long range jumps. We establish the law of iterated logarithms for the distance and occupation time, in terms of two functions concerned with the first exit time from balls, and uniform bounds on the tails of jumping measure. Especially, we concentrate on the sharp sufficient conditions for both small and large time. This is joint work with Soobin Cho and Panki Kim.

This is the joint work with Soobin Cho and Panki Kim.

**Tabea Tscherpel (Bielefeld)**: On the Sobolev stability of the $L^2$-Projection mapping to finite element spaces

Abstract: The $L^2$-projection mapping to Lagrange finite element spaces is a crucial tool in numerical analysis. Especially when dealing with evolution problems Sobolev stability of the $L^2$-projection is needed. While the proof of Sobolev stability is straightforward for uniform meshes, it is much more challenging for graded meshes. We tackle this difficulty by investigating the decay properties of the $L^2$-projection analytically. Inspired by the miracle of extrapolation we introduce a maximal operator. This allows us to obtain equivalence of decay and certain weighted estimates, that eventually lead to Sobolev stability. We present recent results in 2D and 3D for a wide range of polynomial degrees and graded meshes. Those include a large class of meshes used in practical applications.

This is joint work with Lars Diening and Johannes Storn from Bielefeld University (Germany).

**Minhyun Kim (Bielefeld)**: Local regularity theory for nonlocal equations

Abstract: We introduce several nonlocal equations which correspond to classical local models, attempting to develop parallel or unified theory. We study local regularity properties such as local boundedness, weak Harnack inequality, and local Hölder regularity of weak solutions to these nonlocal equations.