Sergei Yakovenko's blog: on Math and Teaching

Monday, November 10, 2014

Lecture 2 (Nov 7, 2014)

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 6:04
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Local theory of Fuchsian singular points

  • Monodromy and holonomy.
  • Growth of multivalued solutions.
  • Tame singularities.
  • Principal example: the Euler system \dot X=\frac At X, A\in\mathfrak{gl}(n,\mathbb C). Solution:
    X(t)=t^A=\exp (A \ln t), monodromy \Delta X(t) =X(t)M, M=\mathrm e^{2\pi\mathrm i A}.
  • Fuchsian condition.
  • Gauge classification of linear systems, A(t)\Longleftrightarrow \dot H(t)H^{-1}(t)+H(t)A(t)H^{-1}(t).
  • Meromorphic gauge classification of tame (regular) systems.
  • Holomorphic gauge classification of Fuchsian singularities: A(t)=\frac 1t(A_0+tA_1+t^2A_2+\cdots),
    A_0=\Lambda+\mathrm N, \Lambda=\mathrm{diag}(\lambda_1,\dots,\lambda_n), \mathrm N^n=0.
  • Resonances (integer differences between eigenvalues of A_0.
  • Holomorphic Eulerization of non-resonant Fuchsian singularities.

Reference: [IY], section 16.

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