Date 
Time 
Room 
Speaker 
Title and description 
Taken Notes 
Thursday, March 3, 2011. 
10:30 AM 
Lunt 1st Floor 
Sam Gunningham 
Introduction to nonAbelian Hodge Theory. Discussed example of NarasimhanSeshadri. Identified space of flat, unitary connections with space of polystable complex vector bundles on a complex curve. 
Notes 
Tuesday, March 8, 2011. 
2 PM 
Lunt 1st Floor 
Vladimir Kotov 
Introduction to Higgs Bundles.
I will be talking about augmented bundles (with a focus on Higgs bundles)
on compact Riemann surfaces. I will give all the definitions and examples,
construct corresponding moduli stacks and course moduli spaces. Also if
time permits, I will discuss their geometry. 

Wednesday, April 6, 2011. 
3 PM 
Lunt 1st Floor 
Sam Gunningham 
Variations of Hodge Structure. 
Notes 
Wednesday, April 20, 2011. 
3 PM 
Lunt 1st Floor 
Chris Elliot 
Twistor Spaces.We have seen that one can associate to any variation of Hodge structure on a compact Kahler manifold X a C^x invariant harmonic bundle. Is there an abstraction of the notion of variation of Hodge structure that gives us a description of all harmonic bundles? Simpson showed that the answer is yes, by defining socalled 'Twistor structures': Hodge structures that have 'forgotten' a C^x action. In the context of nonabelian Hodge theory, this is particularly useful when describing a nonabelian Hodge theorem for noncompact manifolds. We will introduce these objects, with examples coming from the twistor spaces associated to hyperkahler manifolds, as studied by Hitchin and others. 

Friday, April 29, 2011. 
3 PM 
Lunt 1st Floor 
Jesse Wolfson 
NAHC for Principal Bundles.
Abstract: Much of the interest in the nonAbelian Hodge correspondence is
that it provides powerful tools for studying the moduli of representations
of the fundamental group of our Kahler manifold. A priori, these
representations take values in GL(n), but the example of Teichmuller space
(equivalently the moduli of representations of a surface group into SL(2)
or the moduli of holomorphic structures on the surface) motivates us to
set up a theory for representations into other groups as well. Setting up
the definitions of principal Higgs bundles, stability, harmonic
structures, etc. requires some care and looks a little abstract at first.
In my talk, I'll try to motivate the definitions and indicate how we
should think about the NAHC, starting with examples, and then moving on to
the abstract definitions.


Friday, April 29, 2011. 
5 PM 
Lunt 1st Floor 
Hiro Tanaka 
Deformation Theory via GoldmanMillson. "Lie algebras control deformation problems." This philosophy has become a big motivator for developing derived algebraic geometry. It's also had great applications including Kontsevich's work on deformation quantization (part of his Fields Medal work). In this talk I'd like to talk about (what I view as) the most geometric and intuitive of examplesthe problem of deforming a flat principal Gbundle. This is due to work of Goldman and Millson, inspired by ideas from Deligne. Here a Lie algebra pops out naturally, and allows us to conclude that singularities of the representation variety into a compact group G are at worst quadratic. There is also a sufficient criterion for seeing where the representation variety is smooth. 
