List of Titles and Abstracts
Monday, July 14, 2025
9:00 AM - 9:50 AM Professor Hiroaki Aikawa, Chubu University
Title: Potential Analysis on Nonsmooth DomainsⅠ
Abstract: The Laplace and heat equations are classical and often considered well understood. However, many questions about the boundary behavior of solutions and supersolutions in general domains remain open. In this talk, we explore how domain complexity affects potential-theoretic properties, including the integrability of positive superharmonic functions, the Martin boundary, elliptic and parabolic boundary Harnack principles, intrinsic ultracontractivity, and related notions. To this end, we examine various nonsmooth domains, such as Lipschitz, NTA, uniform, inner uniform, John, Hölder, and $L^p$ domains, among others. The first lecture presents the main results, while the second provides an outline of the key techniques and proof strategies.
10:00 AM – 10:50 AM Professor Hiroaki Aikawa, Chubu University
Title: Potential Analysis on Nonsmooth DomainsⅡ
Abstract: The Laplace and heat equations are classical and often considered well understood. However, many questions about the boundary behavior of solutions and supersolutions in general domains remain open. In this talk, we explore how domain complexity affects potential-theoretic properties, including the integrability of positive superharmonic functions, the Martin boundary, elliptic and parabolic boundary Harnack principles, intrinsic ultracontractivity, and related notions. To this end, we examine various nonsmooth domains, such as Lipschitz, NTA, uniform, inner uniform, John, Hölder, and $L^p$ domains, among others. The first lecture presents the main results, while the second provides an outline of the key techniques and proof strategies.
11:00 AM – 11:50 AM Professor Zhou-ping Xin, Chinese University of Hong Kong, Hong Kong
Title and Abstract: TBA
2:00 PM – 2:50 PM Professor Xuwen Chen, University of Rochester, USA
Title and Abstract: TBA
( 2:50 PM – 3:30 PM Coffee Break )
3:30 PM – 4:00 PM TBA
4:00 PM – 4:30 PM Postdoctoral Scholar Zhongyang Gu, University of Tokyo, Japan
Title: The Helmholtz decomposition of a BMO type vector field in a domain
Abstract: The Helmholtz decomposition in the \(Lp\)-setting was well-studied for. However,it is not suitable to investigate this decomposition for the vector fields. In this talk, we will introduce a BMO type of vector fields in a domain, whose normal component to the boundary is well-controlled, and present its Helmholtz decomposition as a substitute theory for the setting. This talk is based on a series of joint works with Professor Yoshikazu Giga (The University of Tokyo).
Tuesday, July 15, 2025
9:00 AM - 9:50 AM Professor Edriss Titi, Texas A & M University, USA
Title: Mathematical Analysis of Atmospheric and Oceanic Dynamics Models: Cloud Formation and Sea-ice Models Ⅰ
Abstract: In these talks we will present rigorous analytical results concerning global regularity, in the viscous case, and finite-time singularity, in the inviscid case, for oceanic and atmospheric dynamics models. Moreover, we will also provide a rigorous justification of the derivation of the Primitive Equations of planetary scale oceanic dynamics from the three-dimensional Navier-Stokes equations as the vanishing limit of the small aspect ratio of the depth to horizontal width. In addition, we will also show the global well-posedeness of the coupled three-dimensional viscous Primitive Equations with a micro-physics phase change moisture model for cloud formation. Eventually, we will also present short-time well-posedness of solutions to the Hibler’s sea-ice model.
10:00 AM – 10:50 AM Professor Edriss Titi, Texas A & M University, USA
Title: Mathematical Analysis of Atmospheric and Oceanic Dynamics Models: Cloud Formation and Sea-ice Models Ⅱ
Abstract: In these talks we will present rigorous analytical results concerning global regularity, in the viscous case, and finite-time singularity, in the inviscid case, for oceanic and atmospheric dynamics models. Moreover, we will also provide a rigorous justification of the derivation of the Primitive Equations of planetary scale oceanic dynamics from the three-dimensional Navier-Stokes equations as the vanishing limit of the small aspect ratio of the depth to horizontal width. In addition, we will also show the global well-posedeness of the coupled three-dimensional viscous Primitive Equations with a micro-physics phase change moisture model for cloud formation. Eventually, we will also present short-time well-posedness of solutions to the Hibler’s sea-ice model.
11:00 AM – 11:50 AM Professor Zoran Grujic, University of Alabama at Birmingham, USA
Title: On ruling out a class of type II blow-up scenarios in the hyper-dissipative Navier-Stokes equationsⅠ
Abstract:
It has been known since the pioneering work of J.L. Lions in 1960s that 3D hyper-dissipative (HD) Navier-Stokes (NS) system does not permit formation of singularities as long as the hyper-dissipation exponent, say beta, is greater or equal to 5/4. Recall that at 5/4 the system is in the critical regime — the energy level and the scaling-invariant levels coincide — while for beta greater than 5/4 the system is in the sub-critical regime. The question of global-in-time regularity in the super-critical regime, beta strictly between 1 and 5/4, has remained a fundamental open problem in mathematical fluid dynamics.
The main goal of the two lectures is to present a mathematical framework — built around a suitably defined scale of sparseness of the super-level sets of the components of the higher-order velocity derivatives — in which a class of `turbulent' blow-up scenarios can be ruled out as soon as the hyper-dissipation exponent is greater than 1. In particular, a class of type II generalized self-similar blow-ups is ruled out which — in turn — rules out approximately self-similar blow-ups, a prime candidate for singularity formation, in all 3D HD NS systems. This is a joint work with L. Xu.
2:00 PM – 2:50 PM Professor Zoran Grujic, University of Alabama at Birmingham
Title: On ruling out a class of type II blow-up scenarios in the hyper-dissipative Navier-Stokes equations Ⅱ
Abstract:
It has been known since the pioneering work of J.L. Lions in 1960s that 3D hyper-dissipative (HD) Navier-Stokes (NS) system does not permit formation of singularities as long as the hyper-dissipation exponent, say beta, is greater or equal to 5/4. Recall that at 5/4 the system is in the critical regime — the energy level and the scaling-invariant levels coincide — while for beta greater than 5/4 the system is in the sub-critical regime. The question of global-in-time regularity in the super-critical regime, beta strictly between 1 and 5/4, has remained a fundamental open problem in mathematical fluid dynamics.
The main goal of the two lectures is to present a mathematical framework — built around a suitably defined scale of sparseness of the super-level sets of the components of the higher-order velocity derivatives — in which a class of `turbulent' blow-up scenarios can be ruled out as soon as the hyper-dissipation exponent is greater than 1. In particular, a class of type II generalized self-similar blow-ups is ruled out which — in turn — rules out approximately self-similar blow-ups, a prime candidate for singularity formation, in all 3D HD NS systems. This is a joint work with L. Xu.
( 2:50 PM – 3:30 PM Coffee Break )
3:30 PM – 4:00 PM Postdoctoral Scholar Marvin Weidner, Universitat de Barcelona, Spain
Title: Optimal regularity for nonlocal free boundary problems
Abstract: Free boundary problems have been a central topic of research in PDE theory for the last fifty years. An increasingly prominent class is that of nonlocal free boundary problems, which naturally arises in models where long range interactions need to be taken into account. In this talk, I will present a recent result on the optimal regularity of solutions to obstacle problems for general nonlocal operators. Interestingly, our approach draws on insights from a seemingly distinct problem, the nonlocal one-phase free boundary problem. This talk is based on joint works with Xavier Ros-Oton.
4:00 PM – 4:30 PM Postdoctoral Scholar David Reynolds, Universidad de Granada, Spain
Title: Lyapunov stability and exponential phase-locking of Schrödinger-Lohe oscillators
Abstract: In this talk based off of joint works with Paolo Antonelli (GSSI) we ill discuss some basics of synchronization dynamics. Then we will introduce the Schrödinger-Lohe model for quantum synchronization. The model is described by a system of Schrödinger equations, coupled through nonlinear, non-Hamiltonian interactions that drive the system towards phase synchronization. The model can be viewed as a quantum generalization of the famous Kuramoto model of phase-synchronization. Despite enjoying similar structural qualities, until recently stability and conver-ence to phase-locked state for nonidentical oscillators has been elusive. e present such stability and convergence results which brings the state f the art for the Schrödinger-Lohe model closer to that of other models within the Kuramoto family. Keywords: emergence, quantum synchronization, Schrödinger-Lohe model.
Wednesday, July 16, 2025
9:00 AM - 9:50 AM Professor Ugur G. Abdulla, Okinawa Institute of Science and Technology, Japan
Title and Abstract: TBA
10:00 AM – 10:50 AM Professor Ugur G. Abdulla, Okinawa Institute of Science and Technology, Japan
Title and Abstract: TBA
11:00 AM – 11:50 AM Professor Irina Mitrea, Temple University, USA
Title and Abstract: TBA
2:00 PM – 9:00 PM Excursion
Thursday, July 17, 2025
9:00 AM - 9:50 AM Professor Giuseppe Mingione, University of Parma, Italy
Title and Abstract: TBA
10:00 AM – 10:50 AM Professor Giuseppe Mingione, University of Parma, Italy
Title and Abstract: TBA
11:00 AM – 11:50 AM Professor Bruno Poggi, University of Pittsburgh, USA
Title: The Dirichlet problem as the boundary of the Poisson problem.
Abstract: We review certain classical quantitative estimates (known as non-tangential maximal function estimates) for the solutions to the Dirichlet boundary value problem for the Laplace equation in a smooth domain in Euclidean space, when the boundary data lies in an $L^p$ space, $p>1$. A natural question that arises is: what might an analogous estimate for the inhomogeneous Poisson problem look like? We will answer this question precisely, and in so doing, we will unravel deep and new connections between the solvability of the (homogeneous) Dirichlet problem for the Laplace equation with data in $L^p$ and the solvability of the (inhomogeneous) Poisson problem for the Laplace equation with data in certain Carleson spaces. We employ this theory to solve a 20-year-old problem in the area, to give new characterizations and a new local T1-type theorem for the solvability of the Dirichlet problem under consideration, and to furnish a bridge to the mathematical physics theory of the Filoche-Mayboroda landscape function. The new results are the product of joint work with Mihalis Mourgoglou and Xavier Tolsa.
2:00 PM - 2:50 PM Professor Zongyuan Li, City University of Hong Kong, Hong Kong
Title: Optimal Liouville theorems for conformally invariant PDEs
Abstract: The celebrated result of Caffarelli, Gidas, and Spruck (1989) classified all nonnegative solutions to a class of semilinear elliptic PDEs, establishing a cornerstone Liouville-type theorem. In this talk, we present an optimal generalization to the fully nonlinear, conformally invariant setting. Time permitting, we will also discuss some applications in conformal geometry. This is joint work with B. Z. Chu and Y. Y. Li (Rutgers).
( 2:50 PM – 3:30 PM Coffee Break )
3:30 PM – 4:20 PM Professor Armin Schikorra, University of Pittsburgh, USA
Title: On well-posedness of $s$-Schrödinger maps
Abstract: I am going to present recent progress on well-posedness of a nonlinear Schrödinger system with loss of derivatives that is a model equation for the $s$-Schrödinger map system \(\partial_t u = u \wedge (-\Delta)^s u\) - which for $s = 1/2$ is the halfwave map equation, for $s = 1$ it is the Schrödinger map equation. We consider the case \(s \in (1/2,1)\).
Joint work with Ahmed Dughayshim and Silvino Reyes-Farina.
4:30 PM – 5:00 PM Professor Minhyun Kim, Hanyang University, South Korea
Title: Recent advances in nonlocal potential theory
Abstract: Nonlocal potential theory is the study of $L$-harmonic functions with respect to nonlocal operators $L$ modeled on the fractional Laplacian. In this talk, I will present recent results on local and boundary behavior of $L$-harmonic functions. The main topics include the removability theorem, isolated singularity theorem, boundary regularity, Wiener criterion, and Green function estimates. This talk is based on joint works with Anders Björn, Jana Björn, Ki-Ahm Lee, Se-Chan Lee, and Marvin Weidner.
5:00 PM – 5:30 PM Graduate Student James Warta, University of Missouri, USA
Title: Carleson Measure Estimates Imply the Parabolic Measure is Muckenhaupt Infinity in the Case of a Graph Domain That's Lipschitz With Respect to the Parabolic Metric
Abstract: The weak solutions to the parabolic Dirichlet problem on a domain whose boundary can be described locally the graph of a function that is Lipschitz with respect to the parabolic metric obey a Carleson measure estimate, then the corresponding parabolic measure on the boundary will belong to the Muckenhaupt class infinity. This improves the existing literature which places additional assumptions on the parabolic uniform rectifiability of the boundary or, equivalently, on the half-order time derivative.
Friday, July 18, 2025
9:00 AM - 9:50 AM Professor Juan Manfredi, University of Pittsburgh, USA
Title and Abstract: TBA
10:00 AM – 10:50 AM Professor Juan Manfredi, University of Pittsburgh, USA
Title and Abstract: TBA
11:00 AM – 11:50 AM Professor Torbjorn Lundh, Chalmers Inst of Technology, Sweden
Title: To play around in a numerical sandbox to generate and illustrate potential theory conjectures
Abstract: A useful classical method in analysis to think and generate new ideas is to sketch and doodle on paper or a black/white–board as an experimental sandbox to enhance our mental processes and to generate new ideas. I would like to exemplify a way to augment this classical method by using numerical methods, while hopefully still preserve our playfulness and creativity. As a first example, I would like to talk about some old, but still unpublished work, initiated with the collaboration, on the so-called 3G-inequality, with our here present delegate, professor Hiroaki Aikawa, resulting in “The 3G inequality for a uniformly John domain” (Kodai Mathematical Journal, 28(2): 209–219, 2005) extending an earlier result of Cranston, Fabes and Zhao: “Conditional gauge and potential theory for the Schrödinger operator” (Trans. Amer. Math. Soc. 307,1988). The second example of this numerical “sandbox technique” will be about an ill-posed free-boundary problem inspired from a biological process that could be seen as an inverse Heley-Shaw flow process. To conclude, the presentation will be focused how one could use high-levelnumerical tool boxes, such as Comsol Multiphysics, to play around to generate conjectures, to be later proven by classical analytic methods.
2:00 PM – 2:50 PM Professor Eliot Fried, Okinawa Institute of Science and Technology, Japan
Title and Abstract: TBA
( 2:50 PM – 3:30 PM Coffee Break )
3:30 PM – 4:00 PM Professor James McCoy, University of Newcastle, Australia
Title: Isoperimetric inequality for multiply winding curves
Abstract: We’ll discuss some old and new results for the isoperimetric inequality in the plane, for closed curves of positive integer winding number. In particular, I will outline some work in progress with Yong Wei and Glen Wheeler on a new nonlinear fourth order parabolic curvature flow that can be used to prove an isoperimetric inequality for multiply winding curves of sufficiently small oscillation of curvature. Time permitting, I will outline how this flow can also be used to prove a similar result for embedded curves in the sphere and multiply-winding curves in hyperbolic space.
4:00 PM – 4:30 PM Ph.D. Abderrahmane Lakhdari, Tunis El Manar University, Tunisia
Title: Existence and Regularity Result for a Heisenberg 𝜑-Laplacian Problem Without Space Reflexivity
Abstract: This presentation aims to investigate the existence of at least one weak solution for a nonlinear problem governed by the Heisenberg 𝜑-Laplacian under Dirichlet boundary conditions. We examine this problem in both reflexive and non-reflexive cases of the associated Heisenberg Orlicz-Sobolev space. Additionally, we seek to establish certain regularity and uniqueness results under suitable assumptions. To achieve this, we employ variational and topological methods. Furthermore, we provide an illustrative example to support our findings.
4:30 PM – 5:00 PM Graduate Student Artur Andrade, Temple University, USA
Title: Fredholm Theory for the Dirichlet Problem for \(\Delta^3\) in Infinitesimally Flat AR Domains.
Abstract: Elliptic boundary value problems naturally arise in modeling various physical phenomena, such as electrostatics, elasticity, steady-state incompressible fluid flow, and electromagnetism. The layer potential method serves as a powerful tool for addressing these problems by reducing them to a boundary integral equation involving a singular integral operator associated with the domain and a coefficient tensor corresponding to the underlying PDE. When this singular integral operator is compact, the boundary integral equation can be treated using Fredholm Theory. While significant progress has been made in studying second-order elliptic systems through this approach, higher-order elliptic systems remain less understood.
In this talk, I will present a distinguished coefficient tensor for the polyharmonic operator $\Delta^3$ in all dimensions, and illustrate how the associated singular integral operator is compact on $L^p$ Lebesgue-type spaces, for all integrability exponents $p\in(1,\infty)$, thus opening the door for the employment of Fredholm Theory for the solvability of the Dirichlet Problem for $\Delta^3$ in infinitesimally flat AR domains.
This is an ongoing work with Dorina Mitrea (Baylor University), Irina Mitrea (Temple University), and Marius Mitrea (Baylor University)