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
 Youtube The 6th GTSS Geometry-Topology Summer School
FGE - Week 1
Aug 9-14
 Videos Summer School
FGE - Week 2
Aug 20
Fri, 3:00		 Riemann Surfaces in Einstein-Hermitian Spaces
Aug 27
Fri, 3:00		 Riemann Surfaces in Einstein-Hermitian Spaces 2
Sep 3
Fri, 3:00
Riemann Surfaces in Einstein-Hermitian Spaces 3
Sep 10
Fri, 3:00
No Seminar
Sep 13-18
The 7th GTSS Geometry-Topology Summer School
Nesin Mathematics Village - Week 1
Sep 20-25
Summer School
Nesin Mathematics Village - Week 2
Oct 1
Fri, 3:00		 No Seminar
Oct 8
Fri, 3:00
Riemann Surfaces in Einstein-Hermitian Spaces 4
Oct 11-15
Special Geometries on Riemannian Manifolds
Hybrid workshop at CRM-Montréal
Oct 15
Fri, 3:00
Riemann Surfaces in Einstein-Hermitian Spaces 5
Counting Jacobi fields on minimal submanifolds
Oct 22
Fri, 3:00
Riemann Surfaces in Einstein-Hermitian Spaces 6
Chern-Weil Theory approach to Characteristic Classes
Oct 29
Fri, 3:00
Riemann Surfaces in Einstein-Hermitian Spaces 7
An application of the Riemann-Roch Formula to Minimal Surfaces A
Nov 5
Fri, 3:00
Riemann Surfaces in Einstein-Hermitian Spaces 8
An application of the Riemann-Roch Formula to Minimal Surfaces B
Nov 12
Fri, 3:00
Riemann Surfaces in Einstein-Hermitian Spaces 9
An application of the Riemann-Roch Formula to Minimal Surfaces C
Nov 19
Fri, 3:00
Riemann Surfaces in Einstein-Hermitian 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 Einstein-Hermitian 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
Chern-Weil 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: 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

A minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature. They are 2-dimensional 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 area-minimizing.

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 three-manifold 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.
  1. Mean curvature vector field on a Riemannian submanifold.

  2. First variational formula for the volume functional.

  3. Second variation of energy for a minimally immersed submanifold.

  4. Stability of minimal submanifolds.
We will be using the following resources.

References:

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



Riemann Surfaces in Einstein-Hermitian 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: Chern-Weil Theory approach to Characteristic Classes.

RS 7: An application of the Riemann-Roch formula to the minimal surfaces.

RS 8: Index of minimal immersions of a sphere into higher dimensional spheres.

RS 9: Holomorphic curves in the 6-dimensional sphere.

RS 10: Higher osculating spaces and higher fundamental forms of a submanifold.

RS 11: Holomorphic curves in the 6-dimensional sphere.

RS 12: Higher fundamental forms of a submanifold.

We will be using the following resources.

References:
  1. N. Ejiri - The Index of Minimal Immersions of S^2 into S^{2n}.
    Mathematische Zeitschrift. (1983).

  2. Kühnel, Wolfgang - Differential geometry. Curves—surfaces—manifolds. Third edition.
    Translated from the 2013 German edition. American Mathematical Society. 2015.

  3. J. Madnick - The Second Variation of Null-Torsion Holomorphic Curves in the 6-Sphere.
    ArXiv:2101.09580 (Jan 2021) 35 pages.

  4. J. Madnick - Free-Boundary Problems for Holomorphic Curves in the 6-Sphere.
    Arxiv:2105.10562 (May 2021). 17 pages.

  5. S. Montiel and F. Urbano - Second Variation of Superminimal Surfaces into Self-Dual Einstein 4-Manifolds.
    Trans. AMS. (1997).



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)



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: Chern-Weil homomorphism and invariant polynomials.

CW 3: Algebra of the algebra of the invariant polynomials, Chern-Weil homomorphism and Pontrjagin Classes.

We will be using the following resources.

References:
  1. Shigeyuki Morita - Geometry of differential forms.

  2. Loring W. Tu - Differential Geometry.



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.


Differential Geometry Seminar Archive

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