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 TBA

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