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User:Wundzer

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Rando on the internet interested in derived categories, string theory, and algebraic geometry. Interested in helping me build out wikipedia in these sections and write content with examples? Put a message on my talk page! I can give instructions on stuff todo. Another option is to start a weekly reading group and write up the results on wikipedia. Thoughts?

Pages I've edited

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  • Pursuing Stacks – working on writing up a sketch of material covered in the manuscript, writing out articles on related homotopy theory constructions
  • Postnikov tower – rewrote most of the article and gave relation to homotopy theory and computing homotopy groups
  • Derived Noncommutative Algebraic Geometry – wrote entire article
  • Chern classes – gave computational examples
  • Pontryagin class – gave computation of K3 surfaces
  • Todd genus – gave examples here too
  • Homotopy groups – added homotopy groups of orthogonal groups and relation to sphere bundles
  • Local systems – discussed bivariant topologicla theory with examples
  • Coherent sheaf cohomology – Added Kunneth formula and related computation for genus g curves
  • Algebraic curve – gave computations for genera of plane curves and curves in .
  • Quot scheme – updated text and gave constructive examples
  • Hilbert scheme – updated examples section and added some insightful examples
  • Grothendieck riemann roch – added examples of vector bundles on curves, smooth proper morphisms and moduli of curves example, and closed embeddings
  • Convexity (algebraic geometry) – Wrote first article with lots of examples.
  • Proper morphism – Added examples and geometric intuition for valuative criterion of properness.
  • Schubert calculus – Wrote up construction, examples, and lines on a cubic surface
  • Field norm – added examples and reorganized page
  • Linear system of divisors – added examples from curves, mentioned hyperelliptic curves, trigonal curves, g.r.d.'s and Brill noether, and improved other examples
  • Dual number – added examples of tangent vectors on the scheme to show how this technology works.
  • Fiber functor – gave definitions and additional references
  • Galois group – refactored the examples section and added some additional examples and a computational proposition. Also added some much needed references
  • Opposite ring – added examples of the oppositve algebra for the free algebra and quaternion algebra
  • Formally smooth map – added examples and non-examples
  • Locally compact field – rewrote most of article, added structure theorems, intuition, and examples from p-adic numbers
  • Solvable group – added motivation for definition and reformatted examples section to be more readable. In addition, I added some more examples
  • Noetherian scheme – added examples and non-examples, including one which motivates the study for schemes over a non-Noetherian base
  • Deformation theory – added deformations of germs of analytic functions and mentioned tangent cohomology
  • Derived algebraic geometry – added examples of derived schemes and spectral schemes
  • Kodaira–Spencer map – rewrote article, added constructions of map and examples of it
  • Exalcomm – updated page with definition of square-zero extension, construction, and structure theorems.
  • Derivator – added sections on motivation, definition, etc.
  • Jacobian ideal – added relations to hodge theory and deformation theory
  • Solvable Lie algebra – updated examples section to be more illuminating, also made it more organized
  • Nilpotent Lie algebra – reorganized and added examples
  • Lie algebra – helped make this page more user friendly by adding examples and explanations
  • Unipotent – fixed up definitions, added example section, added classification using nilpotent lie algebras
  • Normal scheme – added cusp example of normalization and formatted examples section
  • Minimal polynomial – explained tool to compute minimal polynomial and gave examples
  • Integral element – reorganzied and added examples
  • Quadratic integer – explained computation of ring of integers
  • Annihilator (ring theory) – explained annihilators for commutative rings and gave techniques required for their calculation. In addition, added complete calculation for all finite modules over the integers
  • Quintic threefold – rewriting the article...
  • Length of a module – updated article, added references, examples, and related topics, esp to intersection theory
  • Moduli of abelian varieties – created article
  • Moduli of algebraic curves – updated article with construction, properties, and examples of low genera
  • Moduli stack of elliptic curves – expanded upon article by constructing moduli space over characteristic 0 and gave detailed description of the points
  • Gerbe – Added examples of root stacks
  • Stack (mathematics) – reorganized examples section to it's more readable for beginners, added reference to local structure of algebraic stacks, also added examples
  • Azumaya algebra – added examples
  • Homotopy Lie algebra – updated with references and examples
  • Homotopy associative algebra – created page
  • Mirror Symmetry Conjecture – created page
  • Mixed Hodge Structure – created page
  • Intersection homology – added example of intersection cohomology sheaf
  • Milnor map – improved definition with references, added some main theorems
  • Mixed Hodge module – created page
  • Kan complex – expanded and improved article with better organization, examples, application, and structure of Kan complexes
  • Dold–Kan correspondence – stated functorial equivalence + sketched construction
  • ∞-groupoid – adding material
  • Symplectic matrix - reformatted and added theorem about generating set of matrices

General references

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L-functions of motives

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Jacobians of hyperelliptic curves and other arithmetic

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Picard group + Moduli of vector bundles

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  • Adelic presentation (theorem of Weil) for a curve (On the Geometry of Higher Tate Spaces - Aron Heleodoro - Northwestern)
  • For a reductive group and a curve , can be described as
  • There also exists a similar result for arbitrary varieties, but this uses a co-simplicial ring
  • Adelic Descent Theory - https://arxiv.org/abs/1511.06271
  • Residues and adeles - Beilinson (2 pages)

Coherent sheaves

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Modular forms

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Abelian varieties

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General

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Moduli of abelian varieties

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Schottky

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Intermediate Jacobians

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Pathologies in nature

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This will contain references to pathological objects which occur in nature and not by manual hacking.

Representation theory

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Groupoids

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GIT

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Pages needing work

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Projection valued measures

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Algebraic stacks

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There really needs to be a separate algebraic stacks page which is focused entirely on that subset of stacks. This should include definitions, recent theorems (slice theorem), applications, and morphisms of different stacks.

Quasi-coherent sheaves on algebraic stacks

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Deformation theory

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Deligne cohomology

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Picard groups of moduli spaces

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Level structures

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Mention how gives infinitely many bases for , hence we need to consider level structures to get finite etale coverings of moduli spaces

Drinfeld modules

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Moduli of Abelian varieties

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Hodge

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Gromov–Witten invaritants

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Motivic

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Picard–Fuchs

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Ring of integers

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Jacobian ideal

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  • Page 84 of Mirror Symmetry book by Cox Katz has a *much* better explanation for reductions of pole orders
  • Also, this has a lot of results/explanations on computing Gauss-Manin connections

Jacobians and Periods

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Computing period matrices and Hodge theory

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Numerical aspects

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Period Matrices and jacobians of higher dim varieties

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  • Computing Periods of Hypersurfaces – https://arxiv.org/abs/1803.08068
  • Singularities of Differentiable Maps, Volume 2 Chapter (Integrals and differential equations) (MHS) (Period Map and Intersection Form)
  • PERIODS OF ALGEBRAIC VARIETIES -> Oliver Debarre

Coherent sheaf cohomology

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Look at theorem's 4.4 and 4.5 in Altman-Kleiman's book on Grothendieck Duality for useful results of computations for sheaf cohomology

Dualizing complexes

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Atyiah class

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Intersection forms

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There should be a page discussing the intersection forms of manifolds and varieties. In addition, it should reference the Todd index theorem as a tool for computing the intersection forms using the decomposition of integral binary forms.

Brieskorn lattice

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Maybe add this to the Gauss-Manin page...

Algebraic Number theory

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Locally compact field

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Dessins d'enfants

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Riemann's existence theorem

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There really should be a page on Riemann's existence theorem. Here are some references

Galois group

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Galois groups of polynomials

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  • Milne's section on galois groups of polynomials
  • Dummit + Foote

Weil conjectures

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Arithmetic of K3 surfaces

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https://math.rice.edu/~av15/Files/AWS2015Notes.pdf

Stable vector bundles

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Voevodsky Motives

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Tautological Ring of Kontsevich Spaces

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Moduli of curves

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Notes: https://deopurkar.github.io/teaching/moduli/

https://mathoverflow.net/questions/76585/moduli-space-of-genus-2-curves

There should be examples of the moduli of curves page. This could include the stacks genus 2 from Mumford's paper, and genus up-to 6. This paper has a great summary:

https://arxiv.org/abs/1307.6614

https://arxiv.org/abs/1904.08081

Mumford's paper: http://www.dam.brown.edu/people/mumford/alg_geom/papers/1983b--EnumGeomModuli-NC.pdf

https://www.math.brown.edu/~bhassett/papers/genus2/logmodel3.pdf

Riemann-Hurwitz theory

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This is closely related to the moduli of curves. Here are some resources

https://deopurkar.github.io/research/papers/thesis.pdf

Hilbert polynomial

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Add examples of hilbert polynomial for hypersurfaces. Reference is Kollar Rational curves on algebraic varieties. In addition, mention RR and HRR as tools for computing the hilbert polynomial.

Etale topology

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There should be discussions about the local rings, strict henselization, and unramified extensions. Also, there should be discussions about geometric interpretations of Etale topology, Henselian traits, and what the points in the topology sees. The example given here https://math.stackexchange.com/questions/2321214/grothendiecks-vanishing-cycles is excellent!

Embedded points

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There should be a page discussing embedded points and cohen-macaulay schemes. Reference: https://stacks.math.columbia.edu/tag/05AJ

Examples

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Consider the scheme

which is the axis with an embedded point at the origin. Then, this gives a non-example of a Cohen-Macaulay scheme.

Stability Conditions

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Add examples an stuff from

Algebraic geometry pages

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Checkout the exercises in https://amor.cms.hu-berlin.de/~soldatea/alggeom_V4A2_SS16.html https://amor.cms.hu-berlin.de/~soldatea/V4A2/

Algebraic curves stuff...

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https://link.springer.com/book/10.1007/978-3-540-69392-5 (stable reduction exercises are awesome!)

Formal schemes

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Log geometry

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There should be a page on log schemes and log geometry. Checkout Log structure for links to pages not yet created.

Euler sequence

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Add in relative Euler sequence for projective bundles

https://amor.cms.hu-berlin.de/~soldatea/V4A2/AGUebungII3.pdf

Deformations of curves

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There should be a page dedicated to the deformations of curves. This could include discussions of Kodaira-Spencer theory and applications, pointed curves, maps of pointed curves in Kontsevich moduli spaces.

Hilbert Schemes

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  • https://arxiv.org/abs/1512.07363 is a great reference with discussions about the virtual tangent sheaf. Material from this paper and its references could be used in other pages as well, such as enumerative geometry, equivariant k-theory, virtual fundamental classes, and others.

Azumaya algebras

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Checkout this link

and construct examples of azumaya algebras. As a corollary, the quot scheme will give some moduli space of modules of this azumaya algebra.

Kontsevich moduli spaces

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There should be page discussing the Kontsevich moduli spaces of curves. Some references are

Quintic threefold

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Let be a quintic threefold defined by a degree 5 homogeneous polynomial , a section of . Using the map

fiber of at a point is the rational curve .

we can pullback and the push-forward is . This glues to a vector bundle of rank on . There is an associated section whose vanishing locus is the orbifold .

A infinity algebras

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Fourier–Mukai transforms

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Perverse sheaves

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This page is in need of an upgrade. It should include results such as the decomposition theorem and examples of perverse sheaves. http://people.mpim-bonn.mpg.de/geordie/perverse_course/lectures.pdf has a ton of useful info for this, also https://web.math.princeton.edu/~smorel/faisceaux_pervers.pdf

Grothendieck–Riemann–Roch

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There should be some example computations here. This could include some basic examples, like computations related to HRR, GRR on curves https://math.stanford.edu/~vakil/245/245class18.pdf, and mumford's results about tautological classes. Also, the extension to equivariant theories would be nice https://arxiv.org/abs/1205.4742.

Homotopy groups

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There should be ample discussion about applications of the long exact sequence in homotopy theory. This should include the simply connectedness of lie groups, such as , discussion of bundles on classified by . Husemollers fiber bundles book contains useful info about this too. It would be nice if the exotic spheres milnor constructed were accessible through wikipedia articles.

Symplectic groups

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Symplectic groups have a nice decomposition into a few matrix subgroups which are multiplied together. Checkout the books

  • Introduction to Symplectic Dirac Operators
  • Folland: Harmonic analysis in phase space (which gives the proofs)

Symplectic structures

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There should also be also be a discussion about what the standard symplectic structure "does" on using inner products. Again "Introduction to Symplectic Dirac Operators" has a nice discussion :)

Algebraic curves

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Mumford gives a complete list of ways to find algebraic curves. This should be included somewhere to give beginners a look at how to construct any genus of algebraic curve and where to look for more advanced examples.

Postnikov tower

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The whitehead tower should be constructed, explained, and applications with spectral sequences given. This should have similar applications to computing homotopy groups.

Also, there should be these constructions for spectra as well.

Eilenberg–Maclane spaces

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This page should have computations of the rational cohomology ring and the partial computations of the integral cohomology ring. The book "Homotopical Topology" has an *excellent* overview of how to accomplish this feat

Atyiah–Singer index theorem

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This page should have some computations on it! There is an excellent reference giving some easily accessible formulas – https://www.maths.ed.ac.uk/~v1ranick/papers/gilkey3.pdf in particular, it could be computed for smooth complex projective hypersurfaces.

Chern–Weil theory

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Lectures on Chern–Weil Theory and Witten Deformations by Weiping Zhang has a lot of great results for Chern-Weil theory. He gives an overview of Bott localization formula as an application

Also, he discusses 3-manifolds which apparently all have trivial tangent bundle and the Chern-Simons functional

Dupont fiber bundles – contains calculation for CP^n – https://data.math.au.dk/publications/ln/2003/imf-ln-2003-69.pdf

Flat vector bundle

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This page should be edited to include the case of flat vector bundles over whose monodromy is determined by a map , this could also include flat principal bundles, so . This article has a good description https://arxiv.org/pdf/1501.00730.pdf

Holomorphic vector bundles

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The constructions of vector bundles in https://arxiv.org/pdf/1501.00730.pdf should be discussed, including discussions about theta functions as sections of line bundles on elliptic curves.

Mirror Symmetry

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https://math.berkeley.edu/~auroux/papers/cp2mirror.pdf