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Symplectic and Poisson geometry seminar: On the symplectic equivalences for parabolic orbits of integrable Hamiltonian systems

Event Type
Seminar/Symposium
Sponsor
n/a
Virtual
wifi event
Date
Mar 28, 2022   12:00 pm  
Speaker
Nikolay Martynchuk
Contact
Joey Palmer
E-Mail
jpalmer5@illinois.edu
Views
18
Parabolic orbits are the simplest examples of degenerate singularities of integrable two degree of freedom Hamiltonian systems. Yet until recently, their symplectic (and even C^\infty smooth) classification was not known. We fill in this gap and show that the action variables corresponding to such an orbit form a complete set of symplectic invariants (up to the fiberwise symplectic equivalence). This generalises an earlier result by A.V. Bolsinov, L. Guglielmi and E.A. Kudryavtseva proving this in the analytic category. The smooth case that we present is more complicated and has useful consequences for more global symplectic classification problems. 
 
 
We shall also discuss a new classification result for parabolic orbits in the analytic category; specifically, we shall give a simple normal form for such orbits up to the right symplectic equivalence. This type of equivalence is different, but closely related to the usual fiberwise (or right-left) symplectic equivalence and interestingly enough, the normal form that we obtain is not given in terms (of the asymptotics) of the action variables.
 
 
This talk is based on a recent work with Prof. E.A. Kudryavtseva.
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