\documentclass[11 pt]{article} \usepackage{amsmath,amsthm,amsfonts,amssymb,titlesec} \usepackage{hyperref} \usepackage{tikz} \usepackage{verbatim} \usepackage{accents} \usepackage[citestyle=alphabetic,bibstyle=alphabetic,backend=bibtex]{biblatex} \usepackage{todonotes} \usepackage[american]{babel} \usepackage{fancyhdr} \hypersetup{colorlinks=false} \usetikzlibrary{calc, decorations.pathreplacing,shapes.misc} \usetikzlibrary{decorations.pathmorphing} \usepackage[left=1in,top=1in,right=1in]{geometry} \usepackage[capitalize]{cleveref} \newcommand{\mathcolorbox}[2]{\colorbox{#1}{$\displaystyle #2$}} \newcommand{\xxx}{T base with combinatorial potential data } \newcommand{\Xxx}{T base with combinatorial potential data } \newcommand{\xxxc}{combinatorial potential stratified space } \newcommand{\Xxxc}{combinatorial potential stratified space } \newcommand{\argument}{symplectic character } \newcommand{\arguments}{symplectic characters } \newcommand{\snip}[2]{#1} 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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{\Coh}{Coh} \DeclareMathOperator{\CritVal}{CritV} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\FS}{FS} \DeclareMathOperator{\Vect}{Vect} \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} \title{welcome to symplectic snippets} \maketitle \thispagestyle{firstpage}\emph{Symplectic snippets} is a proof-of-concept for organizing some of my mathematical notes on symplectic geometry. It was initially put together for the PIMS Summer School on Floer Homotopy theory. Notes are written as whole articles in LaTeX and then processed into mathematical tags -- snippets -- which can be individually referenced by other articles. Snippets can also process tikz code into SVG files for viewing through your browser. LaTeX code for any snippet can be accessed through the menu on the right. The code for creating this webpage can be \href{https://github.com/jeffsthicks/Snippets}{downloaded here}, while the mathematical tags can be \href{https://github.com/jeffsthicks/tags}{found here}. Here are a few introductory articles which were written with this platform. \begin{itemize} \item \underline{\href{https://jeffhicks.net/snippets/index.php?tag=art_basicSymplectic}{ An introduction to symplectic geometry}} \item \underline{\href{https://jeffhicks.net/snippets/index.php?tag=art_lagrangianSubmanifolds}{ Basics on Lagrangian submanifolds}} \item \underline{\href{https://jeffhicks.net/snippets/index.php?tag=art_tropicalGeometrySeminarOutline}{ Seminar notes on Tropical Chow groups}} \item \underline{\href{https://jeffhicks.net/snippets/index.php?tag=art_exactSequencesInSymplecticGeometry}{ A survey of some of the ways exact sequences appear in symplectic geometry}} \item \underline{\href{https://jeffhicks.net/snippets/index.php?tag=art_heegaardFloer}{ A very brief introduction to Heegaard-Floer theory}} \item \underline{\href{https://jeffhicks.net/snippets/index.php?tag=art_symplecticCohomologyExposition}{ An introduction to Symplectic cohomology}} \end{itemize} \printbibliography \end{document}