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.
2/27: Student and Faculty Q&A
Abstract: Students are encouraged to come with questions related to the
seminar topics. (Great questions! We proved the d-dbar Lemma for compact Kahler manfiolds,
defined Bott-Chern cohomology and showed it is finite dimensional for a
compact complex manifold by
identifying it with the kernel of a fourth order elliptic operator, and
discussed the relationship between the Godbillon-Vey and Chern-Simons invarinats.)
Title: Jam session
3/6: Qian Chen
In this talk I will describe a cohomology group for almost complex manifolds
that was recently introduced by Chan, Karigiannis, and Tsang. The underlying
complex for this cohomology, at least in the integrable case, appears in the
formality theorem of Kahler manifolds by Deligne, Griffiths,
Morgan, Sullivan. I'll explain how this complex is defined and
describe some finite dimensionality results of this cohomology for complex and
almost complex manifolds. In my current research I've shown this cohomology
has a spectral sequence converging to it, whose study is ongoing work.
Title: L_J Cohomology on Almost Complex Manifolds
3/13: Prof. Luis Fernandez (Bronx, CUNY)
Abstract: A map is pseudoholomorphic (aka J-holomorphic, aka almost complex)
if its differential commutes with the almost complex structures. For the
classical almost complex structure of the 6-sphere it is easy to see that
pseudoholomorphic implies harmonic. One can then use some techniques of
harmonic map theory (twistor lifts, harmonic sequences) to study these
pseudoholomorphic maps and obtain existence results, as well as compute the
dimension of the moduli space of such maps.
Title: Pseudoholomorphic 2-spheres in the 6-sphere
3/20: Tobias Shin (Stony Brook University)
Abstract: We will describe Demailly and Gaussier's differential relation
corresponding to integrability of almost complex structures. By using a
theorem of Sommese, we will be able to show that there are no homotopy
obstructions to integrability up to complex dimension 77.
Moreover, using the method of holonomic approximation, we will be able to
approximate any formal solution by a holonomic section, under certain
constraints. This in particular implies that, in the even complex dimensions
up to 77 (and dimension 3), any almost complex manifold admits a sequence of
almost complex structures so that the pointwise supremum norms of the
Nijenhuis tensors become arbitrarily small.
Title: Almost complex manifolds are complex, almost
3/27: Prof. Caner Koca (City Tech, CUNY)
Abstract: Einstein-Maxwell metrics, which originate from Physics, have
recently become a popular subject in Mathematics because of their interesting
ties with complex geometry. In this talk, I will give an introduction to these
Riemannian metrics, explain their relation to Kahler geometry, construct some
non-trivial examples, and outline the recent results.
Title: Introduction to Einstein-Maxwell Metrics
4/3: Luis Fernandez
Abstract: I'll describe how the space of almost complex structures can be
thought of as a homogeneous space, Grassmannian, or a choice of spinor (up to scale).
Title: Three equivalent models of the space of ACS on R2n.
4/10: Scott Wilson
Abstract: Work of Donaldson, recently expanded on by D. Salamon et al,
describes the space of almost complex structures on R2n as a co-adjoint orbit
of SL(2n,R). There is a symplectic structure easily described in this
model (which agrees with our previous
presentations). This induces a symplectic structure on the space of ACS's of
any manifold with volume form. I'll describe this, as well as an induced vector
field given by the action functional which integrates the norm squared of the
Title: Another model of the space of ACS on R2n, and a symplectic structure
on the space of all almost complex structures on a manifold with volume form.
4/24: No meeting, spring break.
5/8: Prof. Bora Felengez (Hunter College)
We will discuss isotropic subspaces of C^n and as an application, we
will prove the statement in the title by identifying the manifold of positive
maximal isotropic subspaces of C^3 and the homogeneous space SO(6)/U(3).
Title: The space of complex structures on R^6 compatible with the
standard metric and orientation is diffeomorphic to the complex projective