Fall 2018:

date: 9/12
Organizational and introductory material
Welcome new students!

date: 9/19
No seminar or classes due to holiday

9/26: Scott Wilson
Title: Dolebault 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: Dolebault 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 6manifolds.

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 pseudoholomorphic 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
CalabiYau 6manifolds. 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. socalled almost Kahler manifolds. From
this we will deduce several identities of various Laplacians, Bettinumber
bounds on the dimensions of certain spaces of harmonic forms, and
generalizations (to the nonintegrable 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.

11/21: No meeting this week

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 KodairaSpencer. 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.

Past seminars:
Spring 2018
Fall 2017
