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

• https://web.archive.org/web/20170921213130/https://www.ma.utexas.edu/users/rhughes/TopicsInD-modules.pdf

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