Almost Complex Geometry Seminar

Department of Mathematics
The Graduate Center of CUNY

Fridays 11:45am - 1:45pm
Room 3212
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.

Fall 2019:

8/23: Prof. Luigi Vezzoni (U. di Torino)
Title: Geometric flows of Balanced metrics
Note: Room 3309 for this meeting

Abstract: An Hermitian metric g on a complex manifold (M, J) is called balanced if its fundamental form is co-closed. Typically examples of balanced manifolds are given by modifications of Kahler manifolds, twistor spaces over anti-self-dual oriented Riemannian 4-manifolds and nilmanifolds. In the talk it will be discussed two new geometric flows of balanced structures. The first of them was introduced in [1] and consists in a generalisation of the Calabi flow to the balanced context. The flow preserves the Bott-Chern cohomology class of the initial structure and in the Kahler case reduces to the classical Calabi flow. The other flow still preserves the Bott-Chern class of the initial structure but, in contrast with the first one, it is a potential flow and it does not preserve the Kahler condition. For the both flows it will be discussed the well-posedness and the stability around Kahler-Einstein metrics.

8/30: Organizational meeting
Faculty and students welcome

Abstract: We will make a plan on topics to cover, as well as potential papers to read and work through on days when no speakers are scheduled. Come with questions and/or ideas.

9/6: Prof. Scott Wilson
Title: Harmonic symmetries for Hermitian manifolds

Abstract: I will report on some recent work that generalizes the symmetries of the Hodge numbers of Kahler manifolds to the more general setting of compact complex manifolds with compatible metric. I will review some introductory material and the talk will be self contained. New attendees and students are especially welcome. Here is a preprint: arxiv:1906.02952.

9/13: Prof. Luis Fernandez (BCC CUNY)
Title: Basics on almost complex manifolds

Abstract: Given that some of the regular attendees to the seminar will be away, I will do a general review of basic facts about almost complex, complex, almost Kahler, and Kahler manifolds, including metrics, connections, and questions about integrability. I will look at some particular examples of these objects in some detail (especially the 6 sphere). The talk is intended for a general audience with some knowledge of (real) differential geometry. Students are very welcome.

9/20: Santiago Simanca
Title: Elementary Almost Hermitian Properties of Products of Spheres

Abstract: We consider products of two odd dimensional spheres, or S^2 x S^2, S^2 x S^4, S^2 x S^6, and S^6 x S^6, with suitable almost complex structures. In each case, we describe families of conformal classes of almost Hermitian metrics relative to them, and some tensorial quantities of relevance associated such structures. In the latter three cases, we show that with the metrics we use, these Riemannian manifolds do not carry integrable orthogonal almost complex structures.

9/27: Prof. Zhixu Su and Aleksandar Milivojevic
Title: Smooth or almost complex manifolds with prescribed Betti numbers

Abstract: Prescribing a sequence of Betti numbers (or more specifically the rational homotopy type), is there any almost complex manifold realizing the algebraic data? We will discuss the most basic nontrivial case where the sum of Betti numbers is three. Rational surgery reduces the problem to finding characteristic numbers satisfying certain integrality conditions. We will firstly discuss the smooth case, and then prove the non-existence of any almost complex structure on such a manifold if the dimension is greater than 4. We will also provide a general realization theorem for the almost complex case.

10/4: Prof. Mehdi Lejmi (BCC CUNY)
TBA

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10/11: Xujia Chen
Title: TBA

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10/18: Qian Chen
Title: TBA

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10/25: Prof. Fei Ye (Queensborough)
Title: TBA

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11/8: Prof. Michael Albanese
Title: TBA

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11/15: Prof. Yury Ustinovskiy (NYU)
Title: TBA

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Spring 2019
Fall 2018
Spring 2018
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