Geodesic
Stokes Cocycle and Differential Galois Groups
Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Differential and Functional-Differential Equations — Satellite of International Congress of Mathematicians ICM-2002 (Moscow, MAI, 11–17 August, 2002). Part 2, Tome 2 (2003), pp. 103-115.

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The classification of germs of ordinary linear differential systems with meromorphic coefficients at 0 under convergent gauge transformations and fixed normal form is essentially given by the non-Abelian 1-cohomology set of Malgrange–Sibuya. (Germs themselves are actually classified by a quotient of this set.) It is known that there exists a natural isomorphism $h$ between a unipotent Lie group (called the Stokes group) and the 1-cohomology set of Malgrange–Sibuya; the inverse map which consists of choosing, in each cohomology class, a special cocycle called a Stokes cocycle is proved to be natural and constructive. We survey here the definition of the Stokes cocycle and give a combinatorial proof for the bijectivity of $h$. We state some consequences of this result, such as Ramis, density theorem in linear differential Galois theory; we note that such a proof based on the Stokes cocycle theorem and the Tannakian theory does not require any theory of (multi-)summation. 
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M. Loday-Richaud. Stokes Cocycle and Differential Galois Groups. Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Differential and Functional-Differential Equations — Satellite of International Congress of Mathematicians ICM-2002 (Moscow, MAI, 11–17 August, 2002). Part 2, Tome 2 (2003), pp. 103-115. http://geodesic.mathdoc.fr/item/CMFD_2003_2_a6/

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