12 Encuentro Jessica Jaurez

This talk presents a detailed study of holomorphic foliations on complex surfaces, specifically focusing on the analytic classification of pairs of foliations and the properties of their tangency curves. It explores how these tangency curves serve as analytical invariants, how they behave under coordinate changes, and their relationship with the analytic classification spaces of the foliations themselves. The results address fundamental questions for mathematicians working in complex dynamical systems, foliation theory, and singularity theory.

High-Level Overview

The presentation delves into the analytic classification of pairs of holomorphic foliations defined near the origin in complex two-dimensional spaces, examining the geometric and analytic properties of their tangency curves—curves where two distinct foliations meet tangentially. These tangency curves are shown to be powerful invariants that reflect deep structural information about the foliations. The talk systematically develops frameworks to understand how these invariants behave under analytic equivalences and coordinate changes, connecting these ideas to the broader classification problem of complex foliations. It highlights recent advances in describing these classes via infinite-dimensional functional spaces and explores how the tangency curves' analytic classes relate to the foliations’. This is crucial for researchers interested in complex geometry, singular foliations, and dynamical systems.