What is the current situation of the second part of the Hilbert 16th problem? What are the most updated news on this problem?
An update from April 2018 is given by Patrick Speissegger.
The idea, going back to Poincaré, is to reduce the two-dimensional counting problem (counting limit cycles in the plane) to a one-dimensional counting problem (counting certain points on a line). Roussarie (1998) showed that Hilbert’s 16th problem follows if a certain "finite cyclicity conjecture" holds. A tameness condition called "o-minimality" allows to reformulate Roussarie's conjecture as a conjecture of o-minimality. Speissegger discusses special cases where o-minimality can be proven and proposes this approach as a promising way to prove Hilbert's 16th problem.