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{\Large SYMPLECTIC INVARIANTS AND HAMILTONIAN DYNAMICS}\\
U.T.K. Mathematics Department, Spring 1996
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Beginning in 1985, the area of symplectic geometry and Hamiltonian
dynamical systems has experienced intense activity, centered primarily around
the following topics: pseudo-holomorphic curves on symplectic manifolds
(Gromov), symplectic capacities and non-existence of symplectic
embeddings (Ekeland-Hofer, Gromov), the Arnold conjecture on fixed points
of symplectic diffeomorphisms (A. Floer), existence of closed orbits on compact
level sets for the energy (Weinstein's conjecture). This activity has
generated developments in low-dimensional topology, such as Floer homology,
the study of four-dimensional symplectic manifolds and the restrictions
on Seiberg-Witten invariants of symplectic 4-manifolds discovered recently
by C.Taubes.
{\bf Description.} This proposed research-level seminar aims to present
an introduction to some of these developments, based primarily on the text
with the same title by H.Hofer and E.Zehnder (Birkh\"auser 1994). The contents
of the book are as follows:
\begin{itemize}
\item{(i)}Introduction to symplectic geometry;
\item{(ii)}Symplectic capacities;
\item{(iii)}Existence of closed orbits;
\item{(iv)}The Arnold conjecture, Floer homology and symplectic homology.
\end{itemize}
The precise choice of topics covered will (naturally) depend on the interests,
background and energy of the participants. The undersigned expects to
present most of the topics himself, but will eagerly seek volunteers.
In particular, graduate students taking the course for credit will be
asked to present at least one section of the book. {\it Note: at least
three students have to be enrolled for this seminar to run.}
{\bf Prerequisites.}Calculus and differential equations on manifolds,
including differential forms.
Tentative time: 3:30-5:00,MW. Interested graduate students: please
send e-mail to freire@math.utk.edu,
or drop a not in my mailbox. This seminar will run as M539,
Seminar in Differential Equations; section number TBA in early January.
Alex Freire
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