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2018-02-18  16:14 hrs.

The topology of solutions of polynomial vector fields in \(\mathbb{C}^2\)

Nataliya Goncharuk
Higher School of Economics

The phase portrait of a polynomial equation in \(\mathbb{C}^2\) with complex time is a holomorphic foliation of \(\mathbb{C}^2\). The talk is devoted to the topology of (generic or some specific) leaves of (generic or some specific) polynomial foliations.

For a generic analytic foliation of \(\mathbb{C}^2\), at most countable number of leaves are topological cylinders, all other leaves are topological discs. In the space \(A_n\) of foliations of \(\mathbb{C}^2\) given by polynomials of degree n, the analogous result is not known. However, there are some partial results.

Limit cycles (Ilyashenko, 1978; improved by others):
For a generic vector field from \(A_n\), there is a countable number of (homologically independent) non-contractible loops on the leaves.
Thus either a countable number of leaves are at least cylinders, or there is a leaf of infinite genus.

Separatrix connections (Volk, 2006):
For a dense subset in \(A_n\), there is a separatrix connection, i.e. two different fixed points of the vector field have a common separatrix. This result is not directly related to the topology of the leaves, but the ideas of the proof are used to prove the following statement.

One handle (NG&YK): For a dense subset in \(A_n\), there is a leaf with at least one handle.

Infinite genus (NG&YK): There are examples (with positive dimension in \(A_n\)) of foliations having the leaves of infinite genus.

I will outline the proofs of these four results.


Palapa Guillermo Torres -- Jueves 22 de mayo de 2014, 16:00 horas

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