“Auxiliary Lesson” שעור עזר) #8) December 13, 2007

Finitely generated subgroups of , I. Formal theory. 

1. Formal normal form for a single holomorphic self-map from . Parabolic germs.
2. Bochner theorem on holomorphic linearization of finite groups.
3. Stratification of the subgroup of parabolic germs .
4. Tits alternative for finitely generated subgroups of : every such subgroup is either metabelian (its commutator is commutative, e.g., trivial), or non-solvable (all iterated commutators are nontrivial).
5. Centralizers and symmetries: formal classification of solvable subgroups.
6. Integrable germs and their holomorphic linearizability.

Recommended reading: Section 6 (first part) from the book (printing disabled)

Disclaimer applies, as usual

“Auxiliary Lesson” שעור עזר) #5) November 22, 2007

Formal linearization and obstructions. Poincare theorem

1. Formal equivalence of formal vector fields (total recall)
2. (Additive) Resonances
3. Poincare formal linearization theorem
4. Proof of the Poincare theorem:
   • Homological equation
   • Commutator with diagonal linear vector field
   • Stabilization of the series
5. Resonant monomials. Resonant normal form. Poincare–Dulac paradigm.
6. Formal classification of formal self-maps. Multiplicative resonances.
7. Survey of further results. Formal types of line and planar singularities.

Reading material: Section 4 from the book (printing disabled).Disclaimer (alas, still required) .

“Auxiliary Lesson” שעור עזר) #4) November 15, 2007

Formal series, formal vector fields: Flows and embedding

1. Formal series algebra . Formal vector fields (derivations) . Formal equivalence of vector fields. Truncation. Convergence in the formal algebra  .
2. Formal inverse function theorem. Geometric series.
3. Integration and formal flow of vector fields. Exponent.
4. Embedding in the flow. Linear case. Matrix logarithms.
5. Embedding in the flow and formal logarithms.

Reading Section 3 from the textbook.