# Almost Complex Geometry Seminar

## Department of MathematicsThe Graduate Center of CUNY

### Wednesdays 11:45am - 1:45pm Room 6417 Organizers: Luis Fernandez, Scott WilsonQian 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.

### Fall 2018:

#### date: 9/12 Organizational and introductory material

Welcome new students!

#### 9/26: Scott Wilson Title: Dolbeault cohomology for almost complex manifolds

Abstract: I will report on recent work with Joana Cirici that extends the Dolbeault cohomology and its surrounding theory to the case of arbitrary almost complex manifolds. A spectral sequence will be described and explicit calcualtions will be done in the case of some interesting Lie groups and nilmanfiolds. I'll also describe some degeneration results and an application that shows this cohomology can be used to show certain almost complex manifolds do not admit nearly Kahler metrics.

#### 10/3: Scott Wilson Title: Dolbeault cohomology for almost complex manifolds (part II)

Abstract: This will be a continuation of the previous talk. I'll do some hands on calculations and examples, and describe the harmonic theory that accompanies the Dolbeault theory. If time permits, I'll describe some degeneration results and applications to nearly Kahler 6-manifolds.

#### 10/10: Luis Fernandez Title: Introduction to Kahler manifolds

Abstract: I will describe foundational material concerning Kahler manifolds.

#### 10/17: Aleksander Doan (Stony Brook University) Title: Castelnuovo's bound and rigidity in almost complex geometry

Abstract: I will discuss the question of whether an energy bound implies a genus bound for pseudo-holomorphic curves in almost complex manifolds. After reviewing what is known in dimensions other than 6, I will talk about a new result in this direction in dimension 6. This result is motivated by an idea of defining new invariants of symplectic Calabi-Yau 6-manifolds. The talk is based on joint work with Thomas Walpuski.

#### 10/24: Luis Fernandez Title: Additional properties of Kahler manifolds

Abstract: I'll explain how a Kahler metric is, up to second order, the standard metric in local holomorphic coordinates. From this we deduce the Kahler identities and several consequences.

#### 10/31: Scott Wilson Title: Almost Kahler identities and applications

Abstract: I'll describe a generalization of the Kahler identities that holds for all almost complex manifolds with compatible metric whose associated fundamental two form is closed, i.e. so-called almost Kahler manifolds. From this we will deduce several identities of various Laplacians, Betti-number bounds on the dimensions of certain spaces of harmonic forms, and generalizations (to the non-integrable setting) of the various dualities that occur for Kahler manifolds. This is recent joint work with Joana Cirici.

#### 11/7: Scott Wilson Title: Almost Kahler identities and applications (part II).

Abstract: This will be a continuation of last week's talk that provides more details and applications. I plan to include describe a generalization of Lefschetz duality for almost Kahler manifolds. This is recent joint work with Joana Cirici.

#### 11/14: Collaborative Title: Problems and Techniques (with a view towards calculations)

Abstract: We will state some questions related to the topics presented in this semester's seminar, and discuss some techniques for calculating examples of Dolbeault cohomology and harmonic spaces, as a means to gain insight.

#### 11/28: Matthew Cushman Title: Deformations of Complex Structures

Abstract: We will consider the study of small deformations of complex structures on smooth manifolds, with emphasis on the techniques of Kodaira-Spencer. Examples will be given in complex dimension 1, and other approaches will be discussed as time permits.

#### 12/5: Samuel Hosmer Title: A canonical spin^c structure on almost complex manifolds

Abstract: The complex analogue of a spin structure on a manifold not only can be shown to exist for AC manifolds, but there is a canonical lift of the homomorphism U_n into SO_2n \times S^1 (given by inclusion in the first factor and determinant in the second) over the double covering of SOn \times S^1 by spin^c that we will explicitly construct. This will evidently produce a spin^c structure on an AC manifold.

#### 12/12: Matthew Cushman Title: Maslov Classes Part 1: Lagrangian Subbundles

Abstract: Abstract: We will review the necessary background material to discuss Maslov classes, starting from symplectic linear algebra and proceeding through Lagrangian subbundles and the Lagrangian Grassmanian. Motivating examples will be given. If time permits we will define the Maslov class assigned to a pair of Lagrangian subbundles.