By Claude Mitschi, David Sauzin
Offering an common advent to analytic continuation and monodromy, the 1st a part of this quantity applies those notions to the neighborhood and international learn of complicated linear differential equations, their formal strategies at singular issues, their monodromy and their differential Galois teams. The Riemann-Hilbert challenge is mentioned from Bolibrukh’s viewpoint.
the second one half expounds 1-summability and Ecalle’s conception of resurgence below really normal stipulations. It includes a number of examples and offers an research of the singularities within the Borel aircraft through “alien calculus”, which gives a whole description of the Stokes phenomenon for linear or non-linear differential or distinction equations.
the 1st of a sequence of 3, entitled Divergent sequence, Summability and Resurgence, this quantity is geared toward graduate scholars, mathematicians and theoretical physicists attracted to geometric, algebraic or neighborhood analytic homes of dynamical structures. It contains valuable workouts with recommendations. the necessities are a operating wisdom of uncomplicated complicated research and differential algebra.
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Extra info for Divergent Series, Summability and Resurgence I: Monodromy and Resurgence (Lecture Notes in Mathematics)
Proceedings of the American Mathematical Society 2-6, 860–867 (1951) ´ Cha90. : Introduction a` l’analyse complexe. Editions MIR. Moscou (1990) CoLe55. Coddington, E. : Theory of Ordinary Differential Equations. McGrawHill (1955) Fo81. : Lectures on Riemann surfaces. Graduate Texts in Mathematics vol. 81, Springer (1981). Ha64. : Ordinary differential equations. Joh. Wiley (1964) In56. Ince, E. : Ordinary differential equations. Second edition. , New York (1956) JS87. Jones, G. : Complex Functions, An algebraic and geometric point of view.
N. Note that γ γ s if this holds, then f s = f 0 for all s ∈ u(s 0 ), that is, the map s → f γs is locally constant. Let zk = γs (tk ), z0k = γs 0 (tk ) for all k (see Fig. 3). Fig. 3: Homotopy invariance of analytic continuation s If k = 1, (the pairs) ft10 and fts1 are adjacent. This follows from ε ε |z01 − a)| < , |z1 − a| < 2 2 s which implies that |z01 − z1 | < ε, hence that W = Ut10 ∩Uts1 ∩Ua is nonempty (since s s all radii are ≥ ε) and fts1 and ft10 coincide on W . The FUT implies that fts1 and ft10 s0 s0 s s coincide on Ut1 ∩Ut1 , and ft1 and ft1 are adjacent as pairs.
Let (S) be a linear differential system of order p over k. The differential Galois group of (S) over k is a linear algebraic group over C. Proof. A linear algebraic group over C is by definition a subgroup of some GL(n,C) defined as the zero-set in GL(n,C) of a family of polynomials in n2 variables. In other words, it is a closed subgroup of GL(n,C) for the Zariski topology. We refer to [Hu75], [Bor91], [PSi01],[CH11] for general definitions and properties about linear algebraic groups. Let us first prove that the differential Galois group G can be identified with a subgroup1 of GL(p,C).