Possible Topics and References for Student talks
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Łojasiewicz gradient inequality
Łojasiewicz: Sur les trajectoires du gradient d’une fonction analytique}, Seminari di Geometria 1982-1983, Universit`a di Bologna, (1984), 115—117.
Colding and Minicozzi: Łojasiewicz inequalities and applications, Surveys in Differential Geometry, 19 ,1, (2014).
Draft paper w Rick Moeckel.
Yves Colin de Verdiere
Periodic geodesics for contact sub-Riemannian 3D manifolds
Spectral asymptotics for sub-Riemannian Laplacians
Richard Schwartz.
General mathematical culture. Control Theory and geometry.
Conway's Nightmare: Brahmagupta and Butterflies
page five, first par. of
The stuff that Greg Laughlin got me excited about
Balancing Stick
area. control theory. bio-mechanics. differential delay eqns.
control theory history and background
possible connections
between balancing sticks and market processes:
and the extra-solar system GSL867
Greg suggests a (do-able!) experiment with a cell phone and a yardstick.
Gabriel Paternain
I have in mind this theorem he proved with his brother Miguel.
Theorem: if a natural mechanical system has , at some fixed energy, a Hill region which is compact with boundary then at that energy the system is NOT Anosov.
Paternain, Gabriel P.; Paternain, Miguel: On Anosov energy levels of
convex Hamiltonian systems. Math. Z. 217 (1994), no. 3, 367–376.
There is another, perhaps slicker, proof using the asymptotic Maslov index in:
Contreras, Gonzalo; Gambaudo, Jean-Marc; Iturriaga, Renato;
Paternain, Gabriel P. The asymptotic Maslov index and its
applications. Ergodic Theory Dynam. Systems 23 (2003), no. 5,
1415–1443.
But anything by Paternain is worth reading and covering
Papers by Vivina Barutello or Susana Terracini.
For example
Chaotic dynamics in refraction galactic billiards
Complete integrability
The Pentagram map: a discrete integrable system
Kerr space time geodesic flow is completely integrability
Miscellaney
River and basin dynamics. Papers by Bjorn Birnir. available on arXiv