Geometrical Themes Inspired by the N-body Problem

Presenting a selection of recent developments in geometrical problems inspired by the N-body problem, these lecture notes offer a variety of approaches to study them, ranging from variational to dynamical, while developing new insights, making geometrical and topological detours, and providing historical references.


A. Guillot's notes aim to describe differential equations in the complex domain, motivated by the evolution of N particles moving on the plane subject to the influence of a magnetic field. Guillot studies such differential equations using different geometric structures on complex curves (in the sense of W. Thurston) in order to find isochronicity conditions.



R. Montgomery's notes deal with a version of the planar Newtonian three-body equation. Namely, he investigates the problem of whether every free homotopy class is realized by a periodic geodesic. The solution involves geometry, dynamical systems, and the McGehee blow-up. A novelty of the approach is the use of energy-balance in order to motivate the McGehee transformation.



A. Pedroza's notes provide a brief introduction to Lagrangian Floer homology and its relation to the solution of the Arnol'd conjecture on the minimal number of non-degenerate fixed points of a Hamiltonian diffeomorphism.

Prezentand o selectie de evolutii recente in problemele geometrice inspirate de problema corpului N, aceste note de curs ofera o varietate de abordari pentru a le studia, variind de la variationale la dinamice, dezvoltand in acelasi timp noi perspective, facand ocoliri geometrice si topologice si oferind referinte istorice .A. Notele lui Guillot urmaresc sa descrie ecuatii diferentiale in domeniul complex, motivate de evolutia particulelor de N care se deplaseaza pe planul supus influentei unui camp magnetic.