# Seminar on Geometry

## Spring/Fall 2022, Boğaziçi, İstanbul

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

## Schedule of talks

 TIME SPEAKER TITLE Jan 14Fri, 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 21Fri, 2:30 Minimal Spheres Inside Constant Curvature Spaces 2 Higher osculating spaces and higher fundamental forms of a submanifold Jan 28Fri, 2:30 Minimal Spheres Inside Constant Curvature Spaces 3 Higher fundamental forms of a submanifold Feb 4Fri, 2:30 Minimal Spheres Inside Constant Curvature Spaces 4 Minimal Surfaces Feb 11Fri, 2:30 Minimal Spheres Inside Constant Curvature Spaces 5 Elliptic differential systems Feb 18Fri, 2:30 Minimal Spheres Inside Constant Curvature Spaces 6 Minimal immersions of the 2-sphere Feb 25Fri, 2:30 Minimal Spheres Inside Constant Curvature Spaces 7 The case of N-sphere as the ambient space Mar 4Fri, 2:30 No Seminar Mar 7Mon, 5:00 Algebraic Topology of Lie groups and Grassmannians with applications to Geometry Max-Planck-Institut Topology Seminar, Bonn Mar 11Fri, 2:30 Minimal Spheres Inside Constant Curvature Spaces 8 Elliptic differential systems Mar 18Fri, 2:30 Minimal Spheres Inside Constant Curvature Spaces 9 Minimal immersions of the 2-sphere Mar 25Fri, 2:30 Minimal Spheres Inside Constant Curvature Spaces 10 Frenet-Boruvka formulas for the minimal immersion of a 2-sphere Mar28-30 Workshop Workshop on Manifolds with Symmetries University of Stuttgart - Poster Apr 1Fri, 2:30 Minimal Spheres Inside Constant Curvature Spaces 11 Frenet-Boruvka formulas of a complex curve in a constant curvature space Apr 8Fri, 2:30 Minimal Spheres Inside Constant Curvature Spaces 12 The case of N-sphere as the ambient space Apr 15Fri, 2:30 Minimal Spheres Inside Constant Curvature Spaces 13 Minimal Spheres inside Round Spheres Apr 22Fri, 2:30 Minimal Spheres Inside Constant Curvature Spaces 14 Minimal Spheres inside Round Spheres 2 Apr 29May 1 Conference(Online) 36th Annual Geometry Festival Courant Institute of New York University May 6Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 1 Hopf Problem May 13Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 2 Clifford Torus, Willmore and Lawson Conjectures May 20Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 3 SU(3) structure on the 6-sphere (i.e. almost Hermitian) May 27Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 4 Index and nullity of superminimal curves in the 4-sphere Jun 3Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 5 Superminimal curves in the 2n-sphere Jun 10Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 6 Index and nullity of a minimal surface Jun 17Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 7 TBA Jun20-24 Conference(Hybrid) Differential geometry and geometric analysis Karsten Grove's 75th birthday conference in Florence, Italy Jul 1Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 8 TBA Jul 8Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 9 TBA Jul 15Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 10 TBA Jul 22Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 11 TBA Jul 29Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 12 TBA Aug 5Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 13 TBA Aug 12Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 14 TBA Aug 19Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 15 TBA Aug 26Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 16 TBA Aug 29Sep 2 Conference Spaces, Structures, Symmetries Bari, Italy Sep 9Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 17 TBA Sep 16Fri, 2:30 Null torsion holomorphic curves in 6-dimensional sphere 18 TBA Sep 23Fri, 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.

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.

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

Differential Geometry Seminar Archive

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