Difference between revisions of "Algebraic Geometry Seminar Spring 2012"

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(Spring 2012)
(Abstracts)
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Abstract: Let X be a variety equipped with a G-action and G-invariant regular function, w. The GIT quotient X//G depends on the additional data of a G-linearized line bundle. As one varies the G-linearized line bundle, X//G changes in a very controlled manner. We will discuss how the category of matrix factorizations, mf(X//G,w), changes as the G-linearized line bundle varies. We will focus on the case where G is toroidal. In this case, we show that, as one travels through a wall in the GIT cone, semi-orthogonal components coming from the wall are either added or subtracted.
 
Abstract: Let X be a variety equipped with a G-action and G-invariant regular function, w. The GIT quotient X//G depends on the additional data of a G-linearized line bundle. As one varies the G-linearized line bundle, X//G changes in a very controlled manner. We will discuss how the category of matrix factorizations, mf(X//G,w), changes as the G-linearized line bundle varies. We will focus on the case where G is toroidal. In this case, we show that, as one travels through a wall in the GIT cone, semi-orthogonal components coming from the wall are either added or subtracted.
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===Laurentiu Maxim===
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''Characteristic classes of Hilbert schemes of points via symmetric products''
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I will explain a formula for the generating series of (the push-forward under the Hilbert-Chow morphism of) homology characteristic classes of the Hilbert schemes of points for a smooth quasi-projective variety of arbitrary pure dimension. The  result is based on a geometric construction of a motivic exponentiation generalizing the notion of motivic power structure, as well as a formula for the generating series of homology characteristic classes of symmetric products.

Revision as of 18:23, 6 February 2012

The seminar meets on Fridays at 2:25 pm in Van Vleck B215.

The schedule for the previous semester is here.

Spring 2012

date speaker title host(s)
January 27 Sukhendu Mehrotra (Madison) Generalized deformations of K3 surfaces (local)
February 3 Travis Schedler (MIT) Symplectic resolutions and Poisson-de Rham homology Andrei
February 10 Matthew Ballard (UW-Madison) Variation of GIT for gauged Landau-Ginzburg models (local)
February 17 Arend Bayer (UConn) TBD Andrei
February 24 Laurentiu Maxim (UW-Madison) Characteristic classes of Hilbert schemes of points via symmetric products local
March 2 Marti Lahoz (Bonn) TBD Sukhendu
March 9 Shilin Yu (Penn State) TBD Andrei
March 16 Weizhe Zheng (Columbia) TBD Tonghai
March 23 Ryan Grady (Notre Dame) Twisted differential operators as observables in QFT. Andrei
April 27 Ursula Whitcher (UW-Eau Claire) TBA Matt
May 4 Mark Andrea de Cataldo (Stony Brook) TBA Laurentiu

Abstracts

Sukhendu Mehrotra

Generalized deformations of K3 surfaces

Travis Schedler

Symplectic resolutions and Poisson-de Rham homology

Abstract: A symplectic resolution is a resolution of singularities of a singular variety by a symplectic algebraic variety. Examples include symmetric powers of Kleinian (or du Val) singularities, resolved by Hilbert schemes of the minimal resolutions of Kleinian singularities, and the Springer resolution of the nilpotent cone of semisimple Lie algebras. Based on joint work with P. Etingof, I define a new homology theory on the singular variety, called Poisson-de Rham homology, which conjecturally coincides with the de Rham cohomology of the symplectic resolution. Its definition is based on "derived solutions" of Hamiltonian flow, using the algebraic theory of D-modules. I will give applications to the representation theory of noncommutative deformations of the algebra of functions of the singular variety. In the examples above, these are the spherical symplectic reflection algebras and finite W-algebras (modulo their center).

Matthew Ballard

Variation of GIT for gauged Landau-Ginzburg models

Abstract: Let X be a variety equipped with a G-action and G-invariant regular function, w. The GIT quotient X//G depends on the additional data of a G-linearized line bundle. As one varies the G-linearized line bundle, X//G changes in a very controlled manner. We will discuss how the category of matrix factorizations, mf(X//G,w), changes as the G-linearized line bundle varies. We will focus on the case where G is toroidal. In this case, we show that, as one travels through a wall in the GIT cone, semi-orthogonal components coming from the wall are either added or subtracted.

Laurentiu Maxim

Characteristic classes of Hilbert schemes of points via symmetric products

I will explain a formula for the generating series of (the push-forward under the Hilbert-Chow morphism of) homology characteristic classes of the Hilbert schemes of points for a smooth quasi-projective variety of arbitrary pure dimension. The result is based on a geometric construction of a motivic exponentiation generalizing the notion of motivic power structure, as well as a formula for the generating series of homology characteristic classes of symmetric products.