Seminar on Geometry
Spring/Fall 2021, Boğaziçi, İstanbul
Time / Location: Fridays 3:00 / TB240 
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

8manifolds with locally conformally Spin(7)structure
Over GMeet 
Jan 22 Fri, 3:00 

Locally Conformally Kähler Manifolds 3
cSasakian 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 EinsteinMaxwell 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
CheegerGromoll 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 2325 

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 1213 

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 27

Youtube 
The 6th GTSS GeometryTopology Summer School
FGE  Week 1 
Aug 914

Videos 
Summer School
FGE  Week 2 
Aug 20 Fri, 3:00 

Riemann Surfaces in EinsteinHermitian Spaces

Aug 27 Fri, 3:00 

Riemann Surfaces in EinsteinHermitian Spaces 2

Sep 3 Fri, 3:00 

Riemann Surfaces in EinsteinHermitian Spaces 3

Sep 10 Fri, 3:00 

No Seminar

Sep 1318


The 7th GTSS GeometryTopology Summer School
Nesin Mathematics Village  Week 1 
Sep 2025


Summer School
Nesin Mathematics Village  Week 2 
Oct 1 Fri, 3:00 

No Seminar

Oct 8 Fri, 3:00 

Riemann Surfaces in EinsteinHermitian Spaces 4

Oct 1115


Special Geometries on Riemannian Manifolds
Hybrid workshop at CRMMontréal 
Oct 15 Fri, 3:00 

Riemann Surfaces in EinsteinHermitian Spaces 5
Counting Jacobi fields on minimal submanifolds 
Oct 22 Fri, 3:00 

Riemann Surfaces in EinsteinHermitian Spaces 6
ChernWeil Theory approach to Characteristic Classes 
Oct 29 Fri, 3:00 

Riemann Surfaces in EinsteinHermitian Spaces 7
An application of the RiemannRoch Formula to Minimal Surfaces A 
Nov 5 Fri, 3:00 

Riemann Surfaces in EinsteinHermitian Spaces 8
An application of the RiemannRoch Formula to Minimal Surfaces B 
Nov 12 Fri, 3:00 

Riemann Surfaces in EinsteinHermitian Spaces 9
An application of the RiemannRoch Formula to Minimal Surfaces C 
Nov 19 Fri, 3:00 

Riemann Surfaces in EinsteinHermitian Spaces 10
Higher osculating spaces and higher fundamental forms of a submanifold 
Nov 26 Fri, 3:00 

No Seminar

Dec 3 Fri, 3:00 

Riemann Surfaces in EinsteinHermitian Spaces 11
Higher fundamental forms of a submanifold 
Dec 10 Fri, 3:00 
Erdem Şafak Öztürk

Chern  Weil Theory and Characteristic Classes 1
Preliminaries 
Dec 17 Fri, 3:00 

Chern  Weil Theory and Characteristic Classes 2
Cancelled 
Dec 24 Fri, 2:30 

Chern  Weil Theory and Characteristic Classes 2
ChernWeil homomorphism and invariant polynomials 
Dec 31 Fri, 2:30 

Chern  Weil Theory and Characteristic Classes 3
Invariant polynomials and Pontrjagin classes 
Jan 7 Fri, 2:30 

Chern  Weil Theory And Characteristic Classes 4

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: cSasakian 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:
 S. Dragomir, L. Ornea 
Locally conformal Kähler geometry.
Progress in Mathematics, 155. Birkhäuser Boston, Inc., Boston, MA, 1998.
 Vaisman, Izu. Some curvature properties of complex surfaces.
Ann. Mat. Pura Appl. (4) 132 (1982), 1–18 (1983).
 Vaisman, Izu. On locally and globally conformal Kähler manifolds.
Trans. Amer. Math. Soc. 262 (1980), no. 2, 533–542.
 Gauduchon, Paul. La 1forme de torsion d'une variété hermitienne compacte.
[Torsion 1forms of compact Hermitian manifolds] Math. Ann. 267 (1984), no. 4, 495–518.
 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: CheegerGromoll Metric and the Unit Sphere.
We will be using the following resources.
References:
 S. Dragomir, L. Ornea 
Locally conformal Kähler geometry.
Progress in Mathematics, 155. Birkhäuser Boston, Inc., Boston, MA, 1998.
 Falcitelli, Ianus, Pastore. Riemannian submersions and related topics.
World Scientific Publishing Co., Inc., River Edge, NJ, 2004.
 Şahin, B. Riemannian submersions, Riemannian maps in Hermitian geometry, and their applications.
Elsevier/Academic Press, London, 2017.
Lectures on Stability Of Minimal Submanifolds
A minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature.
They are 2dimensional analogue to geodesics, which are analogously defined as critical points of the length functional.
Minimal surface theory originates with Lagrange who in 1762 considered the variational problem of finding
the surface z = z(x, y) of least area stretched across a given closed contour.
He derived the Euler–Lagrange equation for the solution
He did not succeed in finding any solution beyond the plane. In 1776 Jean Baptiste Marie Meusnier discovered that
the helicoid and catenoid satisfy the equation and that the differential expression corresponds to twice the mean curvature of the surface,
concluding that surfaces with zero mean curvature are areaminimizing.
By expanding Lagrange's equation,
Gaspard Monge and Legendre in 1795 derived representation formulas for the solution surfaces. While these were successfully used by Heinrich Scherk in 1830 to derive his surfaces, they were generally regarded as practically unusable. Catalan proved in 1842/43 that the helicoid is the only ruled minimal surface.
Progress had been fairly slow until the middle of the century when the Björling problem was solved using complex methods. The "first golden age" of minimal surfaces began.
Schwarz found the solution of the Plateau problem for a regular quadrilateral in 1865 and for a general quadrilateral in 1867 using complex methods.
Weierstrass and Enneper developed more useful representation formulas, firmly linking minimal surfaces to complex analysis and harmonic functions.
Other important contributions came from Beltrami, Bonnet, Darboux, Lie, Riemann, Serret and Weingarten.
Between 1925 and 1950 minimal surface theory revived, now mainly aimed at nonparametric minimal surfaces.
The complete solution of the Plateau problem by Jesse Douglas and Tibor Radó was a major milestone.
Bernstein's problem and Robert Osserman's work on complete minimal surfaces of finite total curvature were also important.
Another revival began in the 1980s. One cause was the discovery in 1982 by Celso Costa of a surface that disproved the conjecture that the plane,
the catenoid, and the helicoid are the only complete embedded minimal surfaces in R^3 of finite topological type.
This not only stimulated new work on using the old parametric methods,
but also demonstrated the importance of computer graphics to visualise the studied surfaces and numerical methods to solve the "period problem"
(when using the conjugate surface method to determine surface patches that can be assembled into a larger symmetric surface,
certain parameters need to be numerically matched to produce an embedded surface).
Another cause was the verification by H. Karcher that the triply periodic minimal surfaces originally described empirically by Alan Schoen in 1970 actually exist. This has led to a rich menagerie of surface families and methods of deriving new surfaces from old, for example by adding handles or distorting them.
Currently the theory of minimal surfaces has diversified to minimal submanifolds in other ambient geometries,
becoming relevant to mathematical physics (e.g. the positive mass conjecture, the Penrose conjecture)
and threemanifold geometry (e.g. the Smith conjecture, the Poincaré conjecture, the Thurston Geometrization Conjecture).
In this lecture series we will give an introduction to some topics in minimal submanifold theory.
The topics to be covered are as follows.
 Mean curvature vector field on a Riemannian submanifold.
 First variational formula for the volume functional.
 Second variation of energy for a minimally immersed submanifold.
 Stability of minimal submanifolds.
We will be using the following resources.
References:
Li, Peter. Geometric analysis. Cambridge University Press, 2012.
Riemann Surfaces in EinsteinHermitian Spaces
This is a continuation of the basic minimal submanifold theory lectures.
In particular minimal embeddings of spheres into higher dimensional spheres especially as holomorphic curves.
Topics to be covered are as follows.
RS 1: Jacobi Operator. Higher dimensional fundamental forms.
RS 2: Higher dimensional curvatures.
RS 3: Minimal immersions into higher spheres.
RS 4: A Hermitian structure on the normal bundle.
RS 5: Counting Jacobi fields on minimal submanifolds.
RS 6: ChernWeil Theory approach to Characteristic Classes.
RS 7: An application of the RiemannRoch formula to the minimal surfaces.
RS 8: Index of minimal immersions of a sphere into higher dimensional spheres.
RS 9: Holomorphic curves in the 6dimensional sphere.
RS 10: Higher osculating spaces and higher fundamental forms of a submanifold.
RS 11: Holomorphic curves in the 6dimensional sphere.
RS 12: Higher fundamental forms of a submanifold.
We will be using the following resources.
References:
 N. Ejiri  The Index of Minimal Immersions of S^2 into S^{2n}.
Mathematische Zeitschrift. (1983).
 Kühnel, Wolfgang  Differential geometry. Curves—surfaces—manifolds. Third edition.
Translated from the 2013 German edition. American Mathematical Society. 2015.
 J. Madnick  The Second Variation of NullTorsion Holomorphic Curves in the 6Sphere.
ArXiv:2101.09580 (Jan 2021) 35 pages.
 J. Madnick  FreeBoundary Problems for Holomorphic Curves in the 6Sphere.
Arxiv:2105.10562 (May 2021). 17 pages.
 S. Montiel and F. Urbano  Second Variation of Superminimal Surfaces into SelfDual Einstein 4Manifolds.
Trans. AMS. (1997).
Lectures on 8manifolds with locally conformally Spin(7)structure
In this seminar series we will give elementary talks on Spin(7) geometry on 8manifolds.
References:
 Agricola, Ilka  The Srní lectures on nonintegrable geometries with torsion.
Arch. Math. (Brno) 42 (2006), suppl., 5–84.
 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 8dimensional 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 8dimensional manifolds admitting
Spin(7)structure. In general, torsionfree 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 8dimensional manifold admits locally
conformal parallel Spin(7) structure.
Salur and Yalcinkaya [6] studied almost symplectic structure on
Spin(7)manifold with 2plane field. Then, Fowdar [2] studied Spin(7)
metrics from K ̈ahler geometry. In this research, we introduce 8manifold
equipped with locally conformal Spin(7)structure with 2plane field.
Then, almost Hermitian 6manifold 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)
Chern  Weil Theory and Characteristic Classes
Chern  Weil homomorphism computes topological invariants of vector bundles and principal bundles on a given C^{\infty}manifold via connection forms and curvature forms representing de Rham cohomology classes of the manifold. Considering the concepts connection and curvature, the theory becomes a bridge between algebraic topology and differential geometry.
Topics to be covered are as follows.
CW 1: Preliminaries on vector bundles, connection and curvature.
CW 2: ChernWeil homomorphism and invariant polynomials.
CW 3: Algebra of the algebra of the invariant polynomials, ChernWeil homomorphism and Pontrjagin Classes.
We will be using the following resources.
References:
 Shigeyuki Morita  Geometry of differential forms.
 Loring W. Tu  Differential Geometry.
Seminars
Kalafat 1 :
Yüksek boyutlarda geometri konularıyla ilgili bir tanıtım
konuşmasıdır. Öncelikle, nboyutlu 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 wellknown settings (CalabiYau, KählerEinstein, constantscalarcurvatureKähler, extremal Kähler). In light of recent examples and developments,
I will elucidate a possible role of EinsteinMaxwell 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 finitedimensional slice to the moment map (scalar curvature).
This argument reduces the local picture of constant scalar curvature metrics,
and other geometries, to a finitedimensional GIT problem.
The idea is old, but we consider filling some gaps in the argument.
Reference:
Székelyhidi, Gábor. The KählerRicci flow and Kpolystability. Amer. J. Math. 132 (2010), no. 4, 1077–1090.
Differential Geometry Seminar Archive
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