Talk:Deformation (mathematics)

message: ===Discuss Deformation Functors===
This page should include a discussion of (pre-)deformation functors with examples. A good resource for this material is https://www.math.ucdavis.edu/~osserman/classes/256A/notes/deform.pdf .
Another reference is https://cel.archives-ouvertes.fr/cel-00392119/document
===Examples Needed===
In addition, this should include examples both from geometry and arithmetic:
* Hilbert schemes and deformations
* curves (over finite fields)
* Galois representations
For curves try the following
* take a smooth curve and see if it is smooth over \mathbb{Z}/q^2, then take the system with non-changing coefficients
* take a smooth curve over \mathbb{Z}_p and let one of the coefficients be an infinite series. Each truncation gives a deformation of the curve over \mathbb{F}_p.
===Differential Graded Lie Algebra===
Discuss differential graded lie algebras and their use in deformation theory. Look at page 17 of https://arxiv.org/pdf/1409.5996.pdf
===Add in Kodaira-Spencer Theory===
This page should include the basic definitions from Kodaira-Spencer theory, including the
* definition of deformations in the complex-analytic settings
* kodaira-spencer morphism
* vanishing of H^2(X,T_X) implies there exists a complete family of deformations where the kodaira-spencer map is an isomorphism
* Add Bogomolev-Tian-Todorov theorem that the Kodaira-Spencer map of a local universal deformation of a Calabi-Yau manifold is an isomorphism. (https://static1.squarespace.com/static/57bf2a6de3df281593b7f57d/t/57bf67bf6a49636398ee220e/1472161728174/tian-todorov.pdf)
=== Discuss Applications to Singularity Theory ===
* take examples of miniversal deformations from the milnor number page
=== Formalize KS-Theory for Local Deformations ===
* discuss pro-representability/universal deformation
* give example
* discuss schlessinger's theorem
=== Hilbert Schemes ===
* discuss how deformation theory is the local study of Hilbert schemes  — Preceding unsigned comment added by 97.122.75.155 (talk) 04:47, 2 August 2017 (UTC)
thread_title:  Todo 

message: * discuss how deformation theory is the local study of Hilbert schemes  — Preceding unsigned comment added by 97.122.75.155 (talk) 04:47, 2 August 2017 (UTC)
thread_title:  Hilbert Schemes 

message: This page should also discuss some of the material in http://www.claymath.org/library/monographs/cmim01c.pdf  — Preceding unsigned comment added by 161.98.8.4 (talk) 21:19, 30 July 2017 (UTC)
thread_title: Moduli of Curves/Gromov-Witten Theory

message: I think this page should include a survey of some of the main theorems in deformation theory. This should include
* Schlessinger's theorem (give non pro-representable)
* Grothendieck's theorem on deformations of abelian varieties
* Serre-Tate theorem (already there but should be developed further and moved to the theorems section)
* Deligne Illusie theorem
* T^1 lifting theorem
* BTT theorem
=== Some references ===
* http://web.archive.org/web/20151010220650/https://math.berkeley.edu/~achinger/notes/char-p-5.pdf
* https://math.stackexchange.com/questions/102252/english-translation-or-summary-of-relevements-modulo-p2-et-decomposition-du
Kaptain-k-theory (talk) 04:19, 20 August 2021 (UTC)
thread_title:  Main theorems in deformation theory 

message: * http://web.archive.org/web/20151010220650/https://math.berkeley.edu/~achinger/notes/char-p-5.pdf
* https://math.stackexchange.com/questions/102252/english-translation-or-summary-of-relevements-modulo-p2-et-decomposition-du
Kaptain-k-theory (talk) 04:19, 20 August 2021 (UTC)
thread_title:  Some references