Wednesday January 31, 2018 11:30 AM 5127
 Marlon de Oliveira Gomes, Stony Brook University
Organizational MeetingWe will have a short organizational meeting to set the course for the seminar which will ultimately be aimed at studying stability conditions and wallcrossing phenomena.

Wednesday February 07, 2018 11:30 AM 5127
 Frederik Benirschke, Stony Brook University
Introduction to GITIn this talk, we will give some indication of where quotients arise and are important in geometry and then introduce Geometric Invariant Theory as an algebrogeometric approach to producing them, together with some examples.

Wednesday February 14, 2018 11:30 AM 5127
 Marlon de Oliveira Gomes, Stony Brook University
Projective GIT quotientsIn 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 5127
 Yoonjoo Kim, Stony Brook University
The HilbertMumford Criterion for StabilityIn 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 1parameter subgroups, complete with examples.

Wednesday February 28, 2018 11:30 AM  12:30 AM 5127
 Matthew Lam, Stony Brook University
Symplectic reductionThe talk aims to provide a selfcontained introduction to the MarsdenWeinstein quotient, a.k.a. symplectic reduction.

Wednesday March 07, 2018 11:30 AM  12:30 AM 5127
 John Sheridan, Stony Brook University
The KempfNess theoremIn this talk, we will briefly recall the notions of the GIT quotient of an algebraic Gvariety (G reductive, complex) and the symplectic quotient of a Hamiltonian Kmanifold (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 KempfNess theorem. Timeallowing, 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 5127
 John Sheridan, Stony Brook University
Variations of GIT quotientsIn this talk we will recall the dependence of a projective GIT quotient on a Glinearization 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 wallcrossing phenomenon.

Wednesday April 04, 2018 11:30 AM  12:30 AM 5127
 Alexandra Viktorova, Stony Brook University
Holomorphic structures on complex vector bundles IWe will discuss the relation between holomorphic structures on vector bundles over curves, Dolbeault operators, and unitary connections. We will introduce the AtiyahBott 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 5127
 Michael Albanese, Stony Brook University
Holomorphic structures on complex vector bundles, IILast time we discussed the space of unitary connections and the action of the gauge group. This time we will define a symplectic form (the AtiyahBott 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 5127
 Marlon Gomes, Stony Brook University
Holomorphic structures on complex vector bundles IIILast time we described the AtiyahBott 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 infinitedimensional analog of the KempfNess theorem to describe the algebraic (GIT) counterpart of this construction.

Wednesday April 25, 2018 11:30 AM  12:30 AM 5127
 François Greer, Stony Brook University
Variations of moduli of stable bundles. I will introduce algebrogeometric 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 5127
 Timothy Ryan, Stony Brook University
Wallcrossing and Bridgeland StabilityBridgeland 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 NeronSeveri 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.

