"Hilbert 16-th" is one of the most persistent problems in the famous Hilbert's list. It asks: What may be said about the number and location of limit cycles of a polynomial vector fields of given degree in the plane? Whilst many particular results about the location are obtained the very existence of the Hilbert number H(n), an upper bound for the number of limit cycles for a polynomial vector field of degree n, is not known, even for n = 2.

In these lectures we will discuss two statements:

A polynomial vector field in the plain has but a finite number of limit cycles.

Limit cycles of an analytic vector field cannot accumulate to a polycycle of this field.

The second one requires a thorough analysis of the monodromy maps of polycycles.

This study gives rise to two (relatively) new classes of maps: almost regular germs and functional cochains. In the first lecture I will give a survey of the state of matter with H-16. In the next two lectures I will describe two main tools of the study of functional cochains: the Phragmen-Lindelof theorem for cochains, and super exact asymptotic series.

I will discuss my own approach to the problems and will not touch the Ecalle's one.

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Let these lectures be a modest contribution to my dream of the peaceful mankind living as one family on this Globe that is so small.