Differential Galois theory has the objective to study linear ODEs (or connections) with the help of algebraic groups. Roughly and explicitly, to a matrix
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and a differential system
, we associate a subgroup of
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, the differential Galois group, whose function is to measure the complexity of the solutions. There are three paths to this theory: Picard-Vessiot extensions, monodromy representations and Tannakian categories.
If instead of working with complex coefficients we deal with a discrete valuation ring
, the construction of the differential Galois groups is less obvious and the theory of groups gives place to that of group schemes. This puts forward the Tannakian approach and relevant concepts from algebraic geometry like formal group schemes and blowups. In this talk, I shall explain how to associate to these differential equations certain flat
-group schemes, what properties these may have--what to expect from a group having a generically faithful representation which becomes trivial under specialisation and how to compute with the help of the analytic method of monodromy. The talk is a horizontal report on several works done in collaboration with P.H. Hai and his students N.D. Duong and P.T. Tam over the past years.
