Chennai Mathematical Institute

Seminars




2:30 pm, Seminar Hall
Algebraic aspects of Fuchsian Reduction

Satyanad Kichenassamy
Université de Reims Champagne-Ardenne.
06-12-19


Abstract

Fuchsian Reduction is a general technique for constructing singular solutions of linear or nonlinear partial differential equations, by reducing them to a hierarchy of equations that are reminiscent of problems of Fuchs-Frobenius type, insofar as solutions are uniquely determined by the first few terms of their expansions. However, the reduced equations may involve non-analytic coefficients, even if the coefficients in the problem being reduced do not exhibit branching. Reduction theory may be viewed as the counterpart of uniformization theory for differential equations, except that reduction theory also applies to non-analytic problems, involving Sobolev or H\"older regularity.

After reviewing the main open problems that have been solved by Fuchsian Reduction --- in soliton theory, conformal geometry, relativistic cosmology, detonation theory or control theory --- we focus on the algebraic aspects of the local construction of such "divisors." Typically, solutions admit of uniformization by the introduction of auxiliary variables, that may be interpreted as coefficients of binary forms. More complicated situations involve algebras that do not seem to have been studied or even identified.