1/30: Matthew Cushman
Abstract: We discuss the Weil homomorphism for principal bundles, from
invariant functions on the Lie algebra to characteristic forms on the base.
Title: Introduction to Chern-Weil theory.
2/6: Matthew Cushman
Abstract: We will discuss fiber integration and a generalization of the Weil
homomorphism which gives secondary characteristic classes.
Title: Transgression of characteristic forms and the Bott homomorphism
2/13: Prof. Luis Fernandez (Bronx, CUNY)
I will explain the workings of, and do some examples with, a Mathematica
notebook I recently created to calculate the first two pages of a spectral
sequence on some examples of almost complex manifolds.
Computing examples of the Cirici-Wilson spectral sequence using Mathematica.
2/20: Scott Wilson
A linear operator on a vector space can be extended to the exterior algebra
either as a derivation, or as an algebra map. In the case of an almost complex
structure (operating on the complexified cotangent space of a manifold) these
two choices can be used to encode the integrability condition, in terms of a
compatibility with the exterior differential. This simplifies some interesting
Lie theory. One can similarly (try to) extend complex conjugation in two
ways. I will share some partial progress towards extending this algebra
to a representation of sl(2,R) which is mirror-dual to Lefschetz's
construction, the latter of which plays an important role for symplectic
manifolds with compatible metrics.
Title: The de Rham complex of a complex manifold.
3/27: Caner Koca (City Tech, CUNY)