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\DeclareMathOperator{\Arg}{\mathbf{M}}\begin{filecontents}{references.bib}
@article{ballard2012hochschild,
  title={Hochschild dimensions of tilting objects},
  author={Ballard, Matthew and Favero, David},
  journal={International Mathematics Research Notices},
  volume={2012},
  number={11},
  pages={2607--2645},
  year={2012},
  publisher={OUP}
}

@article{craw2007explicit,
  title={Explicit methods for derived categories of sheaves},
  author={Craw, Alastair},
  publisher={Citeseer}
}
@article{weinstein1971symplectic,
  title={Symplectic manifolds and their {L}agrangian submanifolds},
  author={Weinstein, Alan},
  journal={Advances in Mathematics},
  volume={6},
  number={3},
  pages={329--346},
  year={1971},
  publisher={Academic Press}
}

@article{hanlon2022aspects,
  title={Aspects of functoriality in homological mirror symmetry for toric varieties},
  author={Hanlon, A and Hicks, J},
  journal={Advances in Mathematics},
  volume={401},
  pages={108317},
  year={2022},
  publisher={Elsevier}
}

@article{biran2013lagrangian,
  title={{L}agrangian cobordism. {I}},
  author={Biran, Paul and Cornea, Octav},
  journal={Journal of the American Mathematical Society},
  volume={26},
  number={2},
  pages={295--340},
  year={2013}
}



@article{tanaka2016fukaya,
  title={The Fukaya category pairs with Lagrangian cobordisms},
  author={Tanaka, Hiro Lee},
  journal={arXiv preprint arXiv:1607.04976},
  year={2016}
}


@book{seidel2008fukaya,
  title={Fukaya categories and Picard-Lefschetz theory},
  author={Seidel, Paul},
  volume={10},
  year={2008},
  publisher={European Mathematical Society}
}


@article{seidel2003long,
  title={A long exact sequence for symplectic {F}loer cohomology},
  author={Seidel, Paul},
  journal={Topology},
  volume={42},
  pages={1003--1063},
  year={2003}
}

@article{da2001lectures,
  title={Lectures on symplectic geometry},
  author={da Silva, Ana Cannas},
  journal={Lecture Notes in Mathematics},
  volume={1764},
  year={2001},
  publisher={Springer}
}

@article{polterovich1991surgery,
  title={The surgery of {L}agrange submanifolds},
  author={Polterovich, Leonid},
  journal={Geometric \& Functional Analysis GAFA},
  volume={1},
  number={2},
  pages={198--210},
  year={1991},
  publisher={Springer}
}

@misc{perutz2008handleslide,
  doi = {10.48550/ARXIV.0801.0564},
  
  url = {https://arxiv.org/abs/0801.0564},
  
  author = {Perutz, Timothy},
  
  keywords = {Symplectic Geometry (math.SG), Geometric Topology (math.GT), FOS: Mathematics, FOS: Mathematics, 53D12; 53D40; 57M27; 32U40},
  
  title = {Hamiltonian handleslides for {H}eegaard {F}loer homology},
  
  publisher = {arXiv},
  
  year = {2008},
  
  copyright = {Assumed arXiv.org perpetual, non-exclusive license to distribute this article for submissions made before January 2004}
}


@incollection{audin1994symplectic,
  title={Symplectic rigidity: {L}agrangian submanifolds},
  author={Audin, Mich{\`e}le and Lalonde, Fran{\c{c}}ois and Polterovich, Leonid},
  booktitle={Holomorphic curves in symplectic geometry},
  pages={271--321},
  year={1994},
  publisher={Springer}
}
@phdthesis{oancea2003suite,
  title={La suite spectrale de {L}eray-{S}erre en homologie de {F}loer des vari{\'e}t{\'e}s symplectiques compactes {\`a} bord de type contact},
  author={Oancea, Alexandru},
  year={2003},
  school={Universit{\'e} Paris Sud-Paris XI}
}
@article{abouzaid2010geometric,
  title={A geometric criterion for generating the {F}ukaya category},
  author={Abouzaid, Mohammed},
  journal={Publications Math{\'e}matiques de l'IH{\'E}S},
  volume={112},
  pages={191--240},
  year={2010}
}
@article{viterbo1999functors,
  title={Functors and computations in {F}loer homology with applications, I},
  author={Viterbo, Claude},
  journal={Geometric \& Functional Analysis GAFA},
  volume={9},
  number={5},
  pages={985--1033},
  year={1999},
  publisher={Springer}
}
@misc{stacks-project,
  author       = {The {Stacks project authors}},
  title        = {The Stacks project},
  howpublished = {\url{https://stacks.math.columbia.edu}},
  year         = {2022},
}

@article{wendlbeginner,
  title={A beginner’s overview of symplectic homology},
  author={Wendl, Chris},
  journal={Preprint. www. mathematik. hu-berlin. de/wendl/pub/SH. pdf}
}

@article{seidel2006biased,
  title={A biased view of symplectic cohomology},
  author={Seidel, Paul},
  journal={Current developments in mathematics},
  volume={2006},
  number={1},
  pages={211--254},
  year={2006},
  publisher={International Press of Boston}
}
@article{arnol1980lagrange,
  title={{L}agrange and {L}egendre cobordisms. I},
  author={Arnol'd, Vladimir Igorevich},
  journal={Funktsional'nyi Analiz i ego Prilozheniya},
  volume={14},
  number={3},
  pages={1--13},
  year={1980},
  publisher={Russian Academy of Sciences}
}

@article{fukaya2007lagrangian,
  title={{L}agrangian intersection {F}loer theory-anomaly and obstruction, chapter 10},
  author={Fukaya, K and Oh, YG and Ohta, H and Ono, K},
  journal={Preprint, can be found at http://www. math. kyoto-u. ac. jp/\~{} fukaya/Chapter10071117. pdf},
  year={2007}
}

@article{biran2014lagrangian,
  title={Lagrangian cobordism and Fukaya categories},
  author={Biran, Paul and Cornea, Octav},
  journal={Geometric and functional analysis},
  volume={24},
  number={6},
  pages={1731--1830},
  year={2014},
  publisher={Springer}
}
@article{bourgeois2009symplectic,
  title={Symplectic homology, autonomous {H}amiltonians, and {M}orse-{B}ott moduli spaces},
  author={Bourgeois, Fr{\'e}d{\'e}ric and Oancea, Alexandru},
  journal={Duke mathematical journal},
  volume={146},
  number={1},
  pages={71--174},
  year={2009},
  publisher={Duke University Press}
}


@incollection{auroux2014beginner,
  title={A beginner’s introduction to {F}ukaya categories},
  author={Auroux, Denis},
  booktitle={Contact and symplectic topology},
  pages={85--136},
  year={2014},
  publisher={Springer}
}

@article{singer1933three,
  title={Three-dimensional manifolds and their {H}eegaard diagrams},
  author={Singer, James},
  journal={Transactions of the American Mathematical Society},
  volume={35},
  number={1},
  pages={88--111},
  year={1933},
  publisher={JSTOR}
}

@article{ozsvath2004holomorphic,
  title={Holomorphic disks and three-manifold invariants: properties and applications},
  author={Ozsv{\'a}th, Peter and Szab{\'o}, Zolt{\'a}n},
  journal={Annals of Mathematics},
  pages={1159--1245},
  year={2004},
  publisher={JSTOR}
}

@article{ozsvath2004introduction,
  title={An introduction to {H}eegaard {F}loer homology},
  author={Ozsv{\'a}th, Peter and Szab{\'o}, Zolt{\'a}n},
  journal={{F}loer homology, gauge theory, and low-dimensional topology},
  volume={5},
  pages={3--27},
  year={2004}
}

@article{fet1952variational,
  title={Variational problems on closed manifolds},
  author={Fet, Abram Il'ich},
  journal={Matematicheskii Sbornik},
  volume={72},
  number={2},
  pages={271--316},
  year={1952},
  publisher={Russian Academy of Sciences, Steklov Mathematical Institute of Russian~…}
}\end{filecontents}
\addbibresource{references.bib}\begin{document}

\title{examples of symplectic manifolds }
 \maketitle
 \thispagestyle{firstpage}\begin{example}
\label{exm:linearSymplecticSpace}

    The simplest example comes from $\RR^{2n}$, which we give the coordinates $(q_i, p_i)$. 
    In these local coordinates, we can define a symplectic form by  
        \[\omega_{std}=\sum_{i=1}^n d p_i\wedge d q_i.\] 
    Note that when $n=1$, this gives the standard area form on $\RR^2$.
    In these coordinates, it is easy to check that $\frac{\omega_{std}^n}{n!}=\text{vol}_{\RR^{2n}}$, the standard volume form.

\end{example}
\begin{example}
\label{def:symplecticSurface}

    Let $(X, g)$ be an oriented surface.
    Then $g$ prescribes a volume form $\omega:=\text{vol}_g\in \Omega^2(X;\RR)$, which is an example of a non-degenerate 2-form. 
    Because $\Omega^3(X;\RR)=0$, it trivially follows that $\omega$ is closed. 
    This example raises the possibility of the same space having many different symplectic forms, as an oriented surface can be equipped with several different metrics.

\end{example}
\begin{example}
\label{exm:ccstarcylinder}
    An example that will be especially relevant later is $(\CC^*)^n$.
    We will equip this with a different symplectic form than the one inherited as a subset of $\CC^n=\RR^{2n}$.
    Since $(\CC^*)^n$ is a group, it is natural to ask for a symplectic form on $(\CC^*)^n$ which is invariant under the group action. 
    The symplectic form 
    \[
        \omega=\frac{1}{2\pi} d(\log |z|)\wedge d\theta
    \]
    gives an example of such a symplectic form.
When $n=1$, then this is the area form on $(\CC^*)$ which embeds into three dimensional space as an infinitely long cylinder, as drawn in \cref{fig:ccStarCylinder}.

\begin{figure}
 \label{fig:ccStarCylinder}
 \centering
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\end{tikzpicture}\caption{The symplectic structure that we choose for \(\CC^*\) makes it an infinitely long cylinder.}
 \end{figure}
\end{example}
The most important example of symplectic manifold comes from physics, which is the historical origin of symplectic geometry. 
\begin{example}
\label{def:symplecticCotangentBundle}

    Let $Q$ be a smooth $n$-dimensional manifold.
    We now describe a canonical symplectic form on the cotangent bundle, $T^*Q$.
    At every point $q\in Q$, there exists chart $q\in U\subset Q$ which we can parameterize with coordinates $(q_1, \ldots, q_n)$.
    The cotangent bundle $T^*U$ inherits coordinates $(q_1, p_1, q_2, p_2, \ldots, q_n, p_n)$, where the $p_i$ linearly parameterize the fibers of the cotangent bundle in the direction of the basis element $dq_i$.
    \footnote{ The letter $p$ is chosen for historical reasons. The form $dq_i$ is dual to the vector $\partial q_i$, which represents the velocity of a particle.     The dual to velocity is momentum, which was historically represented by the variable $\rho$.    Therefore, the momentum coordinates have been denoted by $p_i$.  } 
    In these coordinates, the canonical symplectic form on this chart is:
    \[\omega=\sum_{i=1}^n dq_i \wedge dp_i=-d(p dq).\]

\end{example}
In the physical literature, $T^*Q$ is called the ``phase-space'' which encodes both the position and momentum of a particle.
The canonical symplectic form describes the natural pairing of momentum and velocity.
There is also a coordinate-free description of $\omega$.
 To do so, we first define a canonical 1-form  $\eta\in \Omega^1(T^*Q)$.
Let $(q,p)\in T^*Q$ be a point, where the coordinates take values $q\in Q$ and $p\in T^*_qQ$.
The map $\pi: T^*Q\to Q$ induces a map $\pi_*: T(T^*Q)\to TQ$. 
The canonical 1-form is defined by its value on tangent vectors  $v\in T_{(q,p)}(T^*M)$
\[\eta_{(q,p)}(v):=p(\pi_*(v)).\]
This describes the pairing between the $Q$-component (velocity in the base) of $v$  and the momentum coordinate $p$. 
From the canonical form, we obtain a coordinate-free definition of canonical symplectic form as 
\[\omega=-d\eta.\] 
Note that $\omega$ is not simply a closed 2-form, but rather an \emph{exact symplectic form. }  

\underline{\href{https://jeffhicks.net/snippets/index.php?tag=art:weinstein}{ One can also prove}} that the cotangent bundle serves as a general kind of local model for symplectic manifolds. 
We've already seen the cotangent bundle appear in the examples of symplectic structures on $\RR^{2n}$ and $(\CC^*)^n$, which can be interpreted as the symplectic structures on the cotangent bundles $T^*\RR^n$ and $T^*T^n$ respectively. 
This example also gives an example of how an almost complex structure and symplectic structure can interact.
    Let $Q$ be a manifold equipped with a connection.
    The tangent bundle of $Q$ comes with an almost complex structure.
    With a choice of connection we obtain a splitting
    \[T_{(q,v)}TQ= T_v (T_q Q)\oplus T_q Q\]
    with an isomorphism $A: T_v(T_q Q)\to T_qQ.$
    One can then construct an almost complex structure by taking the matrix 
    \[J:=\begin{pmatrix} 0 & A\\ -A^{-1} & 0\end{pmatrix}: T_{(q,v)}TQ\to T_{(q,v)}TQ.\]
    One can similarly (but not canonically) construct an almost complex structure for the cotangent bundle. 
    Pick $g$ a metric for $Q$, which induces a bundle isomorphism between the tangent and cotangent bundle. 
    Let $(q_1, \ldots, q_n, p_1, \ldots p_n)$ be local coordinates for $T^*Q$ chosen so that $\partial q_1, \ldots \partial q_n$ form an orthonormal basis at the origin and the coordinates $p_1, \ldots, p_n$ parameterize the linear coordinates determined by the basis $\{\iota_{\partial q_i}g\}$.
    Then an almost complex structure is specified by 
    \begin{align*}
        J \partial q_i=\partial p_i && J\partial p_i = -\partial q_i.
    \end{align*}
    Note that the resulting almost complex structure depends on the choice of metric, which modifies the ``size'' of the cotangent fiber relative to the base. 
    The metric $\omega(J, -)$ is then the standard induced metric on the cotangent bundle.\printbibliography
 \end{document}