User:Stca74/Maths to do

Individual articles to work on

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

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

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

Sheaf cohomology and related theories

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

derived category

Personally less expertise in the more advanced areas (stable and rational homotopy,...).

Algebraic and analytical geometry

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.