\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\QQ{{\mathbb Q}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \)

Symplectic snippets

welcome to symplectic snippets

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 downloaded here, while the mathematical tags can be found here. Here are a few introductory articles which were written with this platform.

Snippet index

Browse the notes by topic and type

388snippetstotal
56articlesoverview notes
88definitionsvocabulary
17theoremsresults
application2 article56 claim1 conjecture1 construction6 convention4 corollary2 definition88 diagram12 equation2 example43 exercise20 exposition7 figure46 lemma11 proof25 proposition26 remark14 solution5 theorem17

Getting started

Start with local models and core vocabulary.

  1. introduction to symplectic geometry article
  2. symplectic manifolddefinition
  3. Lagrangian submanifolddefinition
  4. Weinstein Neighborhood Theoremtheorem

Lagrangians

Review examples, surgery, and cobordisms.

  1. introduction to Lagrangian submanifoldsarticle
  2. Lagrangians from conormalsexample
  3. Lagrangian connect sum construction
  4. lagrangian cobordismsarticle

Floer theory

Read from symmetric products to Heegaard Floer theory.

  1. a sketch of Heegaard-Floer theoryexposition
  2. Construction of Heegaard Floer cohomologyarticle
  3. Heegaard diagramdefinition
  4. Heegaard moves generate equivalencestheorem

Symplectic cohomology

Review Liouville domains, Reeb flow, and Viterbo restriction.

  1. a sketch of symplectic cohomologyexposition
  2. Liouville domaindefinition
  3. Reeb vector fielddefinition
  4. Viterbo restrictiontheorem

Tropical geometry

Read the tropical material on cycles and rational equivalence.

  1. a short introduction to tropical geometryarticle
  2. tropical chow groupdefinition
  3. tropical pushforwardproposition
  4. rational equivalence in tropical geometryarticle

Topics

Symplectic Geometry142
Lagrangians93
Floer Theory88
Symplectic Cohomology47
Tropical Geometry38
Categories61
Mirror Symmetry34

Browse snippets

application 2
article 56
claim 1
conjecture 1
construction 6
convention 4
corollary 2
definition 88
diagram 12
equation 2
example 43
exercise 20
exposition 7
figure 46
lemma 11
proof 25
proposition 26
remark 14
solution 5
theorem 17