TAGSS I - Summer School in Enumerative Geometry
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
and
Cristina Manolache - Imperial College London