## Seminar on Geometry## Spring/Fall 2022, Boğaziçi, İstanbulTime / Location: Fridays 3:30 / TB-240 |
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TIME |
SPEAKER |
TITLE |

Jan 14 Fri, 2:30 |
Zoom link. Pass: geometry in Turkish.
Meeting ID: 991 1027 7750 |
Minimal Spheres Inside Constant Curvature Spaces 1
Mean curvature from an algebraic Cartan Lemma |

Jan 21 Fri, 2:30 |
Minimal Spheres Inside Constant Curvature Spaces 2
Higher osculating spaces and higher fundamental forms of a submanifold | |

Jan 28 Fri, 2:30 |
Minimal Spheres Inside Constant Curvature Spaces 3
Higher fundamental forms of a submanifold | |

Feb 4 Fri, 2:30 |
Minimal Spheres Inside Constant Curvature Spaces 4
Minimal Surfaces | |

Feb 11 Fri, 2:30 |
Minimal Spheres Inside Constant Curvature Spaces 5
Elliptic differential systems | |

Feb 18 Fri, 2:30 |
Minimal Spheres Inside Constant Curvature Spaces 6
Minimal immersions of the 2-sphere | |

Feb 25 Fri, 2:30 |
Minimal Spheres Inside Constant Curvature Spaces 7
The case of N-sphere as the ambient space | |

Mar 4 Fri, 2:30 |
No Seminar
| |

Mar 7 Mon, 5:00 |
Algebraic Topology of Lie groups and Grassmannians with applications to Geometry
Max-Planck-Institut Topology Seminar, Bonn | |

Mar 11 Fri, 2:30 |
Minimal Spheres Inside Constant Curvature Spaces 8
Elliptic differential systems | |

Mar 18 Fri, 2:30 |
Minimal Spheres Inside Constant Curvature Spaces 9
Minimal immersions of the 2-sphere | |

Mar 25 Fri, 2:30 |
Minimal Spheres Inside Constant Curvature Spaces 10
Frenet-Boruvka formulas for the minimal immersion of a 2-sphere | |

Mar 28-30 |
Workshop |
Workshop on Manifolds with Symmetries
University of Stuttgart - Poster |

Apr 1 Fri, 2:30 |
Minimal Spheres Inside Constant Curvature Spaces 11
Frenet-Boruvka formulas of a complex curve in a constant curvature space | |

Apr 8 Fri, 2:30 |
Minimal Spheres Inside Constant Curvature Spaces 12
The case of N-sphere as the ambient space | |

Apr 15 Fri, 2:30 |
Minimal Spheres Inside Constant Curvature Spaces 13
Minimal Spheres inside Round Spheres | |

Apr 22 Fri, 2:30 |
Minimal Spheres Inside Constant Curvature Spaces 14
Minimal Spheres inside Round Spheres 2 | |

Apr 29 May 1 |
Conference(Online) |
36th Annual Geometry Festival
Courant Institute of New York University |

May 6 Fri, 3:30 |
Null torsion holomorphic curves in 6-dimensional sphere 1
Hopf Problem | |

May 13 Fri, 3:30 |
Null torsion holomorphic curves in 6-dimensional sphere 2
Clifford Torus, Willmore and Lawson Conjectures | |

May 20 Fri, 3:30 |
Null torsion holomorphic curves in 6-dimensional sphere 3
SU(3) structure on the 6-sphere (i.e. almost Hermitian) | |

May 27 Fri, 3:30 |
Null torsion holomorphic curves in 6-dimensional sphere 4
Index and nullity of superminimal curves in the 4-sphere | |

Jun 3 Fri, 3:30 |
Null torsion holomorphic curves in 6-dimensional sphere 5
Superminimal curves in the 2n-sphere | |

Jun 10 Fri, 3:30 |
Null torsion holomorphic curves in 6-dimensional sphere 6
Index and nullity of a minimal surface | |

Jun 17 Fri, 3:30 |
Null torsion holomorphic curves in 6-dimensional sphere
Cancelled | |

Jun 20-24 |
Conference(Hybrid) |
Differential geometry and geometric analysis
Karsten Grove's 75th birthday conference in Florence, Italy |

Jul 1 Fri, 3:30 |
Null torsion holomorphic curves in 6-dimensional sphere
Cancelled | |

Jul 4 Mon, 21:00 |
Akademik Öyküler
İstanbul Üniversitesi Matematik Kulübü | |

Jul 8 Fri, 3:30 |
Null torsion holomorphic curves in 6-dimensional sphere 7
Minimal surfaces in the 6-sphere. First and second normal bundles | |

Jul 15 Fri, 3:30 |
Null torsion holomorphic curves in 6-dimensional sphere 8
Moving frames in the 6-sphere | |

Jul 22 Fri, 3:30 |
Null torsion holomorphic curves in 6-dimensional sphere 9
Moving frames in the 6-sphere 2 | |

Jul 29 Fri, 3:30 |
Null torsion holomorphic curves in 6-dimensional sphere 10
Holomorphic curvature and torsion | |

Aug 5 Fri, 3:30 |
Null torsion holomorphic curves in 6-dimensional sphere 11
Holomorphic Frenet equations | |

Aug 12 Fri, 3:30 |
Null torsion holomorphic curves in 6-dimensional sphere 12
Holomorphic interpretation of the Torsion | |

Aug 15-19 |
Youtube |
The 8th GTSS Geometry-Topology Summer School
IMBM - Week 1 |

Aug 22-26 |
Videos | Summer School
IMBM - Week 2 |

Aug 29 Sep 2 |
Conference |
Spaces, Structures, Symmetries
Bari, Italy |

Sep 9 Fri, 3:30 |
No Seminar
| |

Sep 12-16 |
The 9th GTSS Geometry-Topology Summer School
Nesin Mathematics Village - Week 1 | |

Sep 19-23 |
Summer School
Week 2 | |

Sep 30 Fri, 3:30 |
No Seminar
| |

Oct 7 Fri, 3:30 |
Minimal varieties in higher dimensional spheres 1
Introduction. Riemannian vector bundles. | |

Oct 13-16 |
Conference |
Symmetry and shape
Santiago de Compostela, Spain |

Oct 21 Fri, 3:30 |
Minimal varieties in higher dimensional spheres 2
Laplace operator | |

Oct 28 Fri, 3:30 |
Minimal varieties in higher dimensional spheres 3
Geometry of immersed submanifolds | |

Nov 4 Fri, 3:30 |
Minimal varieties in higher dimensional spheres 4
Curvature of the normal bundle | |

Nov 8 Tue, 2:15 |
Oberseminar Global Analysis and Operator Algebras |
On special submanifolds of the Page space
Location: Endenicher Allee 60, Room 0.008 |

Nov 11 Fri, 3:30 |
Minimal varieties in higher dimensional spheres 5
Mean curvature and minimal varieties | |

Nov 18 Fri, 4:00 |
Minimal varieties in higher dimensional spheres 6
The case of Kähler manifolds | |

Nov 25 Fri, 4:00 |
Minimal varieties in higher dimensional spheres 7
The case of Kähler manifolds 2 | |

Dec 2 Fri, 4:00 |
Minimal varieties in higher dimensional spheres 8
The case of Kähler manifolds 3 | |

Dec 9 Fri, 4:00 |
Minimal varieties in higher dimensional spheres 9
An extension of the Synge Lemma | |

Dec 16 Fri, 4:00 |
Minimal varieties in higher dimensional spheres 10
Laplacian of the Shape Operator | |

Dec 23 Fri, 4:00 |
Minimal varieties in higher dimensional spheres 11
Mean curvature and minimal varieties | |

Dec 30 Fri, 4:00 |
Minimal varieties in higher dimensional spheres 12
Curvature of the normal bundle | |

Jan 6 Fri, 4:00 |
Minimal varieties in higher dimensional spheres 13
Projection of parallel vector fields as the Kernel of the Jacobi operator |

In this learning seminar series, we focus on the minimal immersions of the 2-dimensional sphere in a space of constant sectional curvature. We will present some results and their proofs following the below resources.

Topics to be covered are:

** C1:** Connection and curvature forms. Mean curvature from an application of algebraic Cartan Lemma.

** C2:** Higher osculating spaces and higher fundamental forms of a submanifold.

** C3:** Higher fundamental forms of a submanifold.

** C5:** Elliptic differential systems.

** C6:** Minimal immersions of the 2-sphere.

** C7:** The case of N-sphere as the ambient space.

** C8:** Elliptic differential systems 2.

** C9:** Minimal immersions of the 2-sphere.

** C10:** Frenet-Boruvka formulas for the minimal immersion of a 2-sphere.

** C11:** Frenet-Boruvka formulas of a complex curve in a constant curvature space.

** C12:** The case of N-sphere as the ambient space.

** C13:** Minimal Spheres inside Round Spheres.

** C14:** Minimal Spheres inside Round Spheres 2.

We will be using the following resources.

- Chern, Shiing Shen.
*- On the minimal immersions of the two-sphere in a space of constant curvature.*

Problems in analysis (Lectures at the Sympos. in honor of Salomon Bochner, Princeton Univ., N.J., 1969),

pp. 27–40. Princeton Univ. Press, 1970. - Calabi, Eugenio.
*- Minimal immersions of surfaces in Euclidean spheres.*

J. Differential Geometry 1 (1967), 111–125.

In this learning seminar series we will make an introduction to the
holomorphic curves in the 6-dimensional sphere.
Recall that the 6-sphere is the only sphere
in higher dimesions which has an almost complex structure. See the *Reference-2* for the history of the Hopf problem.

This allows us to talk about holomorphic curves inside it.
Since they are complex curves of a complex dimension 3-space, they
have an analogue of Frenet-Serre frame equations called *Frenet-Boruvka formulas.*
In the round 6-sphere, null-torsion holomorphic curves are fundamental examples of minimal surfaces.

Ingredients of the individual seminars are as follows:

** S-1:** Hopf Problem.

** S-2:** Clifford Torus, Willmore and Lawson Conjectures

** S-3:** SU(3) structure on the 6-sphere (i.e. almost Hermitian).

** S-4:**Index and nullity of superminimal curves in the 4-sphere.

** S-5:**Superminimal curves in the 2n-sphere.

** S-7:** Minimal surfaces in the 6-sphere. First and second normal bundles.

** S-8:** Moving frames in the 6-sphere.

** S-9:** Moving frames in the 6-sphere 2.

** S-10:** Holomorphic curvature and torsion.

** S-11:** Holomorphic Frenet equations.

** S-12:** Holomorphic interpretation of the Torsion.

** S-13:** Characteristic classes of bundles in the presence of null-torsion.

- Madnick, Jesse.
*- The Second Variation of Null-Torsion Holomorphic Curves in the 6-Sphere.*

Available at the arXiv:2101.09580 (Jan 2021) 34 pages. - Agricola, I; Bazzoni, G; Goertsches, O; Konstantis, P; Rollenske, S.
*- On the history of the Hopf problem.*

Differential Geom. Appl. 57 (2018), 1–9. Here is the Volume and the Event web site.

In this learning seminar series we will make an introduction to the
holomorphic curves in the 6-dimensional sphere.
Recall that the 6-sphere is the only sphere
in higher dimesions which has an almost complex structure. See the *Reference-2* for the history of the Hopf problem.

This allows us to talk about holomorphic curves inside it.
Since they are complex curves of a complex dimension 3-space, they
have an analogue of Frenet-Serre frame equations called *Frenet-Boruvka formulas.*
In the round 6-sphere, null-torsion holomorphic curves are fundamental examples of minimal surfaces.

Ingredients of the individual seminars are as follows:

** S-1:** Introduction. Riemannian vector bundles.

** S-3:** Geometry of immersed submanifolds.

** S-4:** Curvature of the normal bundle.

** S-5:** Mean curvature and minimal varieties.

** S-6:** The case of Kähler manifolds.

** S-7:** The case of Kähler manifolds 2.

** S-8:** The case of Kähler manifolds 3.

** S-9:** An extension of the Synge Lemma.

** S-10:** Laplacian of the Shape Operator.

** S-11:** Mean curvature and minimal varieties.

** S-12:** Curvature of the normal bundle.

** S-13:** Projection of parallel vector fields as the Kernel of the Jacobi operator.

- James Simons.
*- Minimal varieties in riemannian manifolds.*

Ann. of Math. (2), 88:62–105, 1968.

** Kalafat2 : **
Page manifold is the underlying differentiable manifold of the complex
surface, obtained out of the process of blowing up the complex
projective plane, only once. This space is decorated with a natural
Einstein metric, first studied by D.Page in 1978.
It is a cohomogeneity-1 manifold.
In this talk, we study some classes of submanifolds of codimension
one and two in the Page space. These submanifolds are totally
geodesic. We also compute their curvature and show that some of them
are constant curvature spaces.
Finally, we give information on how the Page space is related to
some other metrics on the same underlying smooth manifold.
Related paper may be accessed from the following:

Kalafat, Mustafa; Sarı, Ramazan.
On special submanifolds of the Page space.

Differential Geom. Appl. 74 (2021), 101708, 13 pp.

Differential Geometry Seminar Archive

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