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Graduate Workshop on Geometry of Hilbert schemes: November 18 – 22, 2013

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