Lucian Beznea (Bucharest) – Strong Feller semigroups and Markov processes

Our aim is to show that the strong Feller and the joint (space-time) continuity for a semigroup of Markov kernels on a Polish space are not enough to ensure the existence of an associated càdlàg Markov process on the same space. One simple counterexample is the Brownian semigroup on R restricted to R \ {0}, for which it is shown that there is no associated càdlàg Markov process. Using results from potential theory we then prove that the analogous result with càdlàg Markov process replaced by right Markov process also holds.
The talk is based on a joint work with Iulian Cîmpean (Bucharest) and Michael Röckner (Bielefeld).
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Sungsoo Byun (SNU) – Spectral moments and Harer–Zagier type recursion formulas in random matrix theory

Random matrix theory enjoys an intimate connection with various branches of mathematics. One prominent illustration of this relationship is the Harer–Zagier formula for spectral moments, which serves as a well-known example demonstrating the combinatorial and topological significance inherent in random matrix statistics. While the Harer–Zagier formula originates from the study of the moduli space of curves, it also gives rise to a fundamental formula in the study of spectral moments of classical random matrices. In this talk, I will introduce Harer–Zagier type formulas for classical Hermitian Gaussian random matrix ensembles and present recent progress across various models.
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Zhen-Qing Chen (Seattle) – Uniform boundary Harnack principle for non-local operators

Boundary Harnack principle (BHP) is a fundamental principle both in Probability and in Analysis. When it holds, it has many far-reaching implications. In this talk, I will present our recent advances in the study of BHP that establishes a uniform BHP on any open sets for a large class of jump diffusions (or equivalently, non-local operators) under a jump measure comparability and tail estimate condition, and an upper bound condition on the distribution functions for the exit times from balls. These conditions are satisfied by a large class of non-local operators, including those that admit a two-sided mixed stable-like heat kernel bound when the underlying metric measure spaces have volume doubling and reverse volume doubling properties. The results of this paper are new even for non-local operators on Euclidean spaces. In particular, our results give not only the scale invariant but also uniform BHP for the first time for non-local operators on Euclidean spaces of both divergence form and non-divergence form with measurable coefficients. Based on joint work with Shiping Cao.
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Yan Fyodorov (King's College London, Bielefeld) – Superposition of plane waves in high spatial dimensions: from landscape complexity to the ground state value

I will discuss some statistical properties of a class of models of high-dimensional random landscapes defined in a Euclidean space of large dimension $N >>1$ via a superposition of $M$ plane waves whose amplitudes, directions of the wavevectors, and phases are taken to be random. The main efforts are directed towards deriving, and then analysing for $N\to \infty, M\to \infty$ while keeping $\alpha=M/N$ finite,
(i) the rates of asymptotic exponential growth with $N$ of the mean number of all critical points and of local minima known as the annealed complexities and
(ii) the expression for the mean (also expected to be typical) value of the deepest landscape minimum (the ground-state energy).
In particular, for the latter we derive the Parisi-like optimization functional and analyze conditions for the optimizer to reflect various phases: replica-symmetric, one-step and full replica symmetry broken, as well as criteria for the continuous, Gardner and random first order transitions between those phases.
The talk will be based on the joint work with Bertrand Lacroix-A-Chez-Toine, arXiv:2411.09687
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Anna Gusakova (Münster) – Concentration inequalities for Poisson $U$-statistics

Let $\eta$ be a Poisson point process on a general measurable space. A Poisson functional is a random variable $F(\eta)$, such that almost surely we have $F(\eta)=f(\eta)$ for some measurable veal valued function $f$ on the space of counting measures. Poisson functionals have been intensively studied within last years and they play an important role in stochastic geometry since many important geometric functionals of stochastic geometry models are in fact Poisson functionals. Poisson $U$-statistic is an example of Poisson functional, which has particularly nice structure. In this talk we present concentration inequalities for Poisson $U$-statistics under some rather mild conditions. We will discuss their optimality and consider a few applications to stochastic geometry models.
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Seokchang Hong (Bielefeld) – Strichartz estimates for the half Klein–Gordon equation on asymptotically flat backgrounds and applications to Dirac equations

The aim of this talk is to establish the $L^2_t$-endpoint Strichartz estimate for (half) Klein–Gordon equations on a weakly asymptotically flat space-time. Using the results by Metcalfe–Tataru, we construct an outgoing parametrix for the operators via the phase space transform. Although the Klein–Gordon equation does not obey the scaling symmetry, the scaling argument plays a crucial role and hence we can restrict ourselves to the operators localised in the unit scale. Consequently, we obtain the dispersive inequality, which is exactly same as the classical one, and then establish the endpoint Strichartz estimates. As an application of this result, we obtain the global well-posedness and scattering for cubic Dirac equation on a weakly asymptotically flat space-time for $H^s$-data, $s>1$.
This talk is based on a joint work with Sebastian Herr.
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Jonas Jalowy (Münster) – Zeros of Polynomials: Limiting distributions, role of the coefficients and evolutions under differential flows

Everyone learns how to find the zeros of a quadratic polynomial in school, but for a large n-degree polynomial, this remains a challenging task. This talk explores the limiting zero distribution of polynomials as n grows and its connection to their coefficients, focusing on real-rooted and random polynomials.
We begin with a user-friendly approach to determine the limiting zero distribution via the 'exponential profile' of the coefficients and apply it to various classical polynomials and operations such as the Hadamard product, repeated differentiation, heat flow, and finite free convolutions.
For random polynomials with i.i.d. rescaled coefficients, the zeros are complex, and their evolution under such differential operators becomes more intricate and visualizing. In one prominent example of Weyl polynomials undergoing the heat flow, the limiting zero distribution evolves from the circular law into the elliptic law until it collapses to the Wigner semicircle law. More generally, we describe the limiting zero distribution and root dynamics from various points of view such as (optimal) transport, differential equations, and free probability.
Illustrative simulations will accompany the talk, leading to intriguing open questions. This talk is based on joint works with Brian Hall, Ching-Wei Ho, Antonia Höfert, Zakhar Kabluchko, and Alexander Marynych.
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Jinwook Jung (Hanyang) – Modulated energy estimates for singular kernels and their applications

In this talk, we provide modulated interaction energy estimates for the kernel $K(x) = |x|^{-\alpha}$ with $\alpha \in (0,d)$, and their applications. The proof relies on a dimension extension argument for an elliptic operator and its commutator estimates. For the applications, we first discuss the quantified asymptotic limit of kinetic equations with singular nonlocal interactions. We show that the aggregation equations and the isothermal or pressureless Euler system with singular interaction kernels are rigorously derived. Second, we employ the estimates to establish the well-posedness theories in Hölder spaces for the kinetic and fluid equations involving singular interaction kernels, mainly about inviscid $\alpha$-surface quasi-geostrophic ($\alpha$-SQG) equations and kinetic Cucker-Smale model.
This talk is based on the collaboration with Y.-P. Choi (Yonsei Univ.) and J. Kim (Ajou Univ.).
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Myungjoo Kang (SNU) – Implicit Surface Reconstruction: A PDE-Guided Deep Learning Approach

In this presentation, we introduce an advanced deep learning approach for reconstructing surfaces from unorganized point clouds. By leveraging an implicit surface representation through a level set function, our method ensures watertight results and seamlessly adapts to various topologies. We employ the p-Poisson equation to precisely learn the signed distance function (SDF), improving accuracy through a variable splitting strategy that incorporates the SDF gradient as an auxiliary variable. Additionally, we enforce a curl-free condition on the auxiliary variable to exploit the irrotational nature of conservative vector fields. Our numerical results illustrate that this strategic integration of partial differential equations and key vector field characteristics efficiently reconstructs high-quality surfaces without the need for prior surface knowledge.
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Jongmyeong Kim (Academia Sinica) – Holder regularity of fractional Laplacian on manifold

ABP estimate is important key step in Kyrlov-Safonov theory to achieve Holder regularity. I will introduce nonlocal ABP estimate on the hyperbolic space which generalize the result of Caffarelli and Silvestre on oneside and the result of Wang and Zhang on otherside. If time permit I will introduce (ongoing) Holder estimate for jump type Dirichlet form on the small class of Riemannian manifolds which involves the hyperbolic space.
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Kihyun Kim (SNU) – On classification of the rates of concentration of bubbles for the radial critical nonlinear heat equation in large dimensions

In my talk given online last December, I discussed my joint work with Frank Merle (IHES and CY Cergy-Paris Université) on classification of bubble tower dynamics for the radial critical nonlinear heat equation in large dimensions. In this talk, I will recall the main result and talk about a formal derivation of the concentration rates of each bubble. If time permits, I will also discuss the scheme of the proof.
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Kyeongbae Kim (SNU) – Gradient estimates for nonlinear nonlocal equations

We discuss potential estimates of non-homogeneous nonlinear nonlocal equations. We first recall previously known results about pointwise gradient estimates for solutions to the nonlinear generalization of Poisson’s equation. Then, we present pointwise gradient estimates of nonlinear nonlocal equations via linear Riesz potentials.
The talk is based on a joint work with Lars Diening, Ho-Sik Lee, and Simon Nowak.
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Minhyun Kim (Hanyang) – Optimal boundary regularity and Green function estimates

We study the optimal $C^s$ boundary regularity for solutions to nonlocal elliptic equations with Hölder continuous coefficients in divergence form in $C^{1,\alpha}$ domains. As an application of our results, we establish sharp two-sided Green function estimates in $C^{1,\alpha}$ domains.
This talk is based on joint work with Marvin Weidner.
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Panki Kim (SNU) – Stability of Hölder regularity and weighted functional inequalities

In this talk, we first introduce new forms of tails of jumping measures and weighted functional inequalities for general symmetric Dirichlet forms on metric measure spaces under general volume doubling condition. Our framework covers Dirichlet forms with singular jumping measures including ones corresponding to trace processes. Using the new weighted functional inequalities, we establish stable equivalent characterizations of Hölder regularity of parabolic functions for symmetric Dirichlet forms. As consequences of the main result, we can show Hölder-continuity of parabolic functions for a large class of symmetric Markov processes blowing up to infinity at the boundary of state spaces. This talk is mainly based on a joint work with Soobin Cho (University of Illinois, USA).
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Ho-Sik Lee (Bielefeld) – Global estimates for elliptic equations with degenerate weights

In second-order elliptic equations, the matrix-valued coefficients equipped in equations are usually assumed to be uniformly elliptic. One of the generalizations of such an assumption is considering Muckenhoupt weight. Assuming the sharp conditions for the coefficient as well as for the boundary of the domain, we obtained the global Calderon-Zygmund estimates for the corresponding Dirichlet problem. Next, we consider the mixed boundary value problem, allowing the case that the Dirichlet condition is assumed on the Cantor set of the boundary, and the Neumann condition is assumed on the complement of the boundary. Under the certain density condition for the subset where the Dirichlet condition is assumed in the sense of capacity, we obtained the existence and higher integrability result. We are also planning to consider the mixed exterior value problem for the fractional Laplace.
These are joint works with Anna Kh. Balci (Charles Univ. in Prague), Sun-Sig Byun (SNU), Lars Diening (Bielefeld), and Guy Foghem (Dresden).
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Kim-Ahm Lee (SNU) – Degenerate Nonlinear Partial Differential Equations in Curvature Flows

In this talk, we are going to discuss Degenerate Nonlinear Partial Differential Equations in Curvature Flows with a noncompact graph as its initial hypersurface. The solution can be expressed as a graph with infinite height on the boundary of its support. We will discuss uniform estimates of the solution up to the infinite height, the evolution of the boundary of its support, and its geometric properties preserved under the flow.
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Seungwoo Lee (SNU) – Mixing of underdamped Langevin dynamics: from cut-off to Erying–Kramers formula

The underdamped Langevin dynamics is a stochastic model describing evolution of thermostated molecular dynamics. In this talk, we discuss mixing behavior of the underdamped Langevin dynamics in the low temperature regime. We observe the cut-off phenonmenon when there is only one stable equilibrium, while observe the metastability when there are multiple stable equilibria. We explain quantitatively precise analyses for both cases; the main difficulty of the model is the degeneracy of the generator associated with the underdamped Langevin dynamics.
This talk is based on two joint works with Professor Seo Insuk from Seoul National University and Ramil Mouad from INRIA france.
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Yong-Woo Lee (SNU) – The probability of almost all eigenvalues being real for the elliptic real Ginibre ensemble

The elliptic Ginibre orthogonal ensemble (eGinOE) is a one-parameter family of random matrix models that interpolates between Hermitian and non-Hermitian real random matrix ensembles. A notable feature of this model is that its spectrum contains real eigenvalues with non-trivial probability. In this talk, we investigate the large deviation probabilities associated with the number of real eigenvalues in the eGinOE. We present a precise estimation for the probability that the spectrum of the eGinOE consists of all real eigenvalues except for finitely many at the strong and weak non-Hermiticity.
This talk is based on joint work with Gernot Akemann and Sung-Soo Byun.
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Chengcheng Ling (Augsburg) – Quantitative approximation of stochastic kinetic equations: from discrete to continuum

We study the strong convergence of a generic tamed Euler–Maruyama (EM) scheme for the kinetic type stochastic differential equation (SDE) (also known as second order SDE) driven by $\alpha$-stable type noise with $\alpha\in(1,2]$. We show that when the drift exhibits a relatively low regularity: anisotropic $\beta$-Hölder continuity with $\beta >1 - \frac{\alpha}{2}$, the corresponding tamed EM converges with a convergence rate $(\frac{1}{2} + \frac{\beta}{\alpha(1+\alpha)} \wedge \frac{1}{2})$, which aligns with the results of first-order SDEs.
This talk is based on the work arXiv:2409.05706 (joint with Zimo Hao and Khoa Lê) and the work arXiv:2412.05142.
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Jongil Park (SNU) – A study on rational homology projective planes

A normal projective surface with the same Betti numbers of the projective plane $\mathbb{CP}^2$ is called a rational homology projective plane or a $\mathbb{Q}$-homology $\mathbb{CP}^2$. People working in algebraic geometry and topology have long studied a $\mathbb{Q}$-homology $\mathbb{CP}^2$ with possibly quotient singularities. It is now known that it has at most five such singular points, but it is still mysterious so that there are many unsolved problems left.
In this talk, I’ll review some known results and open problems in this field which might be solved and might not be solved in near future. In particular, I’d like to review the following two topics and to report some recent progress:
1. Algebraic Montgomery-Yang problem.
2. Classification of $\mathbb{Q}$-homology $\mathbb{CP}^2$ with quotient singularities.
This is a joint work with Woohyeok Jo and Kyungbae Park.
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Marco Rehmeier (TU Berlin) – $p$-Brownian motion and the $p$-Laplacian

We construct a stochastic process which is related to the fundamental solution of the parabolic p-Laplace equation in the same way as Brownian motion is related to the heat kernel of the heat equation. More precisely, for the p-Laplace equation we identify an associated McKean—Vlasov SDE, and our constructed stochastic process consists of solutions to this SDE and, moreover, constitutes a nonlinear Markov process. We call this process a p-Brownian, which for $p=2$ coincides with standard Brownian motion.
Joint work with Viorel Barbu (A.I. Cuza University) and Michael Röckner (Bielefeld University).
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Andre Schenke (NYU Courant) – Non-uniqueness of Leray–Hopf solutions for the 3$D$ fractional Navier–Stokes equations perturbed by transport noise

For the 3$D$ fractional Navier–Stokes equations perturbed by transport noise, we prove the existence of infinitely many Hölder continuous analytically weak, probabilistically strong Leray–Hopf solutions starting from the same deterministic initial velocity field. Our solutions are global in time and satisfy the energy inequality pathwise on a non-empty random interval $[0,\tau]$. In contrast to recent related results, we do not consider an additional deterministic suitably chosen force f in the equation. In this unforced regime, we prove the first result of Leray–Hopf nonuniqueness for fractional Navier–Stokes equations with any kind of stochastic perturbation. Our proof relies on convex integration techniques and a flow transformation by which we reformulate the SPDE as a PDE with random coefficients.
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Robert Schippa (UC Berkeley) – Quantified decoupling estimates and applications

In 2004 Bourgain proved a qualitative trilinear moment inequality for exponential sums and raised the question for quantitative estimates. Here we show quantitative estimates. The proof combines decoupling iterations with semi-classical Strichartz estimates. Secondly, we improve on Bourgain’s well-posedness result for the periodic KP-II equation. The latter part of the talk is based on joint work with Sebastian Herr and Nikolay Tzvetkov.
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Gerald Trutnau (SNU) – Pointwise and a.e. well-posedness results for degenerate Itô-SDEs with measurable coefficients

We present different kinds of existence and uniqueness in law results for degenerate Itô stochastic differential equations on Euclidean space with measurable coefficients. In a first step, we develop results with respect to almost every starting point of the state space and a given (sub-)invariant measure. For this we use functional analytic tools together with probabilistic techniques. In a second step, building on the previous results, we additionally use elliptic regularity results for PDEs to consider a pointwise analysis for every starting point. If time permits, we will consider applications.
This is joint work with Haesung Lee (Kumoh National Institute of Technology, South Korea).
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Zoran Vondraček (Zagreb) – Markov processes with jump kernels decaying at the boundary

In this talk I will explain some aspects of a general theory for non-local singular operators of the type $$ L^{\mathcal{B}}_{\alpha}f(x)=\lim_{\epsilon\to 0} \int_{D,\, |y-x|>\epsilon}\big(f(y)-f(x)\big) \mathcal{B}(x,y)|x-y|^{-d-\alpha}\,dy, $$ and $$ L f(x)=L^{\mathcal{B}}_{\alpha}f(x) - \kappa(x) f(x), $$ in case $D$ is a $C^{1,1}$ open set in $\mathbb{R}^d$, $d\ge 2$. The function $\mathcal{B}(x,y)$ above may vanish at the boundary of $D$, and the killing potential $\kappa$ may be subcritical or critical.
From a probabilistic point of view we study the reflected process on the closure $\overline{D}$ with infinitesimal generator $L^{\mathcal{B}}_{\alpha}$, and its part process on $D$ obtained by either killing at the boundary $\partial D$, or by killing via the killing potential $\kappa(x)$. The general theory developed in this work (i) contains subordinate killed stable processes in $C^{1,1}$ open sets as a special case, (ii) covers the case when $\mathcal{B}(x,y)$ is bounded between two positive constants and is well approximated by certain Hölder continuous functions, and (iii) extends the main results known for the half-space in $\mathbb{R}^d$. The main results are the boundary Harnack principle and its possible failure, and sharp two-sided Green function estimates. The results on the boundary Harnack principle completely cover the corresponding earlier results in the case of half-space. The Green function estimates extend the corresponding earlier estimates in the case of half-space to bounded $C^{1, 1}$ open sets.
Joint work with Soobin Cho, Panki Kim and Renming Song.
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Marvin Weidner (Barcelona) – Boundary regularity for nonlocal equations

There are significant differences between local and nonlocal problems when it comes to the boundary behavior of solutions. For instance, it is a well known fact that $s$-harmonic functions (i.e. solutions to nonlocal elliptic equations governed by the fractional Laplacian) are in general not better than $C^s$ up to the boundary.
As a consequence, in recent years there has been a huge interest in the boundary behavior of solutions to nonlocal equations. By now, the boundary regularity is well understood for the fractional Laplacian and for $2s$-stable nonlocal operators, however very little is known about the natural class of nonlocal operators with inhomogeneous kernels.
In this talk, I will present recent progress on the study of the inhomogeneous case, achieved in collaboration with Xavier Ros-Oton.
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