Organized by Mark de Cataldo, Alexander Kirillov, Jr., and Robert Lazarsfeld.

Dates: November 18 – 22, 2013

Attendee List Download Talk ScheduleView Videos

Given a smooth algebraic surface X, the collection of all n-tuples of points on X (or, more precisely, finite subschemes of X of length n) are parametrized by the punctual Hilbert scheme of X. Over the past two decades. these remarkable parameter spaces have come up in a surprisingly wide range of mathematical areas ranging from algebraic geometry to representation theory to combinatorics. This workshop will expose graduate students to the geometry of these Hilbert schemes, as well as some of the questions to which they have been applied. In addition to basic theory, the workshop will also cover topics such as computation of cohomology of Hilbert schemes, relation to McKay correspondence and representations of Heisenberg and affine Lie algebras, hyperkahler geometry and the n! conjecture.

The lectures on the basic theory, computation of cohomology and relations to representation theory will be given by Robert Lazarsfeld, Mark Andrea de Cataldo and Alexander Kirillov Jr., respectively. The ones on more advanced topics will be given by outside experts, including Izzet Coskun. We expect that some single-topic lectures will be given by the graduate participants in the workshop. There will also be a daily discussion session.

Please note that this workshop is partially funded by the National Science Foundation. In addition to the acknowledgement guidelines available here, please include the following acknowledgment in any published manuscripts or papers: *This research was supported in part by the National Science Foundation under Grant No. NSF *DMS-10-578.

*All workshop talks and discussion periods will be held in the Simons Center Lecture Hall (Room 102)

*More information of the Food Physics Event on Wednesday November 20th can be seen here.

*The Math Colloquium talk will be held in the Simons Center Lecture Hall (Room 102) on Thursday at 4pm.

#### Graduate Workshop on Geometry of Hilbert schemes Schedule

Time | Title | Presenters | Video |

8:00am | Registration/Breakfast begins (8:00am-9:00am) | SCGP Lobby and Cafe | |

9:00am | Introduction to Hilbert schemes of smooth surfaces I | Robert Lazarsfeld | video |

10:30am | Coffee Break | SCGP Cafe | |

11:00am | Heisenberg algebra and Fock space | Alexander Kirillov | video |

12:30pm | Lunch | SCGP Cafe | |

1:45pm | Picard Group and Ample Cone of Hilbert Scheme of Points on a Surface | David Stapleton | video |

2:30pm | Short Break | ||

2:45pm | Dynkin diagrams and McKay correspondence | Ken Ascher | video |

3:30pm | Tea Time | SCGP Cafe | |

4:00pm | Punctual Hilbert scheme of points on surfaces | Xudong Zheng | video |

4:30pm | An introduction to the deformation of Hilb P2 | Chunyi Li | video |

Time | Title | Presenters | Video |

8:00am | Breakfast begins (8:00am-9:00am) | SCGP Cafe | |

9:00am | Introduction to Hilbert schemes of smooth surfaces II | Robert Lazarsfeld | video |

10:30am | Coffee Break | SCGP Cafe | |

11:00am | Cohomology of Hilbert schemes | Mark de Cataldo | video |

12:30pm | Lunch | SCGP Cafe | |

1:45pm | An intersection number | Zili Zhang | video |

2:30pm | Short Break | ||

2:45pm | Simple and affine Lie algebras | Galyna Dobrovolska | video |

3:30pm | Tea Time | SCGP Cafe | |

4:00pm | Introduction to Hilbert schemes on singular curves | Eugene Gorsky | video |

Time | Title | Presenters | Video |

8:00am | Breakfast begins (8:00am-9:00am) | SCGP Cafe | |

9:00am | The action of the Heisenberg algebra | Mark de Cataldo | video |

10:30am | Coffee Break | SCGP Cafe | |

11:00am | Geometric realization of affine Lie algebras | Alexander Kirillov | video |

12:30pm | Lunch | SCGP Cafe | |

1:45pm | Hilb T*C and Higgs bundles | Daniele Boccalini | video |

2:30pm | Short Break | ||

2:45pm | Tautological bundles on the Hilbert scheme of points | Brooke Ullery | video |

3:30pm | Tea Time | SCGP Cafe | |

4:00pm | Hilbert schemes of singular curves and Catalan numbers | Eugene Gorsky | video |

4:30pm | On the local geometry of Hilbert Schemes | Joachim Jelisiejew | video |

5:30pm | Food Physics Event- optional for participants | SCGP Gallery and 102 |

Time | Title | Presenters | Video |

8:00am | Breakfast begins (8:00am-9:00am) | SCGP Cafe | |

9:00am | Hilbert schemes and Hyperkahler manifolds | Giulia Sacca | video |

10:30am | Coffee Break | SCGP Cafe | |

11:00am | Interpolation and the birational geometry of the Hilbert scheme of points | Izzet Coskun | video |

12:30pm | Lunch | SCGP Cafe | |

2:00pm | Gaeta Resolutions of Ideal Sheaves and Effective Divisors on Relative Jacobians | Matthew Woolf | video |

2:30pm | Short Break | ||

2:45pm | Motivic Analytic Number Theory | Daniel Litt | video |

3:30pm | Tea Time | SCGP Cafe | |

4:00pm | Math Colloquium: Mirror Symmetry: an overview (Room 102) | Aleksey Zinger | video |

6:00pm | Workshop Banquet: Complimentary for all funded particpants | SCGP Cafe |

Time | Title | Presenters | Video |

8:00am | Breakfast begins (8:00am-9:00am) | SCGP Cafe | |

9:00am | Hilbert schemes, Procesi bundles and the n! conjecture I | Ivan Loseu | video |

10:30am | Coffee Break | SCGP Cafe | |

11:00am | Bridgeland stability and the birational geometry of the Hilbert scheme of points | Izzet Coskun | video |

12:30pm | Lunch | SCGP Cafe | |

1:45pm | Hilbert schemes, Procesi bundles and the n! conjecture II | Ivan Loseu | video |

3:30pm | Tea Time | SCGP Cafe |