# Holomorphic Dynamics and Complex Foliations

The theory of complex differential equations began in the nineteenth century with the works of Briot, Bouquet, Poincaré, Picard, Darboux, Painlevé, Halphen and Dulac, among many others. When a differential equation is given by polynomials, it defines naturally a foliation by curves in the Euclidean space or one of its compactifications. The main issue is to analyze the dynamics of solutions (the leaves), both locally and globally.

Modern research in the area was taken up by Reeb in France, who was inspired by the works of Painlevé, and Petrovsky, Landis and Iliashenko in Russia, who were motivated by the 16th Hilbert problem. The 70s brought intense development in France, with contributions from Moussu, Mattei, Cerveau, Martinet and Ramis, and in Brazil with the progress made by the group at IMPA. Since then the group has made fundamental contributions in the establishment of important results, often with the collaboration of researchers from other universities.

The phenomena modeled by real polynomial differential equations generate, in a natural way, complex differential equations. The interface between the real equation and its complexified leads to a better understanding of the modeled phenomena. One reason is that the study of the complexified problem allows the use of Complex Analysis and Algebraic Geometry, revealing non apparent aspects of the real problem and producing results that can be interpreted in their original context. Conversely, the study of differential equations of the Picard-Fuchs type and those arising from Gauss-Manin connection results in rigorous proofs of some fundamental theorems of Algebraic Geometry, such as Noether-Lefschetz theorem. Such equations are satisfied by periods of fibrations and gave origin to Hodge theory in Algebraic Geometry.

The research conducted at IMPA ranges from classical questions on integrability through transcendental functions to modern questions about the dynamics of foliations and applications in Algebraic Geometry and Hodge Theory. Some lines of this research are:

- Limit sets of foliations
- Transversal structure of foliations
- Codimension one projective foliations
- Birational geometry of foliations
- Linearizations and normal neighborhoods
- Algebraic solutions of differential equations
- Uniformization of the leaves of a foliation
- Indexes and invariants of projective foliations
- 16
^{th}Hilbert problem and zeros of Abelian integrals - Picard-Fuchs equations and Picard-Lefschetz theory
- Differential equations of modular forms
- Calabi-Yau varieties
- Hodge cycles and algebraic cycles