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
References:
* 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
Fall 2018 Schedule:
08/29/2018 Planning for Fall 2018
09/12/2018 Dolbeault cohomology for almost complex manfiolds:
https://arxiv.org/abs/1809.01416 (1) -- Scott O. Wilson
09/19/2018 Dolbeault cohomology for almost complex manfiolds:
https://arxiv.org/abs/1809.01416 (2) -- Scott O. Wilson
09/26/2018 Dolbeault cohomology for almost complex manfiolds:
https://arxiv.org/abs/1809.01416 (3) -- Scott O. Wilson
10/03/2018 Dolbeault cohomology for almost complex manfiolds:
https://arxiv.org/abs/1809.01416 (4) -- Scott O. Wilson
10/10/2018 Dolbeault cohomology for almost complex manfiolds:
https://arxiv.org/abs/1809.01416 (5) -- Scott O. Wilson
10/17/2018 Dolbeault cohomology for almost complex manfiolds:
https://arxiv.org/abs/1809.01416 (6) -- Scott O. Wilson
10/24/2018 Topological and geometric aspects of almost Kahler manifolds via
harmonic theory:
https://arxiv.org/abs/1809.01414 (1) -- Scott O. Wilson
10/31/2018 Topological and geometric aspects of almost Kahler manifolds via
harmonic theory:
https://arxiv.org/abs/1809.01414 (2) -- Scott O. Wilson
11/07/2018 Topological and geometric aspects of almost Kahler manifolds via
harmonic theory:
https://arxiv.org/abs/1809.01414 (3) -- Scott O. Wilson
11/14/2018 Topological and geometric aspects of almost Kahler manifolds via
harmonic theory:
https://arxiv.org/abs/1809.01414 (4) -- Scott O. Wilson
11/21/2018 Topological and geometric aspects of almost Kahler manifolds via
harmonic theory:
https://arxiv.org/abs/1809.01414 (5) -- Scott O. Wilson
11/28/2018
12/05//2018
12/12/2018
Summer 2018 Schedule:
5/23/2018 Spheres of 4k-dimension do not admit ACS -- Aleksandar Milivojevic
5/28/2018 On the minimal sum of Betti numbers of an ACM (1) -- Aleksandar Milivojevic
https://arxiv.org/abs/1805.04751
6/1/2018 On the minimal sum of Betti numbers of an ACM (2) -- Aleksandar Milivojevic
https://arxiv.org/abs/1805.04751
6/8/2018 Spectral sequences -- Aleksandar Milivojevic
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
Abstract:
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
http://simonsmathfest2018.ws.gc.cuny.edu
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 None orthogonal almost complex structure is integrable on six-sphere -- Luis Fernandez
https://www.ams.org/proc/1987-101-01/S0002.../S0002-9939-1987-0897084-7.pdf
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