Poincaré-Reeb Graphs of Real Algebraic Domains
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Overview
Syllabus
Intro
Setting and goals
Algebraic domains
Overview
The Poincaré-Reeb graph
The asymptotic Poincaré-Reeb tree
Reformulation
The polar curve
Choosing a generic projection
The generic asymptotic Poincaré-Reeb tree
The discriminant locus
Genericity hypotheses
1. Positive asymptotic snake
2. Arnold's snake (one variable)
Idea of the proof
The construction
Separable permutations
Nonseparable permutation - example
Separable tree
Passing to the univariate case
The contact tree
Answer in the univariate case
Construction of the desired bivariate polynomial f
Flip-Flop Algorithm
Vertical planes
The topological critical set
Domain of finite type
Vertical equivalence
Complete invariant
Invariance of the Euler characteristic
Strategy of the proof
Non-compact Poincaré-Reeb graphs
Taught by
Centre International de Rencontres Mathématiques