From now on the announcements will be posted to the Departmental Calendar.
Ivan will be explaining the Langlands Program and Kapustin-Witten approach thereto until further notice
Tuesday, November 7.
Ivan Mirkovic
Title: Langlands program VI
Tuesday, October 31.
Ivan Mirkovic
Title: Langlands program V
Tuesday, October 24.
Ivan Mirkovic
Title: Langlands program IV
We explain how loop Grassmannian of a group $G$ gives rise to Hecke operators and to the dual group, via fusion. Then we hopefully move to the structure of the Kapustin-Witten approach.
Tuesday, October 17.
Ivan Mirkovic
Title: Langlands program III
We start with loop Grassmannians and pass to a sketch of the Witten-Kapustin theory.
Tuesday, October 10.
Ivan Mirkovic
Title: Geometric Langlands Conjecture II
Last time we formulated the GL conjecture in its categorical form and discussed a more classical formulation of the correpondence of local systems on a curve with Hecke eigensheaves on the moduli of budles. We examined the meaning of the clasical formulation for the multipicative group. This time we introduce the Hecke operators for the general case of reductive groups $G$ and relate the Langlands conjectures to the Hitchin system, working towards a sketch of the Kapustin-Witten approach, i.e., differential geometric reformulation based on physics ideas.
Tuesday, October 3.
Ivan Mirkovic
Title: Geometric Langlands Conjecture
This is an introductory talk on the geometric content (on complex curves) of Langlands conjectures in number theory, with a sketch of the Kapustin-Witten approach, i.e., differential geometric formulation based on physics ideas.
Tuesday, September 26.
Eyal Markman will continue his talk.
Tuesday, September 19.
Eyal Markman, University of Massachusetts, Amherst
Title: Higgs bundles, an introduction
Abstract: Let X be a compact oriented surface of genus g and
M the moduli space of isomorphism classes of
n-dimensional irreducible complex representations of
the fundamental group of X.
M is a complex manifold of dimension n^2(2g-2)+2.
When X is endowed with a complex structure, M is
endowed with a one-parameter family of complex structures;
a hyperkahler structure; via work of Hitchin and Donaldson.
One of the new complex structures exhibits M as a moduli space of
holomorphic pairs (E,f), called Higgs pairs,
consisting of a holomorphic vector bundle E on X
and a 1-form valued endomorphism f. This description reveals a
new C^* action on M, a holomorphic symplectic structure,
and a Lagrangian fibration of M by Jacobians of spectral curves,
known as the Hitchin system. We will survey the above, as well as
variants for principal bundles and for higher dimensional
manifolds, constructed by Simpson and others.