Current Seminars‎ > ‎

Almost Complex Geometry

Organizer: Professor: Scott. Wilson, Luis Fernandez 
                   Student: Matthew Cushman, Qian Chen
Time:         Wednesday 11:45am - 1:45pm
Location:   Room 6417
                   Graduate Center
                   365 Fifth Avenue
                   New York, NY 10016


Daniel Huybrechtz. -- Complex Geometry An Introduction, Springer Berlin Heidelberg, New York  2005.
* Shoshichi Kobayashi, Katsumi Nomizu --  Foundations of Differential Geometry Volume II, Wiley Classics Library Edition, 1996.
* Daniele Angella -- Cohomological Aspects in Complex Non-Kahler Geometry, Springer International, 2014.
* Claire Voisin -- Hodge Theory and Complex Algebraic Geometry, Cambridge University Press, 2004.
* Raoul Bott, Loring W. Tu -- Differential Forms in Algebraic Topology, Springer-Verlag New York, 1982.
* Dusa McDuff, Dietmar Salamon--- Introductionn to Symplectic Topology, Oxford Science Publications, 1998.
 M.Field -- Several Complex Variables and Complex Manifolds II, Cambridge university press, 1982

Spring 2018 Suggested Schedule:

1/31/2018    Planning for Spring 2018

2/07/2018    Almost complex structures on six-sphere through octonions -- Luis Fernandez

2/14/2018 Almost complex structures on six-sphere through octonions (continued) -- Bora Ferlengez

2/21/2018      Non-integrable ACS on six-sphere -- Luis Fernandez

2/28/2018      Metric on the space of ACS on six-sphere (continued) -- Luis Fernandez

3/07/2018      cancelled

3/14/2018      cancelled

3/21/2018      cancelled

3/28/2018      Relations among Hodge numbers on a hypothetical complex six-sphere -- Aleksandar Milivojevic
(see attached notes)
3/29/2018       Cohomologies on almost complex manifolds and their applications -- Spiro Kargiannis 
We define three cohomologies on an almost complex manifold (M, J), defined using the Nijenhuis-Lie derivations induced from the almost complex structure J and its Nijenhuis tensor N, regarded as vector-valued forms on M. One of these can be applied to distinguish non-isomorphic non-integrable almost complex structures on M. Another one, the J-cohomology, is familiar in the integrable case but we extend its definition and applicability to the case of non-integrable almost complex structures. The J-cohomology encodes whether a complex manifold satisfies the "del-delbar-lemma", and more generally in the non-integrable case the J-cohomology encodes whether (M, J) satisfies a generalization of this lemma. We also mention some other potential cohomologies on almost complex manifolds, related to an interesting question involving the Nijenhuis tensor.
4/18/2018      Relations among Hodge numbers on a hypothetical complex six-sphere(2) -- Aleksandar Milivojevic
(see attached notes)
4/25/2018      Joining Simons mathfest 2018

5/02/2018    Results concerning almost complex structures on the six-sphere(1), arxiv:1610.09620  -- Scott Wilson

5/09/2018    Results concerning almost complex structures on the six-sphere (2), arxiv:1610.09620  -- Scott Wilson

5/16/2018   Results concerning almost complex structures on the six-sphere (3), arxiv:1610.09620  -- Scott Wilson

5/23/2018    S^(4k) do not admit almost complex structures -- Samuel Hosmer

Fall 2017 Schedule

8/30/2017  Almost Complex Structure - by Scott Wilson 

9/6/2017    Differential Operator over an ACM - by Scott Wilson 

9/13/2017  Integrability of an ACS- by Scott Wilson  

9/20/2017  Holiday - No meeting

9/27/2017  Exterior derivative of ACS - Scott Wilson  

10/4/2017  Metric on ACS - Luis Fernandez 

10/11/2017 Metric on ACS - continued - Luis Fernandez

10/18/2017 The Frolicher - Nijenhuis Bracket - Scott Wilson

10/25/2017 Real Analytic Integrable ACS - Qian Chen

11/1/2017    Real Analytic Integrable ACS (ctn.) - Qian Chen

11/8/2017   Symplectic Reduction - Matthew Cushman

11/15/2017  The Twisting Tennis Racket - Richard Cushman

                                                        Title: The twisting tennis racket

Abstract: This talk gives a mathematical explanation of the twisting phenomenon exhibited in the following experiment. Take a tennis racket and mark its faces so that they can be distinguished.  Call one rough and the other smooth. Hold the racket horizontally so that the amoorh face is up. Toss the racket attempting to make it rotate about the intermediate principal axis, which is through the face 
and perpendicular to the handle. After one rotation catch the racket by its handle. The rough face will almost always be up! Thus the racket has made a near half twist around its handle.

11/22/2017  Meeting cancelled

11/29/2017  Space of Almost Complex Structure on S^6 - Bora Ferlengez

12/6/2017    TBA- Mahmound Zeinalian

qian chen,
Sep 30, 2017, 2:07 PM
qian chen,
Feb 28, 2018, 3:59 PM