Seminar on Geometry

Spring/Fall 2022, Boğaziçi, İstanbul

Time / Location: Fridays 2:30 / TB-240

Schedule of talks

 

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, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 1
Hopf Problem
May 13
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 2
Clifford Torus, Willmore and Lawson Conjectures
May 20
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 3
SU(3) structure on the 6-sphere (i.e. almost Hermitian)
May 27
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 4
Index and nullity of superminimal curves in the 4-sphere
Jun 3
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 5
Superminimal curves in the 2n-sphere
Jun 10
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 6
Index and nullity of a minimal surface
Jun 17
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 7
TBA
Jun
20-24
Conference(Hybrid) Differential geometry and geometric analysis
Karsten Grove's 75th birthday conference in Florence, Italy
Jul 1
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 8
TBA
Jul 8
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 9
TBA
Jul 15
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 10
TBA
Jul 22
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 11
TBA
Jul 29
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 12
TBA
Aug 5
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 13
TBA
Aug 12
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 14
TBA
Aug 19
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 15
TBA
Aug 26
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 16
TBA
Aug 29
Sep 2
Conference Spaces, Structures, Symmetries
Bari, Italy
Sep 9
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 17
TBA
Sep 16
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 18
TBA
Sep 23
Fri, 2:30
Null torsion holomorphic curves in 6-dimensional sphere 19
TBA

Abstracts/Notlar


Lectures on the Minimal Spheres Inside Constant Curvature Spaces

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.

C4: Minimal Surfaces.

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.


References:
  1. 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.

  2. Calabi, Eugenio. - Minimal immersions of surfaces in Euclidean spheres.
    J. Differential Geometry 1 (1967), 111–125.




Null torsion holomorphic curves in 6-dimensional sphere


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-6: Jacobi Operator.

We will be using the following resources.

References:
  1. Madnick, Jesse. - The Second Variation of Null-Torsion Holomorphic Curves in the 6-Sphere.
    Available at the arXiv:2101.09580 (Jan 2021) 34 pages.

  2. 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.




Seminars

Kalafat : In this talk, we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group G_2. Some of these results are new. One often encounters these spaces when studying submanifolds of manifolds with calibrated geometries. As an application, we deduce the existence of certain special 3 and 4-dimensional submanifolds of G_2 holonomy Riemannian manifolds with special properties. These are called Harvey-Lawson(HL) pairs. Which appeared first in the work of Akbulut & Salur about G_2 dualities. Another application is to the free embeddings. We show that if there is a coassociative-free embedding of a 4-manifold into the Euclidean 7-space then the signature vanishes along with the Euler characteristic. As a more recent application, we exhibit a family of complex manifolds, which has a member at each odd complex dimension and which has the same cohomology groups as the complex projective space at that dimension, but not homotopy equivalent to it. We also compute various cohomology rings. (Joint work with S.Akbulut et al.)



Differential Geometry Seminar Archive


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