Math 6370: Topics in Deformation Theory

For a course description, prerequisites, and requirements, please see the syllabus. Please email me if you do not have the Overleaf link to the course notes!

Selected References:

  • I think the following will be the most useful references for this course, although one could supply many others. I've given preference to references that are readable rather than comprehensive.
  • For classical deformation theory in algebraic geometry, there are a couple of textbooks: Hartshorne, Deformation Theory, and Sernesi, Deformations of Algebraic Schemes. For a survey of some aspects, see Illusie, Grothendieck's Existence Theorem in Formal Geometry (in the volume FGA Explained). For the Schlessinger criteria, see Schlessinger, Functors of Artin Rings.
  • For the deformation theory of profinite group representations, see Mazur, Deforming Galois Representations (in Galois Groups over Q). To see the deformation theory of Galois representations in action, see the survey article Darmon-Diamond-Taylor, Fermat's Last Theorem (we won't discuss this material). For a beautiful application (which we will discuss) of deformation theory to finite group theory, see Serre, Exemples de plongements des groupes PSL_2(F_p) dans des groupes de Lie simples.
  • For simplicial homotopy theory, see Goerss-Jardine, Simplicial Homotopy Theory, and the survey article of Goerss-Schemmerhorn, Model Categories and Simplicial Methods (in Interactions between Homotopy Theory and Algebra). For model categories, see the above, Quillen, Homotopical Algebra, and May-Ponto, More Concise Algebraic Topology.
  • For introductions to the cotangent complex, see Quillen, Homology of Commutative Rings, the survey article of Iyengar, Andre-Quillen homology of commutative algebras (again in Interactions between Homotopy Theory and Algebra), and relevant sections of The Stacks Project.
  • For derived algebraic geometry, see Lurie, Derived Algebraic Geometry (this includes the derived Schlessinger theorem).
  • For applications to the study of derived deformations of profinite group representations, see Galatius-Venkatesh, Derived Galois Deformation Rings.