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\DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\TropB}{TropB} \DeclareMathOperator{\weight}{wt} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Fuk}{Fuk} \DeclareMathOperator{\str}{star} \DeclareMathOperator{\Ob}{Ob} \DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\Supp}{Supp} \DeclareMathOperator{\Bl}{Bl} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Tw}{Tw} \DeclareMathOperator{\Int}{Int} \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} \begin{definition} \label{def:hamiltonianVectorField} Let $(X, \omega)$ be a symplectic manifold, and $H: X\to \RR$ is a smooth function. The \emph{Hamiltonian vector field of $H$} is the unique vector field $V_H$ such that $dH=\iota_{V_H}\omega$. \end{definition} \printbibliography \end{document}