Abstract: A classical problem in enumerative geometry is to determine the number
of plane algebraic rational curves of degree d passing through 3d-1
points (eg the number of lines passing through two points). We will
address several generalizations of this problem in real, tropical, and
symplectic geometries. The first part will mainly be devoted to tropical
refined invariants defined by Block and Göttsche, and generalized by
Göttsche and Schroeter. Relations discussed there are suggested by a
formula relating enumerative invariants of two real symplectic
4-manifolds differing by a surgery along a real Lagrangian sphere. This
formula, whose origin can be traced back to a work by Abramovich and
Bertam, will be discussed in the second part. We will give as well
applications to Welschinger invariants of rational symplectic 4-manifolds.
This talk is the second part of in a series of lectures by Brugallé in relation to his BFS invited professorship.