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- Session 1 : Power series and analytic functions and the solutions
- Session 2 : Holomorphic functions and conformal mappings and the solutions
- Session 3 : The Stokes formula and Cauchy theory and the solutions
- Session 4 : The maximum principe and the Schwarz lemma and the solutions
- Session 5 : Méromorphic functions, Laurent series and the residue theorem and the solutions
- Session 6 : Back to the residue theorem, and Rouché's theorem and the solutions
- Session 7 : Special functions and the solutions
- Session 8 : Spaces of holomorphic functions and the solutions
- Session 9 : Conformal mappings and revisions and the solutions
- Session 10 : Approximation and factorization theorems and the solutions
- Session 11 : Harmonic and subharmonic functions and the solutions
- Session 12 : Potential theory and the solutions (unfinished for now)
- 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 reading group studies the first three parts of the book Differential Galois Theory through Riemann-Hilbert Correspondence by Jacques Sauloy. The aim is to build an understanding of Riemann-Hilbert correspondence, which links analytic/algebraic objects (regular singular differential equations on an open set \(\Omega\) of \(\mathbf{P}^1\)) and topological objects (local systems / representations of \(\pi_1(\Omega)\)).
The courses take place every wednesday from 4:45pm to 6:15pm in Salle Bourbaki. If you have any questions, you can write me an email or come directly at the office T9 in the DMA.
Approximate list of courses
- Week 0 : Holomorphic and analytic functions. Isolated zeros theorem, principle of analytic continuation. Holomorphic functions, meromorphic functions, germs. Exp and log functions.
- Week 1 : Analytic continuation along paths, homotopy. Fundamental group of an open subset of \(\mathbf{C}\), action on germs (Chap 5).
- Week 2 : First example, \(z^\alpha\) and the equation \(y'= \alpha y/z\), explicit computation of the monodromy. Introduction to the sheaf of solutions (Chap 6).
- Week 3 : Riemann sphere, definition of scalar and matrix differential systems, wronskian, fundamental solution. Cauchy existence theorem (Chap 7, 7.1 - 7.3).
- Week 4 : Definition of the solution sheaf, local systems. General definition of the monodromy representation (Chap 7, 7.4 - 7.5).
- Week 5 : Holomorphic and meromorphic equivalence of differential systems, cyclic vector lemma. Properties of the monodromy representation and the solution sheaf (Chap 7, 7.6).
- Week 6 : Definition of regular singularities, characterization and resolution of regular singular systems (Chap 9, 9.1 - 9.3 + Théorème 9.18).
- Week 7 : Surjectivity of the matrix exponential, local Riemann-Hilbert correspondence. Regular singular scalar equations, Fuchs criterion (Chap 4, 4.4 + Chap 9, 9.4 - 9.5).
- Week 8 : Fuchsian systems and operators. First case with 3 singularities : the hypergeometric function and its monodromy (Chap 11, 11.1 - 11.4).
- Week 9 : Global Riemann-Hilbert correspondence, statement and proof (Chap 12).
- Week 10 (Categorical intermezzo 1) : Categories and functors, examples. Category of differential systems, solutions functor. Local systems and representations (Chap 8).
- Week 11 (Categorical intermezzo 2) : Category of systems with regular singularities, categorical restatement of the local correspondence (Chap 10).
- Week 12 : Differential algebras, differential automorphisms. Differential Galois group (Chap 13).