Refined invariants for real elliptic curves

The talk is devoted to several real and tropical enumerative problems. We suggest new invariants of the projective plane (and, more generally, of toric surfaces) that arise as results of an appropriate enumeration of real elliptic curves. These invariants admit a refinement similar to the one introduced by Grigory Mikhalkin in the rational case. We discuss tropical counterparts of the elliptic invariants under consideration and prove a recursive formula allowing one to compute them.
This is a joint work with Eugenii Shustin.