Poincaré-Reeb Graphs of Real Algebraic Domains

Explore a conference talk on Poincaré-Reeb graphs of real algebraic domains, delivered by Miruna-Stefana Sorea at the Centre International de Rencontres Mathématiques in Marseille, France. Delve into the intricacies of algebraic domains, the Poincaré-Reeb graph, and the asymptotic Poincaré-Reeb tree. Examine concepts such as polar curves, generic projections, and discriminant loci. Investigate Arnold's snake, separable permutations, and the Flip-Flop Algorithm. Learn about vertical planes, topological critical sets, and domains of finite type. Understand the invariance of the Euler characteristic and explore non-compact Poincaré-Reeb graphs. Access this 59-minute recording, complete with chapter markers, keywords, abstracts, and bibliographies, through CIRM's Audiovisual Mathematics Library for a comprehensive exploration of real algebraic geometry.

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