Seminar on Geometry 

Spring 2015, ODTÜ, Ankara

Time / Location: Fridays 2:00 / M-214

Schedule of talks

 

Jan 21
Wed, 1:00
Eyüp Yalçınkaya
Potential theory on calibrated submanifolds 2
Plurisubharmonic functions

2:00
Mustafa Kalafat
Constant scalar curvature metrics on Hirzebruch surfaces 1
Jan 23
2:00
Constant scalar curvature metrics on Hirzebruch surfaces 2
Jan 30
Fri, 11:00

Constant scalar curvature metrics on Hirzebruch surfaces 3

2:00

Potential theory on calibrated submanifolds 3
Plurisubharmonic functions

3:00
Constant scalar curvature metrics on Hirzebruch surfaces 4
Feb 4
Wed, 2:00


Constant scalar curvature metrics on Hirzebruch surfaces 5
Feb 6
Fri, 2:00

Southern California Geometric Analysis Conference
Feb 13
Fri, 2:00

No Seminar
Feb 18
Wed, 2:00
Constant scalar curvature metrics on Hirzebruch surfaces 6
Feb 20
Fri, 2:00
Potential theory on calibrated submanifolds 4
Hessian on the calibrated submanifold

3:00
Constant scalar curvature metrics on Hirzebruch surfaces 7
Feb 27
Fri, 2:00
Potential theory on calibrated submanifolds 5
Hessian on the calibrated submanifold

3:00
Constant scalar curvature metrics on Hirzebruch surfaces 8
Mar 6
Fri, 2:00
Potential theory on calibrated submanifolds 6
Cousin problem

3:00
Constant scalar curvature metrics on Hirzebruch surfaces 9
Mar 13
Fri, 2:00
Muazzez Şimşir
Forms on flat affine manifolds
Mar 20
Fri, 2:00
Constant scalar curvature metrics on Hirzebruch surfaces 10

3:40
Algebraic Geometry Seminar

4:40
Potential theory on calibrated submanifolds 7
Free submanifolds
Mar 27
Fri, 11:00
Geometry of Grassmann manifolds 1

2:00
Geometry of Grassmann manifolds 2

3:40
Geometry of Grassmann manifolds 3
Apr 2
Thu, 15:40
Mahir Bilen Can
Tulane
Equivariant K-theory of smooth spherical varieties
Colloquium in İkeda
Apr 3
Fri, 11:00
Minimal surfaces in R^3 8

2:00
Constant scalar curvature metrics on Hirzebruch surfaces 11

3:40
Algebraic Geometry Seminar
Apr 10
Fri, 2:00
Constant scalar curvature metrics on Hirzebruch surfaces 12
Apr 17
Fri, 2:00
Minimal surfaces in R^3 10

3:40
Algebraic Geometry Seminar
Apr 20
Mon, 3:40
GT Seminar at İkeda Conformally Kähler surfaces and orthogonal holomorphic bisectional curvature 1
Apr 21
Wed, 2:00
Conformally Kähler surfaces and orthogonal holomorphic bisectional curvature 2
Apr 24
Fri, 2:00
Constant scalar curvature metrics on Hirzebruch surfaces 13
May 1
Fri, 2:00
Introduction to the h-principle 11

3:00
On the scalar curvature of complex surfaces 1
May 6
Wed, 3:00
Aysel Vanlı
(Gazi)
Fraktal Geometri
(Yer: Hacettepe Üniversitesi, Yaşar Ataman salonu)
May 8
Fri, 2:00
Introduction to the h-principle 12

3:00
On the scalar curvature of complex surfaces 2
May 13
Wed, 2:00
Introduction to the h-principle 13

3:00
On the scalar curvature of complex surfaces 3
May 15
Fri, 2:00

No Seminar
May
18-22

Recent Advances in Kähler Geometry
Vanderbilt University, Tennessee
May 28
Fri, 2:00

Gökova Geometry/Topology Conference

Abstracts/Notlar


On the scalar curvature of complex surfaces
Bu seminer serisinde aşağıdaki makaleyi çalışacağız. Referans: On the Scalar Curvature of Complex Surfaces.
Geom. Func. An. 5 (1995) 619--628.

OSCCS 1: Introduction.

OSCCS 2: Seiberg-Witten equations.



Einstein-Maxwell 4-manifoldlar
Riemann manifoldları üzerinde Einstein-Maxwell denklemleri: Claude LeBrun - The Einstein-Maxwell Equations, Extremal Kähler Metrics, and Seiberg-Witten Theory,
in The Many Facets of Geometry: a Tribute to Nigel Hitchin, Bourguignon, Garcia-Prada & Salamon, editors, Oxford University Press, 2010, pp. 17--33.
makalesiyle soyut matematiksel temele oturtulmuştur. Burada bu denklemlerin bazı özel çözümlerinden bahsedeceğiz. Özellikle Kähler manifoldlarında.

Maxwell 1: Einstein-Maxwell equations on a Riemannian manifold.




Special holonomy Riemannian Manifolds and Mirror Symmetry

Potential Theory 2: 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.

Referans: An Introduction to Potential Theory in Calibrated Geometry.
Amer. J. Math. 131 (2009), no. 4, 893-944.

Potential Theory 4: Hessian on the calibrated submanifold. Subharmonic functions defined on the calibrated manifold can be induced to Phi-submanifold. This theorem is the center of this talk. We will define hessian on the calibrated manifold which is cousins of Hessian of Riemannian manifold. Bu the hessian on the calibrated manifold , we can prove the theorem.

Potential Theory 7: Referans: Ünal, İbrahim - Topology of φ-convex domains in calibrated manifolds.
Bull. Braz. Math. Soc. (N.S.)42 (2011), no. 2, 259–275.

Minimal surfaces 8: Introduction to minimal surface equation. Surfaces in R^3
Referans: Colding, Minicozzi - A course in minimal surfaces.
Graduate Studies in Mathematics, 121. American Mathematical Society, Providence, RI, 2011. xii+313 pp. ISBN: 978-0-8218-5323-8.

H-Principle 1: Jet bundles. Referans:
Ünal, İbrahim - h-Principle and ø-free Embeddings in Calibrated Manifolds.
International Journal of Mathematics. 2015.

Geometry of Grassmann manifolds
Referans: Shi, Jin; Zhou, Jianwei Characteristic classes on Grassmannians.
Turkish J. Math. 38 (2014), no. 3,492–523.

GGM1: Submanifolds of G_3 R^7.

GGM2-3: Applications to free immersions.



Constant scalar curvature metrics on Hirzebruch surfaces
Otoba metrics on Hirzebruch surfaces.

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

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

CscH 2: Norm of the Weyl curvature.

CscH 3: 4th order derivatives. Bach Tensor.

CscH 4: Ordinary differential equations. Elliptic integrals.

CscH 5: Volumes. Yamabe Invariants.

CscH 6: Solving the ODE. Elliptic integrals. Jacobi elliptic functions

CscH 7: Yamabe constants. Quadratic curvature functionals.

CscH 8: Yamabe minimizers. Conformal geometry.

CscH 9: Quadratic curvature functionals. B_t functional.

CscH 10: Bach-flat equation.

CscH 11: U(2) Invariant metrics.

CscH 12: Relation to Einstein-Maxwell metrics.

CscH 13: Bach-flat equation.


Şimşir: Using a flat connection we define a certain cohomology similar to Dolbeault Cohomology which plays an important role in the study of flat affine manifolds. In this talk, we prove fundamental identities of the exterior and interior product operators and the star operator on the space of all (p,q) forms.

Mahir: In this talk we present our work on equivariant K-theory of spherical varieties. After explaining our general result, a description of the equivariant K-rings for smooth complete spherical varieties, we present its applications to the wonderful compactifications of minimal rank symmetric varieties. In particular, we will work out our theorem in the case of complete collineations. This is joint work with Soumya Banerjee (Ben Gurion Univ).

Kalafat: We show that a compact complex surface which admits a conformally Kähler metric g of positive orthogonal holomorphic bisectional curvature is biholomorphic to the complex projective plane. In addition, if g is a Hermitian metric which is Einstein, then the biholomorphism can be chosen to be an isometry via which g becomes a multiple of the Fubini-Study metric. This is a joint work with C.Koca. Reference:
Kalafat, Mustafa; Koca, Caner - Conformally Kähler surfaces and orthogonal holomorphic bisectional curvature. Geom. Dedicata 174 (2015), 401-408.

Vanlı : Benoit Mandelbrot, IBM Laboratuvarlarında çalışırken doğada ifade edilemeyen nesnelerin geometrisini tanımlayabileceğini belki de hiç düşünmemişti. 1870’de Greg Cantor, 1890’da da Giuseppe Peano, nokta kümelerinden oluşan şekiller tanımlamışlardı. 1904’de Helge von Koch kar tanesini 1915’de ise Waclaw Sierpinski, Sierpinski eleğini tanımladı. 20. yüzyılın başlarında, Gaston Maurice Julia polinom denklemlerinin tekrar edilmesiyle oluşan matematiksel sistemi tanımladı. Bu buluşlar bir asır sonra 1975 yılında Benoit Mandelbrot’un tanımladığı Fraktal Geometriye birer örnek oldular. Bu seminerde Fraktal Geometrinin doğuşu ve gelişimi hakkında bilgiler verilerek gizemli Fraktal Geometri Dünyasından görsel sunumlar yapılacaktır.



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