User:Username6330
Hello
and welcome to my wiki page! Here I will discuss plans for where I want to see wikipedia (specifically the maths pages) go in the future. My core belief about mathematics is that it should be absolutely democratized. By this, I mean anyone should be able to look up a topic, read the gist of what it is about, get a sense of examples, and be guided through the literature. Unfortunately, the mathematics community has largely failed on this front and I think it is somewhat deliberate. For example, consider this blog post https://web.archive.org/web/20170929060614/https://alexandercoward.com/writing/2017/4/17/blowing-the-whistle-on-the-uc-berkeley-mathematics-department
Even Grothendieck admitted later on in his life that the lack of examples and guiding intuition in mathematics is oppressive (although I cannot find the page where he said this, I think it was one hosted by Ronnie Brown).
The moral of the story is that I want to see more mathematics openly explained on this site which includes a plethora of examples.
Pages I Have Worked On
Here are some pages that I've worked on
- Chern class
- Discrete valuation ring
- Stable curve
- Deligne–Mumford stack
- Regular sequence
- Serre duality
- Constructible sheaf
- Primary decomposition
- Hilbert series and Hilbert polynomial
- Derived scheme
- Borel–Moore homology
- Atiyah–Hirzebruch spectral sequence
- Deformation theory
- Algebraic K-theory
If you see todo lists, or examples needed sections on the talk pages for a math wikipage there is a good chance that I put it up there. These notes are both there to remind myself of things I want to put on wikipedia while giving an open invitation to others for improve wikipedia in a directed manner.
Pages Needing Help
There are several pages on wikipedia which need attention. This includes
- Vanishing cycle (https://www.math.u-psud.fr/~illusie/Illusie-Grothendieck-nearby-cycles5.pdf)
- Lefschetz pencil
- Abelian variety
- Semistable abelian variety
- Serre duality
- Derived functor
- Sheaf cohomology
- Spinor bundle
- Clifford module bundle
- Heat kernel
- Dirac operator
Wish List
I have a wish list of topics which I would like to see discussed on this site including
- equivariant cohomology with applications to computing the cohomology of orbifolds/DM-stacks
- motivic spectral sequences
- derived stacks
- spectral schemes
- chromatic homotopy theory (including the motivic version)
- brown peterson homology
- moduli of curves
- computations with characteristic classes and GRR
- mixed hodge modules/mixed hodge structures
- more perverse sheaves
- resolutions of singularities and using them to compute intersection cohomology
- computations with adams operations both on topological k-theory and algebraic k-theory
- rational homotopy theory
Papers, Notes, Etc.
This is a section I will keep for organizing notes in different topics. Hopefully it will help guide others to learning material and improving wikipedia
Etale Cohomology
- Etale cohomology of number fields - http://www.numdam.org/article/ASENS_1973_4_6_4_521_0.pdf
Homotopy Theory
- Sheaves and homotopy theory - http://math.mit.edu/~dspivak/files/cech.pdf
Constructible Sheaves/Hodge Theory
String Theory
- D-Braines, Sheaves and Categories - http://www.ulb.ac.be/sciences/ptm/pmif/Rencontres/lectures_sheaves.pdf
Intersection Theory
- GRR - https://www.math.leidenuniv.nl/scripties/MasterJavanpeykar.pdf
- Hilbert Series + Chern Polynomials - http://www-personal.umich.edu/~takumim/HilbertandChernPolynomials.pdf
- Steenrod operations and quadratic forms (contains nice overview of intersection theoretic machinery using homological approach) - http://www.mathematik.uni-muenchen.de/~haution/these.pdf