The Reading Group
The goal of this reading group is to go through the various technical steps that appear in the definition of Gromov-Witten invariants and related notions such as quantum cohomology. This includes:
The construction of the moduli space of stable maps
Intersection theory on algebraic stacks
The construction of the virtual fundamental class
and more!
After that, the plan is to spend some time exploring results related to GW invariants (e.g. the Quantum Lefschetz theorem, etc) and work our way towards the theory of Hodge atoms its application to irrationality questions.
Where?
Tuesdays at 2pm in Huxley 658.
Session 1 (Dan): Intersection theory I.
Cones, deformation to the normal cone
Chern/Segre lasses
Session 2 (TBD): Intersection theory II.
Gysin maps
Definition of the virtual fundamental class for zero locii of sections of vector bundles.
A few initial worked examples of virtual fundamental classes.
Session 3 (TBD): Behrend-Fantechi I.
Intrinsic cones
Making the virtual fundamental class definition.
Session 4 (TBD): Behrend-Fantechi II.
Intro to the Cotangent complex, axioms, etc
Perfect obstructions theories
The virtual fundamental class, general definition.
Session 5 (Ines): Application to stable maps
Compute with examples!
Definition of GW invariants
Basic properties (string equation, etc)
Session 6 (Paul): Properties of the virtual fundamental class
Functoriality, compatibilities
Session 7 (TBD): Perfect obstruction theories from deformation theory.
Session 8 (TBD): Quantum cohomology
Session 9 (TBD): The Givental formalism
Session 10 (TBD): Classical mirror symmetry
The quintic threefold
Toric varieties
Session 11 (TBD): Atoms
Introduction to the theory of atoms
Applications to rationality problems
Session 11 + n: Still TBD! A few topics we might touch upon.
More deformation theory
Intersection theory on algebraic stacks
Orbifolds
Connections to string theory and topological quantum field theories.