JISD 2025 – Where Dynamical Systems Meet PDEs

JISD 2025, held at the Centre de Recerca Matemàtica (CRM) from June 30 to July 4, 2025, featured four advanced minicourses delivered by Dmitry Dolgopyat (on averaging and Fermi acceleration in dynamical systems), Serena Dipierro (on the theory of nonlocal minimal surfaces and their connections to phase transitions), Luciano Mari (on Bochner-type techniques in Riemannian geometry and rigidity phenomena), and Sylvain Crovisier (on dynamical decompositions of surface diffeomorphisms). The program also included a lively poster session covering a wide range of topics highlighting the diversity and depth of current research in Dynamical Systems and PDEs.

The 2025 edition of the School on Interactions between Dynamical Systems and Partial Differential Equations (JISD) once again brought together a vibrant community of mathematicians at the Centre de Recerca Matemàtica (CRM). Since its inception in 2002, JISD has grown into a premier international summer school, and this year’s edition reaffirmed its role as a catalyst for collaboration and innovation in the fields of Dynamical Systems and Partial Differential Equations (PDEs).

With a strong cohort of early-career researchers and a diverse international presence, JISD 2025 offered a dynamic environment for learning, networking, and discovery. The school featured four advanced minicourses, a poster session showcasing cutting-edge research, and countless informal discussions that sparked new ideas and potential collaborations.

The heart of JISD 2025 lay in its four intensive minicourses, each led by a distinguished expert:

• Dmitry Dolgopyat(University of Maryland) explored Averaging and Fermi Acceleration, revisiting classical tools in dynamics and presenting new insights into energy evolution for a particle with the environment.

• Serena Dipierro(University of Western Australia) introduced the theory of Nonlocal Minimal Surfaces, focusing on interior regularity, boundary behavior, and connections to phase transitions.

• Luciano Mari(Università degli Studi di Milano Statale) presented Bochner-type techniques in Riemannian Geometry, highlighting how the stability/finite index of a Schrodinger operator naturally appearing in the problem forces the rigidity phenomenon.

• Sylvain Crovisier(Université Paris-Saclay) delivered a captivating course on Dynamical Decompositions of Surface Diffeomorphisms, examining infinite decompositions and their properties.

The poster session was a highlight of the week, offering a platform for young researchers to present their work and engage with peers and senior mathematicians. The diversity of topics reflected the richness of the field:

• Stefano Baranzini(University of Turin) explored Periodic Points for Singular Systems, analyzing the N-centre problem, one-dimensional atom model, and integrable deformations of the Kepler problem.
• Lorenzo Baroni(Universit`a degli Studi Roma Tre) discussed Linear Flows on the Infinite Torus, showing that classical characterizations hold to infinite dimensions when the frequency set generates a free abelian group. He also described orbits whose closures are locally homeomorphic to a product of an interval and a Cantor set.
• Huub de Jong(University of British Columbia) examined Least Period Sets of Finitely Presented Systems, linking periodicity in dynamical systems to zeta functions and symbolic dynamics.
• Sallah Eddine Boutiah(Laval University & Setif-1 University) presented work on Keller-Segel Type Particle Systems, focusing on stochastic differential equations with critical interactions.
• Davide Giovagnoli(University of Bologna) addressed Free Boundary Regularity in the Inhomogeneous Stefan Problem, extending classical results to more general settings with heat sources.
• Marc Homs-Dones(University of Warwick) explored the Minimal Bifurcation Diagram of Torus Flows, identifying the simplest bifurcation n diagram of such family would contain a certain list of bifurcation points and curves.
• Alberto Maione(CRM) introduced a novel compactness result for Nonlocal Linear Operators in Fractional Divergence Form. the symmetric case, he established an equivalence between nonlocal H-convergence and Γ-convergence, leading to a fully variational proof of H-compactness.
• Edhin Mamani(Federal University of Minas Gerais) presented a deep investigation into Lagrangian Foliations Invariant by the Geodesic Flow of Manifolds Without Conjugate Points. Their work generalizes classical results from negatively curved manifolds to more general settings, showing that under certain conditions, horospherical foliations are the unique Lagrangian invariant continuous foliations of the unit tangent bundle.
• Juhee Sohn(Korea National University of Transportation) studied an elliptic system arising from the Maxwell-Chern-Simons model, involving two distinct parameters.