Localisation in Enumerative Geometry

This is a PhD course I teach at SISSA in Spring 2021. If you wish to access the material (recordings of lectures, slides), or attend the course, please drop me an email.

Unless specified otherwise, we meet on Zoom Thursday morning from 11:00 - 13:00.

TOPIC: After an introduction to the classical moduli spaces in Algebraic Geometry (such as Grassmann bundles, Quot and Hilbert schemes), we will cover the foundations of equivariant cohomology and the Atiyah-Bott localisation formula, which will be applied to solve classical enumerative problems in Algebraic Geometry. Then we shall move to the study of virtual invariants defined via virtual fundamental classes, and we will prove the Graber-Pandharipande virtual localisation formula, that will be applied to computations in Donaldson-Thomas theory. If time permits, we shall explore a special case of the Gromov-Witten/Donaldson-Thomas correspondence for the resolved conifold.