Enumerative Geometry
Enumerative Geometry is concerned with enumerating objects with certain geometric properties. Historically, this field was concerned with answering questions about the number of solutions to a given system of constraints, say for example
How many lines are there on a smooth cubic surface?,
How many smooth conics are tangent to five general plane conics?
and so forth.
However this field has drastically changed, ever since the introduction of virtual fundamental classes based on the paper of Behrend and Fantechi. The modern perspective on what this field of mathematics is about can be summed up by the quote of Okounkov:
Modern enumerative geometry is not so much
about numbers as it is about deeper
properties of the moduli spaces that
parametrize the geometric objects being
enumerated.
This reading group will be concerned with giving an overview on the current topics and trends of enumerative geometry. We will for the most part work through the book Invitation to Modern Enumerative Geometry by A. Ricolfi.
Schedule:
- Monday, 11am-1pm in Bayes, Meeting Room 1.35, starting October 6th
- Week 1: Introduction to Enumerative Geometry, Noah Dizep
- Week 2: Background Material, Emanuel Roth
- Week 3: Schubert Calculus, Raisa Serova (with Appendix by Sebastián Fuentes)
- Week 4: Hilbert Schemes of points, N/A
- Week 5: Localization, N/A
- Week 6: Virtual Classes, N/A
- Week 7: Degree zero DT invariants, N/A
- Week 8: DT/PT correspondence, N/A
- Week 9: GW/DT correspondence, N/A
- Week 10: Modern Topics in EG, N/A