RTG Seminar

from Monday
January 01, 2018 to Thursday
May 31, 2018
Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars

Wednesday
January 31, 2018

11:30 AM
5-127
Marlon de Oliveira Gomes, Stony Brook University
Organizational Meeting

We will have a short organizational meeting to set the course for the seminar which will ultimately be aimed at studying stability conditions and wall-crossing phenomena.


Wednesday
February 07, 2018

11:30 AM
5-127
Frederik Benirschke, Stony Brook University
Introduction to GIT

In this talk, we will give some indication of where quotients arise and are important in geometry and then introduce Geometric Invariant Theory as an algebro-geometric approach to producing them, together with some examples.


Wednesday
February 14, 2018

11:30 AM
5-127
Marlon de Oliveira Gomes, Stony Brook University
Projective GIT quotients

In this talk, we will continue with our introduction to GIT by looking this time at projective quotients. We will try to give some indication of the relative delicacy of this case compared to the optimal situation we saw with affine quotients.


Wednesday
February 21, 2018

11:30 AM
5-127
Yoonjoo Kim, Stony Brook University
The Hilbert-Mumford Criterion for Stability

In this talk we will focus our attention on the notion of stability previously introduced for these actions by reductive groups. We will state a theorem, due in part to Hilbert and Mumford respectively, which detects the (semi)stable points of an action using an eigenanalysis of the 1-parameter subgroups, complete with examples.


Wednesday
February 28, 2018

11:30 AM - 12:30 AM
5-127
Matthew Lam, Stony Brook University
Symplectic reduction

The talk aims to provide a self-contained introduction to the Marsden-Weinstein quotient, a.k.a. symplectic reduction.


Wednesday
March 07, 2018

11:30 AM - 12:30 AM
5-127
John Sheridan, Stony Brook University
The Kempf-Ness theorem

In this talk, we will briefly recall the notions of the GIT quotient of an algebraic G-variety (G reductive, complex) and the symplectic quotient of a Hamiltonian K-manifold (K compact, real). We will then show that when a space X fits both descriptions, these two quotient spaces coincide - this is the content of the Kempf-Ness theorem. Time-allowing, we will study an application to decomposing tensor products into SL(2,C)-representations, originally due to Clebsch and Gordan.


Wednesday
March 28, 2018

11:30 AM - 12:30 AM
5-127
John Sheridan, Stony Brook University
Variations of GIT quotients

In this talk we will recall the dependence of a projective GIT quotient on a G-linearization of a line bundle. We will then set up a framework in which to vary this linearization and observe that different (but related) GIT quotients arise as we cross walls between chambers in an appropriate space. This will be our first example of a wall-crossing phenomenon.


Wednesday
April 04, 2018

11:30 AM - 12:30 AM
5-127
Alexandra Viktorova, Stony Brook University
Holomorphic structures on complex vector bundles I

We will discuss the relation between holomorphic structures on vector bundles over curves, Dolbeault operators, and unitary connections. We will introduce the Atiyah-Bott symplectic form on the space of connections, and show that the action of the group of changes of gauge is Hamiltonian.


Wednesday
April 11, 2018

11:30 AM - 12:30 AM
5-127
Michael Albanese, Stony Brook University
Holomorphic structures on complex vector bundles, II

Last time we discussed the space of unitary connections and the action of the gauge group. This time we will define a symplectic form (the Atiyah-Bott symplectic form) on this space and show that the gauge group preserves it; that is, the gauge group acts by symplectomorphisms. This is precisely the setting where we expect a moment map, and in this case it has a nice geometric meaning. We will then (attempt to) construct a symplectic reduction.


Wednesday
April 18, 2018

11:30 AM - 12:30 AM
5-127
Marlon Gomes, Stony Brook University
Holomorphic structures on complex vector bundles III

Last time we described the Atiyah-Bott construction of a moduli space of holomorphic structures on a complex vector bundle over a Riemann surface, via symplectic reduction. In this talk, I will discuss an infinite-dimensional analog of the Kempf-Ness theorem to describe the algebraic (GIT) counterpart of this construction.


Wednesday
April 25, 2018

11:30 AM - 12:30 AM
5-127
François Greer, Stony Brook University
Variations of moduli of stable bundles.

I will introduce algebro-geometric notions of stability for vector bundles on higher dimensional bases X, all of which depend on the choice of a polarization (an ample class) H. Next, I will outline Gieseker's construction of the moduli space of stable bundles. As H varies, the moduli space of bundles undergoes birational modifications, according to a wall and chamber decomposition of the ample cone. We will work out examples of this phenomenon on rational and K3 surfaces.


Wednesday
May 02, 2018

11:30 AM - 12:30 AM
5-127
Timothy Ryan, Stony Brook University
Wall-crossing and Bridgeland Stability

Bridgeland generalized the notions of slope/Gieseker stability for vector bundles (sheaves) to stability conditions on the derived category of a variety. He proved that the space of all of these conditions was a manifold which has a wall and chamber structure. When the underlying variety is a surface, this structure has been used to study the geometry of moduli spaces of sheaves on the surface. In particular, there is often a correspondence between these walls and the walls in the Neron-Severi space where the "model" of the moduli space changes. Keeping the technicalities as minimal as possible, we will attempt to explain an example of this phenomenon in the setting of Hilbert schemes of points on the projective plane.


Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars