Deformation Theory, Fall 2021

Professor A.J. de Jong, Columbia university, Department of Mathematics.

Please email me if you are interested in being a part of this and want to be on the mailing list.

The seminar will most likely be held on Friday mornings from 10:30 -- 12:00 in Room 507 in the math building at Columbia.

List of talks

1. Schlessinger's paper [1]. What happens with groupoids? Deformation categories + Inf auts + prorepresentability? Example of group representations? Galois representations? Example of nonsmooth deformation space?
2. Deforming schemes; cohomology and deformations (either with tangent bundle or with cotangent complex); a simple example; T^1 lifting criterion? Example of nonsmooth deformation space?
3. Deforming coherent modules; inf defos + obs; generalization to objects of the derived category; an example of nonsmooth deformation space?
4. Explicit deformations of singularities in terms of lifting equations and relations; an explicit example of nonsmooth deformation space? Burch's theorem? Smoothness of Hilb_n of dim = 2?
5. Deforming schemes via local-to-global method; isolated singularities; ramified Galois covers of curves + Oort's problem, etc
6. Lemma 0E3X and application to contracting curves and M_{g, n + 1} → M_{g, n}
7. Deforming abelian varieties? Deforming group schemes like GL_n? What about G_a? What about \alpha_p?
8. Artin's axioms for algebraic stacks; statement of the theorem including very short intro to algebraic stacks
9. Artin's axioms: example applications for example stack of coherent sheaves a la Lieblich
10. Artin's axioms: openness of versality in a nonstandard way + apply to the stack of coherent sheaves or the stack of complexes on a morphism a la Lieblich

List of meetings

1. September 17: organizational meeting to divide up the talks.
2. September 24: lecture 1 by Johan
3. October 1: lecture 2 by Cailan and Ivan
4. October 8: lecture 3 by Kevin
5. October 15: lecture 4 by Patrick
6. October 22: lecture 5 by Avi
7. October 29: lecture 6 by Caleb and Morena
8. November 5: lecture 6 by Morena, continued
9. November 12: lecture 7 by Baiqing Zhu
10. November 19: lecture 8 by Haodong
11. November 26: no classes.
12. December 3: lecture 9 by Che Shen
13. December 10: lecture 10 by Johan

List of references

1. Schlessinger, Michael, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), pp 208--222
2. Hartshorne, Robin, Deformation Theory, Springer.
3. Stacks project, Chapters on deformation theory, see
   1. Formal deformation theory
   2. Deformation theory
   3. Deformation problems