This is a website to organize the 2011 Pre-Talbot Seminar at Northwestern University. The purpose of these talks is to learn about the basics of non-Abelian Hodge Theory, and also for participants in the 2011 Talbot Workshop to be able to practice their workshop talks.
Date Time Room Speaker Title and description Taken Notes
Thursday, March 3, 2011. 10:30 AM Lunt 1st Floor Sam Gunningham Introduction to non-Abelian Hodge Theory. Discussed example of Narasimhan-Seshadri. 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 so-called 'Twistor structures': Hodge structures that have 'forgotten' a C^x action. In the context of non-abelian Hodge theory, this is particularly useful when describing a non-abelian Hodge theorem for non-compact 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 non-Abelian 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 Goldman-Millson. "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 examples--the problem of deforming a flat principal G-bundle. 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.