Almost Complex Geometry Seminar

Department of Mathematics
The Graduate Center of CUNY

Wednesdays 11:45am - 1:45pm
Room 6417
Organizers: Luis Fernandez, Scott Wilson Qian Chen

Scope: this seminar is devoted to all topics related to almost complex manifolds, including but not limited to: complex manifolds, symplectic topology and geometry, (almost) Kahler geometry, as well the tools of algebraic topology and geometric analysis that have proven useful in studying such structures. The goal is expose students and faculty to foundational material and present day research in a relaxed format that includes traditional lectures with an actively engaged audience.

Spring 2019:

1/30: Matthew Cushman
Title: Introduction to Chern-Weil theory.

Abstract: We discuss the Weil homomorphism for principal bundles, from invariant functions on the Lie algebra to characteristic forms on the base.

2/6: Matthew Cushman
Title: Transgression of characteristic forms and the Bott homomorphism

Abstract: We will discuss fiber integration and a generalization of the Weil homomorphism which gives secondary characteristic classes.

2/13: Prof. Luis Fernandez (Bronx, CUNY)
Title: Computing examples of the Cirici-Wilson spectral sequence using Mathematica.

Abstract: 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.

2/20: Scott Wilson
Title: The de Rham complex of a complex manifold.

Abstract: 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.

2/27: Student and Faculty Q&A
Title: Jam session

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.)

3/6: Qian Chen
Title: L_J Cohomology on Almost Complex Manifolds

Abstract: 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.

3/13: Prof. Luis Fernandez (Bronx, CUNY)
Title: Pseudoholomorphic 2-spheres in the 6-sphere

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.

3/20: Tobias Shin (Stony Brook University)
Title: Almost complex manifolds are complex, almost

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.

3/27: Prof. Caner Koca (City Tech, CUNY)
Title: Introduction to Einstein-Maxwell Metrics

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.

4/3: Luis Fernandez
Title: Three equivalent models of the space of ACS on R2n.

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).

4/10: Scott Wilson
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.

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 Nuijehuis tensor.

4/24: No meeting, spring break.

5/8: Prof. Bora Felengez (Hunter College)
Title: The space of complex structures on R^6 compatible with the standard metric and orientation is diffeomorphic to the complex projective space CP^3.

Abstract: 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).

Past seminars:

Fall 2018
Spring 2018
Fall 2017

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