Event № 606
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 quadratic Plücker relations in Gr(2,4) and some nice Lie algebra. If time permits, we will also discuss new recursive relations for this count (in the spirit of Kontsevich and Gromov-Witten).