Real tropical enumerative invariants
The study of Welschinger invariants of real rational surfaces via tropical geometry naturally leads to real tropical enumerative invariants (TEI for short). We discuss two interesting phenomena in the geometry of real TEI which are not understood yet and which suggest challenging problems. The known correspondence theorems state that the count of real algebraic curves subject to certain constraints can be reduced to the count of some tropical curves. The first observation is that, in many situations, the count of tropical curves is invariant with respect to the variation of (tropical) constraints, whereas the count of real algebraic curves is not invariant. A conjectural explanation is the existence of (still unknown) algebraic enumerative invariant. Another interesting phenomenon has been observed when comparing the complex and real TEI: the complex TEI are related to the structure of a tropical cycle and the tropical intersection theory of the moduli spaces of tropical curves, whereas the real TEI define topological cycles in these moduli spaces. We demonstrate several examples and formulate open problems.