The exercise sessions take place in salle Cartan on wednesdays, from 3:15pm to 5:15pm. Do not hesitate to ask any questions you might have via email or to come to my office (T13, "under the roof"). If you find a mistake in the exercises or in the solutions, please let me know ! All the sheets and solutions are in French. Do not hesitate to ask for clarifications in english.
- Session 1 : Power series and analytic functions and the solutions.
- Session 2 : Cauchy-riemann equations and holomorphy and the solutions.
- Session 3 : Integration and Cauchy's theorem
- A homework problem on Schwarz-Christoffel mappings, it is not mandatory and won't count toward your grade but I will be correcting the papers that are turned in.
- A homework problem on an equivalent to the Riemann Hypothesis. It is not mandatory, but can give you a bonus on the exam. You have until the 5th of may to turn it to me, either in person or in my "mail box" in Espace Cartan. If you need a version in English, please tell me and I will type one.
The aim of the working group is first to understand the theory of differential equations in zero characteristic, in particular we will classify differential equations on \(\mathbb C (\!(z)\!)\). Then, we will study differential equations in positive characteristic with the aim of proving a local-global principle, the Katz-Honda theorem.
Courses take place on every tuesday from 10:45am to 12:15pm in the Bourbaki room. Do not hesitate to email me questions or come to the T13 office to ask me questions directly.
Liste des séances:
Courses take place on every tuesday from 10:45am to 12:15pm in the Bourbaki room. Do not hesitate to email me questions or come to the T13 office to ask me questions directly.
Liste des séances:
- Week 0 : Introduction, the problem of algebraicity of solutions of a differential equation.
- Week 1 : Differential fields, modules with connection and \(D\)-modules. Cyclic vector lemma.
- Week 2 : Constructions on modules with connection : internal Hom, dual, tensor product. Statement of the classification of differential equations on \(\mathbb C (\!(z)\!)\).
- Week 3 : Regular singular equations : definition for operators and modules, classification of regular singular modules.
- Week 4 : Hensel lemma for regular singular equations. Irregular singular equation, definition and Hensel lemma.
- Week 5 : Proof of the classification theorem. Newton polygons, definitions et properties.
- Week 6 : Equations différentielles sur des ouverts de \(\mathbb C\), monodromie et correspondance de Riemann-Hilbert.
- Week 7 : Algebraic solutions of differential equations over \(\mathbb Q(z)\) and \(\mathbb F_p(z)\). Honda's theorem.
- Week 8 : Rational solutions, \(p\)-curvature, Cartier's lemma.
- Week 9 : Regularity and rational exponents of differential equations with vanishing \(p\)-curvature. Hypergeometric equations : regularity and exponents.
- Week 10 : Katz' proof of the \(p\)-curvature conjecture for hypergeometric equations, I.
- Week 11 : Katz' proof of the \(p\)-curvature conjecture for hypergeometric equations, II.