sissa, Trieste, july 3-7 2017

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.