It so happened that at the beginning of 2016 I gave a talk on the conference “Geometric aspects of modern dynamics” in Porto, delivered a minicourse at Journées Louis Antoine in Rennes and wrote an expository paper for the European Mathematical Society Newsletter, all devoted to the same subject. The subject, provisionally dubbed as “Analgebraic geometry”, deals with algebraic-like properties (especially from the point of view of intersection theory) of real and complex analytic varieties defined by ordinary and Pfaffian differential equations with polynomial right hand sides. Thus
analgebraic = un-algebraic + analytic + algebraic (background) + weak algebraicity-like properties.
It turns out that this analgebraic geometry has very intimate connections with classical problems like Hilbert 16th problem, properties of periods of algebraic varieties, analytic number theory and arithmetic geometry.
Any remarks and comments will be highly appreciated.
P.S. Video records (in French) are available from this page.