Saturday, March 1, 2008

Local theory of regular singular points of linear systems

This lecture, in an exceptional way, will take place on Sunday, 16:00-18:00, in the Room 261.

Regular and irregular singularities: growth matters.
Local gauge equivalence (holomorphic, meromorphic, formal). Meromorphic classification of regular singularities.
Fuchsian singularities as a particular class of regular singularities (Sauvage lemma).
Formal classification of Fuchsian singularities (Poincaré-Dulac theorem revisited). Resonances. Levelt upper triangular normal form.
Coincidence of formal and holomorphic classification in the Fuchsian case.
Integrability of the normal form.
Towards global theory of Fuchsian systems on $\mathbb C P^1$ : Monopoles as special classes of rational matrix functions.

Reminder: Today (actually, on Friday) was the deadline for submission of the home exam

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Formal normal form for a single holomorphic self-map from $\text{Diff}(\mathbb C^1,0)$ . Parabolic germs.
Bochner theorem on holomorphic linearization of finite groups.
Stratification of the subgroup of parabolic germs $\text{Diff}_1(\mathbb C^1,0)$ .
Tits alternative for finitely generated subgroups of $\text{Diff}(\mathbb C^1,0)$ : every such subgroup is either metabelian (its commutator is commutative, e.g., trivial), or non-solvable (all iterated commutators are nontrivial).
Centralizers and symmetries: formal classification of solvable subgroups.
Integrable germs and their holomorphic linearizability.

Disclaimer applies, as usual

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