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 school put 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).
The school featured courses by:
Chiu-Chu Melissa Liu - Columbia University
Cristina Manolache - Imperial College London