Pseudo-tropical curves

I will describe joint work with Sergei Lanzat.
Tropical geometry provides a new piece-wise linear approach to algebraic
geometry. The role of algebraic curves is played by tropical curves -
planar metric graphs with certain requirements of balancing, rationality
of slopes and integrality. A number of classical enumerative problems can
be easily solved by tropical methods.
Lately is became clear that a more general approach also makes sense and
seem to appear in other areas of mathematics and physics.
We consider a generalization of tropical curves, removing requirements of
rationality of slopes and integrality and discuss the resulting theory and
its interrelations with other areas. Balancing conditions are interpreted
as criticality of a certain action functional. A generalized Bezout theorem
involves Minkowsky sum and mixed areas. A problem of counting curves passing
through an appropriate collection of points turns out to be related to
quantum tori Lie algebras and quadratic Plücker relations in Gr(2,4).
If time permits, we will also discuss new recursive relations for this count
(in the spirit of Kontsevich and Gromov-Witten).