Seminar on Geometry 

Schedule of talks

Abstracts/Notlar

Cohom 0: Homogeneous spaces. Symmetric spaces. Isometry groups of some basic spaces. Semi-direct products. (Önceki dönem)

Cohom 1: Cohomogeneity tanımlar. Örnekler: Rotationally symmetric surfaces. Spheres under cohomogeneity one action.

Cohom 2: Page Metric.
Bryant-Salamon metrics A: [Duke,89]
Bu metrikleri anlamak için güzel bir kaynak:

Simon Salamon - Riemannian geometry and holonomy groups. (Red book.) (Yazarın sitesinde mevcut.)
Pitman Research Notes in Mathematics Series, 201. Longman Scientific & Technical, Harlow;
copublished in the United States with John Wiley & Sons, Inc., New York, 1989. viii+201 pp. ISBN: 0-582-01767-X

[SS, Ch7 ]: Associated bundles. Expression of the trace free Ricci curvature and anti self dual Weyl curvature.
Dictionary of global invariant forms on the total space of the anti self dual form bundle.

Cohom 3: Bryant-Salamon metrics B:

İnvaryant formlar cinsinden Self-duality ve Einstein şartları.
Twistor uzayına giriş.

Cohom 4: Bryant-Salamon metrics C:

Twistor uzayının hemen hemen kompleks yapısı ve integrallenebilirliği.

Cohom 5: Bryant-Salamon metrics D:

[SS,Ch11]: İnvaryant formları kullanarak G_2 (Lie grubu) yapısı inşa edeceğiz.
Sonrasında da bunlar arasından Holonomi grubu G_2ye eşit olan tam Riemann metriklerini türeteceğiz.

Cohom 6: Special Kähler manifolds: Metrics with holonomy SU(m+1). [or Sp(k)]. Reference: [SS, Ch8 ]

Cohom 7: Metrics on the canonical bundle. 1979 Calabi metrics.

Cohom 8: Calabi metrics and completeness. Examples: Eguchi-Hanson metric. Burns metric. 1988 LeBrun metrics on O(-k) over CP_1.

Cohom 9: Metrics on Hirzebruch surfaces.
İlk ders: S³xS² de çember etkileri. Bölüm olarak Hirzebruch yüzeyleri elde edilmesi.

Referans: Nobuhiko Otoba - Constant scalar curvature metrics on Hirzebruch surfaces.
Ann. Global Anal. Geom. 46 (2014), no. 3, 197–223.

Cohom 10: Hirzebruch yüzeylerinde efektif U(2)/Z_m etkisi. 3 boyutlu kürede U(2) ve SU(2) invaryant vektör alanları.

Cohom 11: Elementary open-dense submanifolds of Hirzebruch surfaces.

Cohom 12: Otoba metrics on Hirzebruch surfaces. Devam.

Cohom 13: U(2) invariant, Riemannian submersion metrics. Boundary conditions for smoothness of the metric.

Cohom 14: Tensor calculations. 1st and 2nd order derivatives. Scalar curvature.

Karigiannis I: This is part one of an introduction to the geometry of G_2 and Spin(7) structures, henceforth called “exceptional structures”. We will begin with a very brief review of Berger’s list of Riemannian holonomy groups and of some non-exceptional structures on manifolds such as almost Hermitian structures. Then we will introduce the octonions, cross products, and the exceptional calibrations on R^7 and R^8, which will allow us to define exceptional structures on manifolds via the structure group of their frame bundles. Next, we will study the concrete representation theory of G_2 and Spin(7), which will allow us to define the torsion forms and define various classes of exceptional structures. Finally, we will express the Ricci tensor in terms of the torsion, and give a concrete computational proof of the theorems of Fernandez-Gray and Fernandez relating parallel and harmonic calibration forms.

Yalçınkaya-123: In this seminar, we define a general normed algebra then construct a special case. Thanks to the Cayley-Dickson process, we will construct a new algebra. Hurwitz theorem shows that there are a few algebras over R. When we define cross product clearly, we may understand of algebraic structure of G2.

Reference book: Spinors and Calibration (Reese Harvey)

Potential Theory 1: Plurisubharmonic functions: In complex analysis, harmonic functions are useful objects since they preserve many properties same as analytic functions (e.g. maximum principle ). In the complex case, differentials can be determined by holomorphic and anti-holomorphic forms. In this seminer, On the other hand, in calibrated geometry, if a space is calibration for the given form phi, then it can be determined whether functions are harmonic. The Harmonicity is powerful tool and many of them are not harmonic. We permit some conditions then we will define subharmonic functions.

LeBrun: While we are very far from being able to completely determine which smooth compact 4-manifolds admit Einstein metrics, the problem becomes much more tractable if we restrict our attention to those 4-manifolds which also admit a symplectic structure. In this context, we now have a complete answer to the question when Einstein constant is also assumed to be non-negative, and we even know know a great deal about the negative case. In this lecture, I will present a new result regarding the question of whether the corresponding Einstein moduli spaces are connected in the positive case. If time allows, I will then survey some interesting open questions regarding the negative case. (1 hour long talk)

Poddar: We will give classifications of torus equivariant principal bundles on toric varieties both in the holomorphic and algebraic settings. In the holomorphic case, we will give some sufficients conditions for splitting, and point out the relations to a classical theorem of Grothendieck and a conjecture of Hartshorne. In the algebraic case, our work implies the triviality of such bundles over (possibly singular) affine toric varieties. This is a joint work with I. Biswas and A. Dey.

C.Özel: In this talk we calculate Reidemeister torsion of flag manifold K/T of compact semi-simple Lie group K = SU(n + 1) using Reidemeister torsion formula and Schubert calculus, where T is maximal torus of K. We find that this number is 1. Also we explicitly calculate ring structure of integral cohomology algebra of flag manifold K/T of compact semi-simple Lie group K = SU(n + 1).

Buket 1: Lokal konformal düz Riemann metriklerinin yapışmasını garanti eden, Kulkarni'nin teoreminin ispatı verilecektir.

Kişisel: We will start by discussing the consequences of the Uniformization Theorem of compact Riemann surfaces and continue by discussing the groups which uniformize Riemann surfaces of genus greater than one. Expect lots of pictures.

Tezer: The purpose of this talk is to present an observation by C. Soland that it is possible to characterize the Euclidean plane as a metric space which obeys three further and natural axioms employing the Cayley-Menger determinants.

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