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Gromov-Witten invariants, which ''count'' curves (with appropriate extra conditions) on smooth projective varieties,
were introduced more than two decades ago;
motivated by high energy physics, they ended up revolutionizing enumerative algebraic geometry
and provided a bridge to other branches of mathematics,
such as integrable systems of differential equations.
Since then, their scope has been expanded in different directions
(e.g. relaxing the smoothness conditions, replacing the variety by a stack, allowing torus actions),
and techniques have been introduced for their computation; moreover, a plethora of other invariants
using the same basic ideas has been introduced,
leading to fruitful investigations on relationships among them.
The Summer School in Enumerative Geometry will bring doctoral students, post-docs, and anyone interested
from a review of the basic construction to current, state-of-the art research in this field, with a special focus on invariants
for Calabi-Yau threefolds, the richest example both in algebraic geometry and physics.
Many important questions about these varieties are still unanswered,
such as giving a mathematically rigorous definition of the Gopakumar-Vafa invariants
(at the moment only available in the language of theoretical physics).