Formal geometry and deformation theory, winter 201718
Organization
Here is the preliminary program.
The topic of this seminar is formal geometry and deformation theory. The idea of the seminar is to get familiarity with formal schemes by studying its basic properties and using them in order to understand different problems.
The first aim of the seminar is to understand Raynaud's generic fiber functor, which establishes an equivalence between some formal schemes and rigid analytic spaces or Berkovich spaces, depending on the chosen approach.
The second aim of the seminar is to work a little bit with deformation theory. The idea is to study the infinitesimal and formal deformations of a geometric object (for example, two lines deform to an hyperbola), and apply the techniques to some problems lifting from positive characteristic to zero characteristic: for example, we can always lift curves and abelian varieties, in the sense that we find one in characteristic zero such that its special fiber is the one we started with. But, in general, we can't always do this, as we will see with the example done by Serre. All these things will be formulated in the language of formal schemes.
Schedule
October 16 
Tanya (+ a little bit Pedro) 
Introduction 
Notes 
October 23 
Tanya 
Formal geometry I: Locally Noetherian Formal Schemes 
October 30 
Tanya 
Formal Geometry II: The Comparison Theorem 
November 6 
Yun 
Formal Geometry III: Grothendieck Existence Theorem 
November 13 
Julian 
Deformation Theory I: Infinitesimal Deformations I 
November 20 
Marcin 
Deformation Theory II: Infinitesimal Deformations II 
November 27 
Marco 
Deformation Theory III: Infinitesimal Deformations III 
December 4 
Efstathia 
Deformation Theory IV: Formal Deformations I 
December 11 
tba 
Deformation Theory V: Formal deformations II 
December 18 
Marcin 
Deformation Theory VI: Formal vs Algebraic Deformation. 
January 8 
Xiaoyu Su 
Lifting theory I: Lifting from Char p to Char 0 
January 15 
Yun 
Lifting theory II: Serre's Example of nonliftable variety 
January 22 
Fei 
Lifting theory III: Lifting Abelian varities from Char p to Char 0 
January 29 
Holger Elbe 
Special session of rigid geometry and Berkovich spaces 
February 5 
Simon and Pedro 
From formal to rigid geometry I (Admissible blow ups) 
February 12 
Simon and Pedro 
From formal to rigid geometry II (Raynaud's fiber functor) 
Literature
Here is the list of references.
