22nd School on Interactions between Dynamical Systems and Partial Differential Equations (JISD 2026)

The liquid drop model

Rupert Frank

Ludwig Maximilian University of Munich

Abstract: The liquid drop model was originally introduced in the nuclear physics literature in 1930 and has recently been studied extensively using techniques from the calculus of variations, geometric analysis, PDE and mathematical physics. The course will provide a gentle introduction to this topic. Among the topics that will be covered are a certain isoperimetric-type question, which is still open in general. We will also discuss an application of the liquid drop model in astrophysics, including a rigorous proof of one of the nuclear pasta phases.

Birkhoff Normal forms, KAM theory and renormalization: an application to the existence of Exotic rotation domains and Herman rings for quadratic complex Hénon maps

Raphaël Krikorian

École Polytechnique and CY Cergy Paris Université

Abstract:  A quadratic Hénon map is an automorphism of of the form . It has a constant Jacobian equal to and has two fixed points. If is on the unit circle (one says is conservative) these fixed points can be both elliptic or both hyperbolic. In the elliptic case, under an additional Diophantine condition, a simple application of Siegel Theorem shows that $h$ admits quasi-periodic orbits with two frequencies in the neighborhood of its fixed points. Surprisingly, in some hyperbolic cases, Shigehiro Ushiki observed numerically what seems to be quasi-periodic orbits belonging to some “Exotic rotation domains” though no Siegel disk is associated to the fixed points.  The aim of these series of lectures  is to develop a theoretical frame work  that explains and proves the existence of these “Exotic rotation domains’’. This framework  also applies  to the dissipative case () and allows to prove the existence of attracting Herman rings. 

The main tools that we shall use are:

• Birkhoff normal forms techniques.
• Approximation by vector fields.
• KAM theory.
• Renormalization.

Geometric flows and applications

Alessandra Pluda

Università di Pisa

Abstract: This course provides an introduction to intrinsic geometric flows and their applications in various mathematical contexts. It begins with the derivation of Mean Curvature Flow, from the area functional to the associated partial differential equation that govern its dynamics. Next, we will conduct a comprehensive analysis of Curve-Shortening Flow, examining its properties and implications in geometric evolution. The course will also cover Network Flow and its relationship to grain boundaries. Finally, we will present the Inverse Mean Curvature Flow, highlighting its original application in proving the Riemannian Penrose Inequality.

CR SINGUALRITIES AND DYNAMICAL SYSTEMS

Laurent Stolovitch

Université Côte d’Azur

Abstract: : In this serie of lectures, we’ll survey some recent results done since the seminal work of Moser and Webster about smooth real analytic surfaces in $C^2$ which are totally real everywhere but at a point where the tangent space is a complex line. Such a point is called a singularity of the Cauchy-Riemann structure. We are interested in the holomorphic classification and the geometry of these surfaces near this singularity. It happens that there is a deep connection with holomorphic classification of some holomorphic dynamical systems near a fixed point so that new results for the later provide new result for the former. In particular, we will present new recent results on KAM-like theory for holomorphic diffeomorphisms at a fixed point, parabolic dynamics in dimension 2 and integrable holomorphic dynamics in higher dimension.