The Monodromy Group - ISBN:3764375353

The Monodromy Group

Date: 14 April 2011, 15:00

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The origins of monodromy theory lie in the works of B. Riemann on functions of complex variables and on complex linear differential equations. Riemann formulated one of the fundamental problems in monodromy theory (now called the Riemann—Hilbert problem): having given singularities and corresponding monodromy transformations, find a differential equation which realizes these data. The monodromy groups of linear differential equations and systems were intensively studied in the nineteen century by F. Klein. G. G. Stokes, H. A. Schwarz. L. Schlesinger, L. Pochhammer, E. Picard, R. Gamier, P. Painleve, H. Poincare, R. Fuchs and others. Schwarz associated the monodromy group of the hypergeometric equation with the spherical triangle groups, generated by inversions with respect to the sides of a spherical triangle. Schlesinger investigated deformations of differential systems with fixed monodromy. Fuchs associated the equation Painleve 6 with the isomonodromic deformation equation. Stokes discovered a strange phenomenon (the Stokes phenomenon) of non-uniqueness of constants in the asymptotic expansions of systems near irregular singularities.

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