Limit cycles of planar vector fields: Hilbert's 16th problem and o-minimality

TL;DR

This paper explores the connection between Hilbert's 16th problem, specifically limit cycles in planar vector fields, and the mathematical framework of o-minimality, highlighting recent developments and their implications.

Contribution

It introduces recent work linking Hilbert's 16th problem to o-minimality, providing insights into the structure and behavior of limit cycles in planar vector fields.

Findings

Establishes a connection between limit cycles and o-minimal structures

Highlights recent progress in understanding Hilbert's 16th problem

Provides a perspective on the application of o-minimality to dynamical systems

Abstract

I discuss some recent work linking certain aspects of the second part of Hilbert's 16th problem to the theory of \hbox{o-minimality}. These notes are adapted from a lecture I gave in the Jour fixe seminar series at the Zukunfts\-kolleg of Universit\"at Konstanz in June 2017.