Cristina Manolache
Boundary contributions to enumerative invariants

Abstract: Enumerative questions have a very long history in Mathematics
and have been revolutionised in the nineties with the construction of the moduli space
of stable maps and the machinery allowing us to integrate on these very singular spaces.
By now we have several moduli spaces on which we can integrate, but the invariants
we obtain are very often not enumerative.
My goal is to investigate how different compactifications of moduli spaces of curves on a given variety
give rise to different invariants.
I will first give several examples of compactifications  such as stable maps, reduced maps, and quasi-maps.
Then, I will explain virtual classes and why it is difficult to see how components
of a moduli space contribute to virtual classes. 
In the end, I will give examples of boundary contributions to enumerative invariants.
More precisely, I will discuss the relationship between Gromov--Witten invariants
and reduced invariants (or Gopakumar--Vafa invariants)
and the relationship between Gromov--Witten invariants and quasi-map invariants.