Seminar on Geometry 

Spring/Fall 2021, Boğaziçi, İstanbul

Time / Location: Fridays 3:00 / TB-240

Schedule of talks

 

TIME              SPEAKER                  TITLE
Jan 3
Sun, 3:20
Zoom bağlantısından yayın yapılacaktır.
ID: 842 5215 9903 Şifre: Geometri
4 ve yukarısı yüksek boyutlarda Geometri ve Topoloji
Türkiye Matematik Kulübü (TMK). Konuşmanın Videosu
Jan 8
Fri, 3:00
Newest Zoom link. Pass: geometry in Turkish.
Meeting ID: 991 1027 7750
Locally Conformally Kähler Manifolds 1
Generalized Hopf Manifolds II
Jan 15
Fri, 3:00
Locally Conformally Kähler Manifolds 2
Vaisman metrics from Cosymplectic manifolds
Jan 20
Wed, 21:00
Eyüp Yalçınkaya
8-manifolds with locally conformally Spin(7)-structure
Over GMeet
Jan 22
Fri, 3:00

Locally Conformally Kähler Manifolds 3
c-Sasakian Manifolds
Jan 29
Fri, 3:00
Locally Conformally Kähler Manifolds 4
Foliations on a generalized Hopf (Vaisman) Manifold
Feb 5
Fri, 3:40
Caner Koca Kähler Geometry and Einstein-Maxwell Metrics
See AG Seminar webpage for Zoom details
Feb 12
Fri, 3:00
Riemannian Submersions 1
Introduction. O'Neill Tensors
Feb 19
Fri, 3:00
Riemannian Submersions 2
Gauss and Codazzi Equations
Feb 26
Fri, 3:00
Riemannian Submersions 3
Curvature under a Riemannian Submersion
Mar 5
Fri, 3:00
Riemannian Submersions 4
Examples of Riemannian Submersions
Mar 12
Fri, 3:00
Riemannian Submersions 5
Warped Products
Mar 19
Fri, 3:00
Riemannian Submersions 6
Sasakian Metric
Mar 26
Fri, 3:00
Riemannian Submersions 7
Sasakian Metric 2
Apr 2
Fri, 3:00
Riemannian Submersions 8
Cheeger-Gromoll Metric and the Unit Sphere
Apr 9
Fri, 3:00
No Seminar
Apr 10
Sat, 12:45
Zoom bağlantısından yayın yapılacaktır.
Katılım için kısa bir ön kayıt gerekmektedir.
Einstein uzayları, Twistorlar ve Karadeliklerin Geometrisi
İstanbul Üniversitesi Fen Fakültesi Matematik Kulübü
Apr 16
Fri, 3:00
Locally Conformally Kähler Manifolds 5
Regular Vaisman Manifolds
Apr
23-25
35th Annual Geometry Festival
Stony Brook University
Apr 30
Fri, 3:00
Locally Conformally Kähler Manifolds 6
LCK_0 manifolds
May 7
Fri, 3:00
Locally Conformally Kähler Manifolds 7
Spectral characterization of Hopf manifolds
May 14
Fri, 3:00
Locally Conformally Kähler Manifolds 8
May 21
Fri, 3:00
Locally Conformally Kähler Manifolds 9
May 28
Fri, 3:00
Locally Conformally Kähler Manifolds 10
Jun 4
Fri, 3:00
Stability of Minimal Submanifolds 1
Jun 11
Fri, 3:00
Stability of Minimal Submanifolds 2
Jun 18
Fri, 3:00
Stability of Minimal Submanifolds 3
Jun 25
Fri, 3:00

Stability of Minimal Submanifolds 4
Jul 2
Fri, 3:00
Craig van Coevering
The slice theorem in Kähler geometry
Jul 9
Fri, 3:00

The slice theorem in Kähler geometry 2
Jul
12-13

18th International Geometry Symposium
İnönü University - Online
Jul 16
Fri, 3:00
Eyüp Yalçınkaya
Spin(7) Geometry with Torsion
Jul 23
Fri, 3:00
Holiday
Kurban Bayramı
Jul 30
Fri, 3:00

Spin(7) Geometry with Torsion 2
Aug 2-7

The 6th GTSS Geometry-Topology Summer School
Week 1
Aug 9-14

Summer School
Week 2

Abstracts/Notlar


Lectures on Locally Conformally Kähler Manifolds

In this learning seminar series we will make an introduction to the locally conformally Kähler (LCK) geometry. A LCK metric is a structure on a complex manifold which falls somewhere between a Hermitian metric and a Kähler metric.

Ingredients of the individual seminars are as follows:

LCK 1: Generalized Hopf Manifolds II.

LCK 2: Vaisman metrics from Cosymplectic manifolds.

LCK 3: c-Sasakian Manifolds

LCK 4: Foliations on a generalized Hopf (Vaisman) Manifold

LCK 5: Regular Vaisman Manifolds.

LCK 6: LCK_0 manifolds.

LCK 7: Spectral characterization of Hopf manifolds.

We will be using the following resources.

References:
  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.

  5. Falcitelli, Ianus, Pastore. Riemannian submersions and related topics.
    World Scientific Publishing Co., Inc., River Edge, NJ, 2004.




Lectures on Riemannian Submersions


In this learning seminar series we will make an introduction to the theory of Riemannian Submersions (RS).


Ingredients of the individual seminars are as follows:

RS 1: Introduction. O'Neill Tensors.

RS 2: Gauss and Codazzi Equations.

RS 3: Curvature under a Riemannian Submersion.

RS 4: Examples of Riemannian Submersions.

RS 5: Warped Products.

RS 6: Sasakian Metric.

RS 7: Sasakian Metric 2.

RS 8: Cheeger-Gromoll Metric and the Unit Sphere.

We will be using the following resources.

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

  2. Falcitelli, Ianus, Pastore. Riemannian submersions and related topics.
    World Scientific Publishing Co., Inc., River Edge, NJ, 2004.

  3. Şahin, B. Riemannian submersions, Riemannian maps in Hermitian geometry, and their applications.
    Elsevier/Academic Press, London, 2017.




Lectures on Stability Of Minimal Submanifolds

In this lecture series we will give an introduction to some topics in minimal submanifold theory.

First and Second variational formula, stability of minimal submanifolds.

We will be using the following resources.

References:

Li, Peter. Geometric analysis. Cambridge University Press, 2012.




Lectures on 8-manifolds with locally conformally Spin(7)-structure


In this seminar series we will give elementary talks on Spin(7) geometry on 8-manifolds.


References:
  1. Agricola, Ilka - The Srní lectures on non-integrable geometries with torsion.
    Arch. Math. (Brno) 42 (2006), suppl., 5–84.

  2. Fernández, Marisa - A classification of Riemannian manifolds with structure group Spin(7).
    Ann. Mat. Pura Appl. (4) 143 (1986), 101–122.

LCSp7 1: Introduction.




Spin(7) Geometry with Torsion


Spin structures have wide applications to mathematical physics, in particular to quantum field theory. For the special class Spin(7) geometry, there are different approaches. One of them is constructed by holonomy groups. According to the Berger classification (1955), the Spin(7) group is one of these holonomy classes. Firstly, it is presented its properties. After that, torsion which is another important term in superstring theory will be geometrically introduced and related to Spin(7) geometry.

Let M be an 8-dimensional manifold with the Riemannian met- ric g and structure group G ⊂ SO(8). The structure group G ⊂ Spin(7), then it is called M admits Spin(7)-structure. M. Fernan- dez [1] classifies the all types of 8-dimensional manifolds admitting Spin(7)-structure. In general, torsion-free Spin(7) manifold are stud- ied considerably.

On the other hand, manifolds admitting Spin(7)-structure with tor- sion have rich geometry as well. Locally conformal parallel structures has been studied for a long time with K ̈ahler condition is the oldest one. By means of further groups whose holonomy is the exceptional, the choices of the G2 and Spin(7) deserves to attention. Ivanov [3], [4], [5] introduces a condition when 8-dimensional manifold admits locally conformal parallel Spin(7) structure. Salur and Yalcinkaya [6] studied almost symplectic structure on Spin(7)-manifold with 2-plane field. Then, Fowdar [2] studied Spin(7) metrics from K ̈ahler geometry. In this research, we introduce 8-manifold equipped with locally conformal Spin(7)-structure with 2-plane field. Then, almost Hermitian 6-manifold can be classified by the structure of M.

Keywords: Spin(7) structure, Torsion , Almost Hermitian structure 2010 Mathematics Subject Classification: Primary 53D15; Sec- ondary 53C29.


References:

[1] M. Fernandez, A Classification of Riemannian Manifolds with Structure Group Spin(7),
Annali di Mat. Pura ed App., vol (143), (1986), 101—122.

[2] U. Fowdar Spin(7) metrics from K ̈ahler Geometry, arXiv:2002.03449, (2020)

[3] S. Ivanov, M. Cabrera, SU(3)-structures on submanifolds of a Spin(7)-manifold,
Differential Geometry and its Applications,V 26 (2), (2008) 113–132

[4] S. Ivanov, M. Parton and P. Piccinni, Locally conformal parallel G2 and Spin(7)-manifolds
Mathematical Research Letters, V 13, (2006), 167–177.

[5] S. Ivanov Connections with torsion, parallel spinors and geometry of Spin(7) manifolds, math/0111216v3.

[6] S. Salur and E. Yalcinkaya Almost Symplectic Structures on Spin(7)-Manifolds,
Proceedings of the 2019 ISAAC Congress (Aveiro, Portugal), 2020)




Seminars

Kalafat 1 : 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.

Caner : A classical problem in Kähler Geometry is to determine a canonical representative in each Kähler class of a complex manifold. In this talk, I will introduce this problem in several well-known settings (Calabi-Yau, Kähler-Einstein, constant-scalar-curvature-Kähler, extremal Kähler). In light of recent examples and developments, I will elucidate a possible role of Einstein-Maxwell metrics in this problem.

Kalafat 2 : Günümüz Diferansiyel Geometrisinde güncel araştırma konularından bir demet takdim edilecektir. Einstein Uzaylarının eğriliği, Karadeliklerin modellenmesi, formülü ve eğriliği. Ricci ve Skaler eğrilik, Spinorlar, Solucan deliği uzayı, Paralel evrenler ve Twistorlar gibi uzaylardan bahsedeceğiz. Konuşmanın sonunda ise vakit elverdiği ölçüde yakın zamanda Nobel ödülü de alan, matematiksel dedem olan Sir Roger Penrose ve çalışma arkadaşı Stephen Hawking'in matematiksel katkılarından bahsedeceğim. Konuşmacının daha önce yaptığı ve internet ortamında da bir sürümü mevcut bulunan:

"4 ve Yukarısı Yüksek Boyutlarda Geometri ve Topoloji"

adlı konuşmaya tamamlayıcı nitelikte olacağı için, tercihen öncesinde veya sonrasında bu konuşmanın da izlenmesi önerilir. Konuşma Türkçe olup, lisans ve üstü öğrencilerine yöneliktir. Konuya uzak olan, ilgilenen öğretim üyeleri de davetlidir.

Craig : This talk considers a familiar argument in Kähler geometry in detail,
where on constructs a finite-dimensional slice to the moment map (scalar curvature).
This argument reduces the local picture of constant scalar curvature metrics,
and other geometries, to a finite-dimensional GIT problem.
The idea is old, but we consider filling some gaps in the argument.

Reference:

Székelyhidi, Gábor. The Kähler-Ricci flow and K-polystability. Amer. J. Math. 132 (2010), no. 4, 1077–1090.



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