2018-12-15  10:10 hrs.

### Topological structure of the leaves of polynomial foliations on $\mathbb C^2$

#### Yury KudryashovNational Research University “Higher School of Economics”

Given a polynomial differential equation in $\mathbb C^2, \dot z=P(z, w), \dot w=Q(z, w)$ with complex time, we study the fibers of the corresponding foliation, i.e., the splitting of $\mathbb C^2$ into (real two-dimensional) trajectories of this vector field.

A typical polynomial foliation on $\mathbb C^2$ has many properties different from those of a typical (polynomial) foliation on $\mathbb R^2$. For example, all fibers are dense in $\mathbb C^2$, and there are infinitely many limit cycles.

However, many questions are still open, for example:

* What are the topological types of the fibers of a typical polynomial foliation?
* Take a polynomial foliation such that all leaves of its restriction to some domain in $\mathbb C^2$ are cylinders (this is called an identical cycle). Is it possible to destroy this picture by a small perturbation of polynomials P,Q?

The talk will start with a brief review of the current state of the art, then I will talk about some recent results by Nataliya Goncharuk and myself.

#### Palapa Guillermo Torres -- Miércoles 14 de mayo de 2014, 12:00 horas

Unidad Cuernavaca del Instituto de Matemáticas UNAM