Abstract

Enumerative Geometry is an old subject and it was the topic of Hilbert's 15'th problem on his list stated in Paris in 1900. This problem gave strong impetus to the subject and motivated new developments in algebraic geometry. The subject fascinates by its connections with other areas of mathematics such as algebraic geometry and mathematical physics. In this lecture we shall look at certain problems on planar polynomial vector fields from the perspective of enumerative geometry which puts these problems and their solutions in a larger context. The wider view helps in solving some classification problems of polynomial vector fields which are significant for applications giving us for example the global classification of Lotka-Volterra systems. The aim is to provide the bifurcation diagrams for these families in their moduli spaces.