What would be the fate of the Hilbert 16th problem?

$$\text{Is} \; H(n)< \infty?$$

When I was a PHD student, working on limit cycle theory, my supervisor encouraged me to learn the elements and basics of Noncommutative Geometry in order to find some new interpretations for the concept of "The number of limit cycles". I have deep admiration to him, since he was a different supervisor among other mathematicians in Iran.

The second part of this note (http://arxiv.org/abs/1302.0001) is an $\varepsilon$- try to find a possible new interpretation for this concept.

Moreover I am interested in the fate of the following question:

Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)

I am also interested to know the answers to the questions listed in this post:

Proposals for polymath projects

BTW my yahoo email is "alitghv".