Adolfo Guillot: Ecuaciones diferenciales complejas y estructuras geométricas en curvas
Richard Montgomery: The N-body problem and Riemannian submersions
Lecture 1: The geometry and topology of the reduced N-body problem
Set-up of N-body problem. Some geometric mechanics and reduction. The reduced configuration space in the planar case : it is the complement of a hyperplane arrangement is complex projective N-2 space.. Its free homotopy class: the (projective, colored) braid group. Pictures, simplifications, and syzygy sequences for N=3. Our (*) big theorem: all free homotopy classes are realized by collision-free solutions when N=3.
(*) joint with Rick Moeckel, U. of Minnesota.
References
[1] primary paper, with Rick Moeckel: Realizing all free homotopy class for the Newtonian three-body problem, with Rick Moeckel: http://arxiv.org/pdf/1412.2263.pdf
[2] with Gil Bor: Poincare y el problema de n-cuerpos: people.ucsc.edu/~rmont/papers/Ncuerpos.pdf
[3] R. Montgomery: The three-body problem and the shape sphere: http://arxiv.org/abs/1402.0841
[4] Scholarpedia article on the 3-body problem:
Lecture 2: Blowing up, making windows.
McGehee blow-up. Near collision manifold. Windows. Importance of `spiralling' near Euler. Sketch proof.
References
[1] above and the other Moeckel references therein.
[5] Chenciner: A l'infini en temps fini.
Lecture 3: Variational versus Window-Blow-up methods.
Introduction to the variational method. Comparison with previous `window-blow-up' methods.
Successes and failures (of both methods?). Work to be done (lots).
References
Montgomery: Hyperbolic Pants fit a three-body problem, in Ergodic Theory and Dynamical Systems, 2007,
available from http://people.ucsc.edu/~rmont/papers/list.html
Annals papers with Chenciner: A remarkable solution to the three-body problem in the case of equal masses http://arxiv.org/abs/math/0011268
Scholarpedia article on N-body choreographies.
Andrés Pedroza: Homologia Lagrangiana de Floer
Empezaremos este curso repasando la Teoría de Morse, esto es como una función suave sobre una variedad rescata la homología de la variedad. En esta teoría los puntos críticos de la función y los espacios de trayectorias que unen puntos críticos son fundamentales.
En el contexto de geometría simpléctica se tienen varias teorías de homologia, entre ellas la homología lagrangiana de Floer (HLF). La HLF, sigue la misma idea que la teoría de Morse con la idea de detectar puntos de intersección de dos subvariedades lagrangianas.