User:Stca74/Maths to do
Individual articles to work on[edit]
Commented articles that can and should be improved irrespective of larger reorganisation.
- Gauss-Manin connection — essentially a rewrite: explain in terms of relative de Rham cohomology
- Gerbe — essentially a rewrite — now in progress (albeit slowly...)
- coherent sheaf — rewiew, correct and expan (definition fixed)
- de Rham cohomology — redo de Rham's theorem and especially its proof (see own Talk page comment)
- Künneth formula — add derived category sheaf version
- Integral — contribute along the lines of the talk page comments
- Elliptic curve — add explanation of group structure wrt Abel-Jacobi map (as per own comments on Talk page)
Algebraic topology and cohomology theme[edit]
The bold face word after title gives planned WP 1.0 importance. Plans for the article follow that.
- Algebraic topology: TOP and VITAL — The main article to organise the whole area. Should give a comprehensive overview with limited technical detail.
"Classical" homology / cohomology theories[edit]
- Homology (mathematics): TOP — to combine with Cohomology; an overview of (co-)homology in all places it is used in, not only algebraic topology
- Cohomology TOP — to combine with Homology; an overview of (co-)homology in all places it is used in, not only algebraic topology
- Homology theory: HIGH - Develop into an overview of topological homology and cohomology theories. Rename accordingly. Treat Eilenberg-Steenrod axioms briefly.
- Cohomology theory: HIGH — change current redirect to point to the (new) Homology theory (above)
- Singular homology: MID — combine with Singular cohomology; review and improve
- Singular cohomology: MID — combine with Singular homology; review and improve
- Simplicial homology: MID —review and improve
- de Rham cohomology: MID —review and improve; link to sheaf cohomology in general and provide pointers to algebraic de Rham cohomology, the Hodge - de Rham spectral sequence and all that;
- cellular homology: MID — review and improve as necessary
- Mayer-Vietoris sequence: MID —review and improve as necessary
- universal coefficient theorem: MID —review and improve
- Künneth formula: MID — expand into a "nexus" to cover the various topological and algebraic "elementary" cases; tie the strings together with showing the derived category formulation for sheaves that generalises all of the special cases
- cup product: MID — expand to cover cup product in its various incarnations; see Künneth above
- cap product: MID — review and expand
- Poincaré duality: HIGH — see Talk:Poincaré duality/Comments
- characteristic class: HIGH TBD
- Whitney class: MID TBD
- Pontryagin class: MID TBD
- Chern class: MID TBD
- Thom class: : MID TBD
[edit]
- sheaf cohomology: HIGH — serious expansion into the nexus of "geometrical" (co)homology theories; most specialist topics into sub-articles; goal: show how Eilenberg-Steenrod fits nicely into a (derived-category) sheaf cohomology (need cohomology with supports / local cohomology)
- Cech cohomology
- local cohomology
- compactly-supported cohomology
- Borel-Moore homology
- Intersection homology
- Intersection cohomology
- Leray spectral sequence
- Local system
- Verdier duality
- six operations
- Künneth formula
- cup product
Homological algebra[edit]
[edit]
Personally less expertise in the more advanced areas (stable and rational homotopy,...).
- covering space
- fundamental group
- homotopy
- homotopy group
- homotopy theory
- fibration
- suspension functor
- smash product
- wedge product
- cofibration
- exact homotopy sequence
- homotopy groups of spheres
- rational homotopy theory
- stable homotopy theory
- spectrum (homotopy theory)
- Eilenberg-MacLane space
- model category
Algebraic and analytical geometry[edit]
Second priority for now. Branches out very fast towards all other parts of AG and category theory (toposes, gerbes, Tannakian formalism, intersection theory,...) To handle together with the larger AG theme once its time comes.