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Resolution of vector fields and applications to parabolic dynamics

Mini-course - 2023-2024 (1st semester) - Master-PhD Maths - Beijing University and BICMR.

All classes will take place in Room 29, Quan Zhai, BICMR (see link).

Time schedule

2023.10.25-12.13: Wednesdays 10h00-11h30 - room 29 Qyan Zhai, BICMR

Abstract

The study of tangent to the identity germs in dimension 2 or higher is strongly related to the resolution of the associated infinitesimal generator.
After introducing the main protagonists to this story, we will show Seidenberg's resolution of singularities of planar foliations, the classification of the reduced singularities, Camacho-Sad's construction of separatrices, and the recent applications to the existence of parabolic curves for planar tangent to the identity germs.
If time allows, we will also comment on the recent developments in dimension 3.

Personal handwritten notes of the classes

Be careful, there could be mistakes, and it could differ from the given classes.

Class 1, 2023.10.25: 1 - Introduction. pdf
Class 2, 2023.11.01: 2 - Foliations. pdf
Class 3-4, 2023.11.08-15: 3 - Reduction of singularities. pdf
Class 5-6, 2023.11.22-12.06: 4 - Elementary singularities. pdf
Class 7, 2023.12.13: 5 - Existence of separatrices. pdf

Suggested references

Brunella - Birational geometry of foliations - Springer IMPA Monog. (2015).
Friedman - Algebraic surfaces and holomorphic vector bundles - Springer Utx (1998).
Griffiths, Harris - Principles of algebraic geometry - Wiley Classic Library (1994).
Il'yashenko, Yakovenko - Lectures on analytic differential equations - Amer. Math. Soc. GSiM (2008).
Kollar - Lectures on resolution of singularties - Princeton Univ. Press AoMS (2007).
Loray - Pseudo-groupe d'une singularité de feuilletage holomorphe en dimension deux - Springer HAL (2021).
Panazzolo - Resolution of singularities of real-analytic vector fields in dimension three - Acta Math. (2006).
Pelletier - - Éclatements quasi-homogènes - Ann. Fac. Sci. Toulouse (1995).
Spivakovsky - Resolution of singularties, an introduction - Springer HAL (2020).