The quest for periodic orbits of dynamical systems - for instance closed geodesics or periodic orbits of particles in a magnetic field - dates back, at least, to the foundational work of Poincaré around 1900, followed by work, among many others, by Lusternik-Schnirelmann around 1930, Kolmogorov-Arnol'd-Moser in the 1960s, and Rabinowitz, Eliashberg and Conley-Zehnder in the early 1980s. Floer's approach to infinite dimensional Morse theory in the mid 1980s, inspired by Gromov's 1985 landmark paper, marked a breakthrough in the efforts to prove the Arnold conjecture: The number of 1-periodic orbits of a Hamiltonian vector field on a closed symplectic manifold N is bounded below by the sum of the Betti numbers of N. At about the same time Hofer entered the stage and together with Wysocki, Zehnder, Eliashberg, among others, contactized the symplectic world eventually leading to the theory of everything: Symplectic Field Theory (SFT).
The lecture course aims to give an overview of methods and
results. Basic knowledge of manifolds is required.
Variedades simpléticas, fluxo Hamiltoniano,
Funcional ação e sua
teoria de variação, curvas J-holomorfas, homologia
de Floer, homologia simplética, conjectura(s) de
Arnold.
Homologia de Rabinowitz-Floer
Variedades de contato, fluxo de Reeb, hipersuperfícies de tipo contato, homologia de contato.
H. Hofer, E. Zehnder, Symplectic invariants
and Hamiltonian dynamics, Birkhäuser, 2011.
D. McDuff, D. Salamon, Introduction to
symplectic topology, Clarendon Press, 1998.
H. Geiges, An introduction to contact
topology, Cambridge University Press, 2008.
U. Hryniewicz, P. Salomão, Uma introdução à geometria de contato e aplicações à dinâmica hamiltoniana, IMPA, 2009.
Métodos Topológicos da Mecânica
Hamiltoniana
Joa Weber
IMECC UNICAMP