Seminar on Geometry 

Schedule of talks

Abstracts/Notlar

LCK 1: Introduction. Lee form. Torsion 1-form.

LCK 2: Weyl connection.

LCK 3: Relation with conformal structures: Hermite-Weyl structures with vanishing distance curvature. (Complex dimension ≥ 3)

LCK 4: Globally conformally Kähler (GCK) manifolds.

LCK 5: Vaisman's conjectures.

LCK 6: Curvature properties.

LCK 7: Blow up.

LCK 8: Adapted cohomology.

LCK 9: Examples. Hopf manifolds.

LCK 10: Inoue surfaces.

LCK 11: A Nilmanifold: Generalized Thurston's manifold.

LCK 12: A 4-dimensional Solvmanifold.

LCK 13: SU(2)xS^1, non-compact examples.

LCK 14: Brieskorn & Van de Ven's manifolds.

LCK 15: Generalized Hopf Manifolds.

1. S. Dragomir, L. Ornea - Locally conformal Kähler geometry.
   Progress in Mathematics, 155. Birkhäuser Boston, Inc., Boston, MA, 1998.


2. Vaisman, Izu. Some curvature properties of complex surfaces.
   Ann. Mat. Pura Appl. (4) 132 (1982), 1–18 (1983).


3. Vaisman, Izu. On locally and globally conformal Kähler manifolds.
   Trans. Amer. Math. Soc. 262 (1980), no. 2, 533–542.


4. Gauduchon, Paul. La 1-forme de torsion d'une variété hermitienne compacte.
   [Torsion 1-forms of compact Hermitian manifolds] Math. Ann. 267 (1984), no. 4, 495–518.

Kalafat : In this talk we will give a discussion on a program which aims to present a combinatorial approach to the exceptional Lie group G2 and its Lie algebra. We give various results about the algebraic structure. We also prove some of the old results in terms of our approach.

Kalafat : Page manifold is the underlying differentiable manifold of the complex surface, obta- ined out of the process of blowing up the complex projective plane, only once. This space is decorated with a natural Einstein metric, first studied by D.Page in 1978.
In this talk, we study some classes of submanifolds of codimension one and two in the Page space. These submanifolds are totally geodesic. We also compute their curvature and show that some of them are constant curvature spaces. Finally we give information on how the Page space is related to some other metrics on the same underlying smooth manifold. This talk is based on a joint work with R.Sarı. Related paper may be accessed from https://arxiv.org/abs/1608.03252 .
Despite working on basic submanifolds, we introduce a variety of mutually-independent techniques, like graphic illustrations, physicist computations, teichmüller space, 3- manifold topology, ODEsystems etc. So that should not be confused with dry, computational diff.geo. involving only symbolic manipulations, meaningless mess of equalities followed by equalities. We always consider the global topology of the submanifold for example, and deal primarily with compact examples.

Kalafat : Yüksek boyutlarda geometri konularıyla ilgili bir tanıtım konuşmasıdır. Öncelikle, n-boyutlu kürenin yüzey alanı ve iç hacminin nasıl hesaplanıldığından bahsedeceğiz. İkinci olarak düğümler teorisi ve geometriyle ilgisinden bahsedilecektir. Örneğin Milnor'un teoremine göre, total eğriliği 4pi den büyük olan sicimlerin düğümlü olması gerektiğinden bahsedeceğiz. Hiperbolik düğüm nasıl olur onu anlatacağız. Son olarak, koni kesitlerinin reel projektif uzayda nasıl doğal olarak yattığından, projektif uzay ve Klein şişesinin yatması için niye üst boyutlara ihtiyaç olduğundan bahsedeceğiz. Ayrıca kompleks projektif uzay ve bazı Lie grupların hacminin, üzerine metrik konarak hesaplanabileceğine değineceğiz. Konuşma Türkçe olup, lisans ve üstü öğrencilerine yöneliktir. Konuya uzak olan, ilgilenen öğretim üyeleri de davetlidir.

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