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\begin{document}

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\section*{symplectic geometry}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry}

[[!include symplectic geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\textbf{Symplectic geometry} is a branch of [[differential geometry]] studying [[symplectic manifold]]s and some generalizations; it originated as a formalization of the mathematical apparatus of [[classical mechanics]] and geometric optics (and the related [[semiclassical approximation|WKB-method]] in [[quantum mechanics]] and, more generally, the method of stationary phase in [[harmonic analysis]]). A wider branch including symplectic geometry is [[Poisson geometry]] and a sister branch in odd dimensions is [[contact geometry]]. A special and central role in the subject belongs to certain real-like half-dimensional submanifolds, called [[Lagrangian submanifold|lagrangian (or Lagrangean) submanifolds]], which are in some sense classical points. Symplectic geometry radically changed after the 1985 article of [[Mikhail Gromov]] on pseudoholomorphic curves and the subsequent work of [[Andreas Floer]] giving birth to symplectic topology or ``hard methods'' of symplectic geometry.

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item In its application to [[physics]], symplectic geometry is the fundamental mathematical language for [[Hamiltonian mechanics]], [[geometric quantization]], geometrical optics.


\item A tremendous amount of insight into higher [[Lie theory]] ([[Lie groupoid]]s, [[Lie ∞-groupoid]]s, [[Lie ∞-algebroid]]s) has derived from [[Alan Weinstein]]`s long-term project of understanding the role of symplectic geometry in [[geometric quantization]]. See there for more details.



\end{itemize}
[[Zoran ?koda]]: it is true that large influence of Weinstein's program can not be overestimated, but it is not the origin of these considerations; it rather builds up on earlier fundamental works of Kirillov, Kostant, Souriau who invented geometric quantization, all of them originally in symplectic context; and the florishing of the subject from mid 1960s till mid 1980-s is related to their work; and other related tracks of Guillemin, Sternberg, Kashiwara, Karasev, Arnold and so on; and vast developments in harmonic analysis and representation theory (Kostant, Auslander, Vogan, Wallach, Stein\ldots{}), [[microlocal analysis]] ([[Kashiwara]], Saito, Hormander, Maslov, Karasev, Duistermaat\ldots{}), [[integrable system]]s/[[quantum group]]s (this is more into more general Poisson geometry: Lie-Poisson groups, classical r-matrices, bihamiltonian systems\ldots{}), and related approaches to quantization (Berezin method, [[coherent state]]s\ldots{}).

\begin{itemize}%
\item [[T-duality]] and (related) [[mirror symmetry]] interchange the symplectic data and complex algebraic geometry data. Some cases of both the symplectic and complex algebraic picture can be unified and studied in a broader concept of [[generalized complex geometry]].

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item There is a [[vertical categorification]] of symplectic geometry to [[higher symplectic geometry]]. This involves [[multisymplectic geometry]] and the geometry of [[symplectic Lie n-algebroid]]s. And their combination.

See [[higher symplectic geometry]]



\end{itemize}
[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

Introductions include

\begin{itemize}%
\item Rolf Berndt, \emph{An introduction to symplectic geometry} (\href{http://www.ams.org/bookstore/pspdf/gsm-26-prev.pdf}{pdf})

\end{itemize}
Discussion from the point of view of [[homological algebra]] of [[abelian sheaves]] is in

\begin{itemize}%
\item C. Viterbo, \emph{An introduction to symplectic topology through sheaf theory} (2010) (\href{http://www.math.polytechnique.fr/cmat/viterbo/Eilenberg/Eilenberg.pdf}{pdf})

\end{itemize}
The notion of symplectic geometry may be understood as the mathematical structure that underlies the physics of [[Hamiltonian mechanics]]. A classical monograph that emphasizes this point of view is

\begin{itemize}%
\item [[Vladimir Arnold]], \emph{[[Mathematical methods of classical mechanics]]}

\end{itemize}
For more on this see [[Hamiltonian mechanics]].

Symplectic geometry is also involved in [[geometric optics]], [[geometric quantization]] and the study of oscillatory integrals and [[microlocal analysis]]. Books concentrating on these topics include

\begin{itemize}%
\item [[Victor Guillemin]], [[Shlomo Sternberg]], \emph{Geometric asymptotics}, Amer. Math. Soc. 1977 \href{http://www.ams.org/online_bks/surv14}{free online}


\item Sean Bates, [[Alan Weinstein]], \emph{[[Lectures on the geometry of quantization]]}, \href{http://www.math.berkeley.edu/~alanw/GofQ.pdf}{pdf}


\item J.J. Duistermaat, \emph{Fourier integral operators}, Progress in Mathematics, Birkh\"a{}user 1995 (and many other references at [[microlocal analysis]]).


\item [[Alan Weinstein]], \emph{Symplectic geometry}, (survey) Bull. Amer. Math. Soc. 5 (1981), 1-13, \href{http://dx.doi.org/10.1090/S0273-0979-1981-14911-9}{doi}


\item N. R. Wallach, \emph{Symplectic geometry and Fourier analysis}, Math. Sci. Press, Brookline, Mass., 1977.


\item wikipedia \href{http://en.wikipedia.org/wiki/Symplectic_geometry}{symplectic geometry} Application: ``Jacobi's elimination of nodes'' is moved to [[equivariant localization and elimination of nodes]].



\end{itemize}


\end{document}