Seminar on Geometry 

Schedule of talks

Abstracts/Notlar

Lecture1 : Bu dönem Jeff Viaclovsky - Topics in Riemannian Geometry 2007
ders notlarını takip edeceğiz.
Bunlar Einstein Manifold'larına giriş niteliğinde elemanter geometri dersleri olacak.
İlk hafta eğrilik tansörünün açılımı ve parçalarını işleyeceğiz. Bu parçaların boyutları ve Weyl tansöründen bahsedeceğiz.

Lecture2 : Kulkarni-Nomizu product. Weyl tansörü.

Lecture3 : Schouten tansörü. Eğrilik tansörünün dik parçalara açılımı.

Einstein-Hermitian: We show that a compact complex surface together with an Einstein-Hermitian metric of positive holomorphic bisectional curvature is biholomorphically isometric to the complex projective plane with its Fubini-Study metric up to rescaling. This result relaxes the Kaehler condition in Berger's theorem (1965), and the positivity condition on sectional curvature in a theorem proved by Koca. Joint work with C.Koca.

4üstGT : Üst 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.

Harmonic Maps 1: A Riemannian metric g on a smooth manifold M gives rise to the notion of a real-valued harmonic function (M,g) --> R . This generalizes the classical situation when the manifold is a flat Euclidean space. One can generalize further to harmonic maps (M,g) --> (N,h) between any two Riemannian manifolds. Harmonic maps are solutions to an elliptic system of partial differential equations, that are in general nonlinear. Harmonic maps are very important both in classical and modern differential geometry. The best known applications are the geodesics and minimal surfaces of Riemannian manifolds. Other important examples are the holomorphic maps between any two Kähler manifolds, generalizing the classical holomorphic maps between complex vector spaces. In these lecture series we will try to investigate the colourful world of harmonic maps. Everybody interested in differential geometry and/or global analysis or in particular, harmonic maps itself is very welcome. Following link gives a very nice description of the bibliography of the harmonic maps.

Harmonic Maps 3: Riemann yüzeyleri arasındaki harmonik gönderimlerden bahsedeceğiz.

General Seminar: A manifold is said to be affine flat if it admits local coordinate systems whose transition maps are affine transformations. For affine flat manifolds it is natural to ask the following question: "Among many Riemannian metrics that may exist on an affine flat manifold, which metrics are most compatible with the flat structure?" In this talk, I will explain that among all others the Kaehler affine metric provides the best compatibility. I will also recall the Kaehlerian manifolds, which are formally similar to the Kaehler affine manifolds noting that the Kaehlerian metric provides the best compatibility with the complex structure. In addition, if I have time I will describe affine harmonic maps which should be a useful tool for studying affine manifolds.

Lecture4 : Eğrilik operatörünün 2-formlara olan etkisi ve 3. boyuttaki özdeğerleri.

Lecture5 : Covaryant türevleri yer değiştirme.

Lecture6 : Laplace ve Hessian operatörlerini yer değiştirmenin bedeli.

Akbulut: Four dimensional smooth manifolds (generalized Euclidean spaces) display very strange behavior than manifolds in any other dimensions. Four is also the dimension of the space-time, which physicist think as a part of a bigger 11-dimensional universe. In this talk, I will review some recent developments of 4-manifolds, in particular discuss corks and plugs, which are small fundamental pieces floating in smooth 4-manifolds determining their smooth structure (twisting along them changes smooth structure), and talk about exotic structures on complex Stein surfaces.

Coşkun: The rationality question of the moduli spaces of vector bundles over an algebraic curve is a longstanding open problem. In this talk, we will talk about the progress toward the solution of this problem and some recent developments.

Harmonic Maps 4: Daha önce yapılanlar detaylandırılacak, ilk tekrar: Space of maps.

Harmonic Maps 5: Connections in the space of maps.

Önsiper: We will try to explain how one defines the "true" fundamental group in algebraic geometry.

Lecture7: İntegral, Hodge yıldız ve adjointler.

Nil: Arf Closure of a local ring corresponding to a curve branch, which carries a lot of information about the branch, is an important object of study, and both Arf rings and Arf semigroups are being studied by many mathematicians, but there is not an implementable fast algorithm for constructing the Arf closure. The main aim of this work is to give an easily implementable fast algorithm for constructing the Arf closure of a given local ring. The speed of the algorithm is a result of the fact that the algorithm avoids computing the semigroup of the local ring. Moreover, in doing this, we give a bound for the conductor of the semigroup of the Arf Closure without computing the Arf Closure by using the theory of plane branches. We also give an exposition of plane algebroid curves and present the SINGULAR library written by us to compute the invariants of plane algebroid curves.

Burak: Volume computations of n-dimensional ball and some measure theory Elementary computations of n-dimensional ball and comparing its volume with volumes of its n-1 dimensional ball slices, and show that it satisfies a gaussian normal distribution. If time permits we will mention the computation of volume o other spaces as complex projective spaces, Lie Groups like SO(n), SU(n). References: Keith Ball, Introduction to Modern Convex Geometry. Gallot, Hulin, Lafontaine - Riemannian Geometry, Zhou,Shi - Characteristic Classes on Grassmann Manifolds.

Önsiper: We will discuss the (partly conjectural) arithmetic and geometric consequences of the existence of hermitian metrics with negatively bounded holomorphic sectional curvature on complex surfaces.

Lecture8: Bochner ve Weitzenböck formülleri 1.

Lecture9: Bochner ve Weitzenböck formülleri 2.

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