# 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.