Seminar on Geometry

Spring/Fall 2024, Boğaziçi, İstanbul

Time / Location: Fridays 4:00 / TB-240

Schedule of talks

 

TIME              SPEAKER                  TITLE
Jan 5
Fri, 4:00
Zoom link. Pass: geometry in Turkish.
Meeting ID: 991 1027 7750
Eigenvalues of the Laplacian on Berger spheres 1
Introduction, metrics on the 2-sphere
Jan 12
Fri, 4:00
Eigenvalues of the Laplacian on Berger spheres 2
Berger sphere metrics
Jan 19
Fri, 4:00

Eigenvalues of the Laplacian on Berger spheres 3
Spectrum of the complex projective space
Jan 26
Fri, 4:00
Eigenvalues of the Laplacian on Berger spheres 4
Jacobi Fields on Spheres
Feb 2
Fri, 4:00
No Seminar
Feb 5-10

The 13th GTSS Geometry-Topology Winter School
Nesin Mathematics Village
Feb 16
Fri, 4:00
No Seminar
Feb 23
Fri, 4:00
Eigenvalues of the Laplacian on Berger spheres 5
Orthogonal, graded decomposition of the Berger Eigenspace
Mar 1
Fri, 4:00

Eigenvalues of the Laplacian on Berger spheres 6
Nontriviality of Eigenspaces
Mar 8
Fri, 4:00
Eigenvalues of the Laplacian on Berger spheres 7
Dimension counting for Eigenspaces
Mar 15
Fri, 4:00

Eigenvalues of the Laplacian on Berger spheres 8
Dimension counting continued
Mar 22
Fri, 4:00
Eigenvalues of the Laplacian on Berger spheres 9
First eigenvalue of the Laplacian
Mar 29
Fri, 4:00
Eigenvalues of the Laplacian on Berger spheres 10
First eigenvalue of the Laplacian on Berger Spheres
Apr 7
Fri, 4:00
Minimal Surfaces and the Bernstein Problem 6
An extrinsic rigidity theorem
Apr 14
Fri, 3:30
Craig van Coevering
Boğaziçi
Extremal Kähler metrics and the moment map
Online AG Seminar
Apr 21
Fri, 4:00
Holiday
Ramazan Bayramı
Apr 28
Fri, 3:00
Minimal Surfaces and the Bernstein Problem 7
Rigidity theorem for higher codimension
May 5
Fri, 3:00
Minimal Surfaces and the Bernstein Problem 8
Rigidity theorem for higher codimension 2
May 12
Fri, 3:00
Minimal Surfaces and the Bernstein Problem 9
Sphere rigidity theorem
May
17-19
Conference A Complex Differential Geometry Meeting at UniTo
Università degli Studi di Torino
May 24
Wed, 2:00
Dies Academicus
May 26
Fri, 4:00
No Seminar
Jun 2
Fri, 4:00
Holiday
Pfingstferien
Jun 9
Fri, 4:00
Minimal Surfaces and the Bernstein Problem 10
Proof of the rigidity theorem
Jun 16
Fri, 4:00
Minimal Surfaces and the Bernstein Problem 11
Reformulation of the rigidity theorem
Jun 23
Fri, 4:00
Minimal Surfaces and the Bernstein Problem 12
Laplacian of the second fundamental form on a sphere
Jun 30
Fri, 4:00
Minimal Surfaces and the Bernstein Problem 13
Laplacian of the second fundamental form on a hypersurface of a sphere
Jul 7
Fri, 4:00
Minimal Surfaces and the Bernstein Problem 14
Estimates on the shape operator
Jul 14
Fri, 4:00
Minimal Surfaces and the Bernstein Problem 15
Scalar curvature rigidity
Jul 17-22
Youtube The 11th GTSS Geometry-Topology Summer School
IMBM - Week 1
Jul 24-29
Videos Summer School
IMBM - Week 2
Jul 31
Aug 4
Conference Workshop on Curvature and Global Shape
Münster
Aug
7-11
Conference Analytic Methods in Complex Geometry
Münster
Aug 18
Fri, 4:00
Minimal Surfaces and the Bernstein Problem 15
Scalar curvature rigidity 2
Aug
20-26
Conference Prospects in Geometry and Global Analysis
Castle Rauischholzhausen, Marburg
Sep 1
Fri, 4:00
Minimal Surfaces and the Bernstein Problem 16
Holomorphic interpretation of the Torsion
Sep 8
Fri, 4:00
No Seminar
Sep 11-16

The 12th GTSS Geometry-Topology Summer School
Nesin Mathematics Village - Week 1
Sep 18-23

Summer School
Week 2
Sep 29
Fri, 4:00
No Seminar
Oct 4-6
Workshop Workshop on Einstein manifolds
Crazy World of Arthur L. Besse, Stuttgart
Oct 13
Fri, 4:00
Minimal Surfaces and the Bernstein Problem 16
Holomorphic interpretation of the Torsion
Oct 20
Fri, 3:30
Minimal varieties in higher dimensional spheres 17
Laplace operator
Oct 27
Fri, 3:30
No Seminar
Nov 3
Fri, 3:30
Minimal varieties in higher dimensional spheres 18
Cone shaped minimal varieties
Nov 10
Fri, 4:00
Minimal varieties in higher dimensional spheres 19
Cone shaped minimal varieties
Nov 17
Fri, 3:30
Minimal varieties in higher dimensional spheres 20
Cone shaped minimal varieties
Nov 24
Fri, 4:00
Minimal varieties in higher dimensional spheres 21
Cone shaped minimal varieties
Dec 1
Fri, 4:00
Minimal varieties in higher dimensional spheres 22
Cone shaped minimal varieties
Dec 8
Fri, 4:00
Plateau problem for minimal surfaces in differential geometry 1
Currents
Dec 15
Fri, 4:00
Plateau problem for minimal surfaces in differential geometry 2
An extension of the Synge Lemma
Dec 22
Fri, 4:00
Plateau problem for minimal surfaces in differential geometry 3
Laplacian of the Shape Operator
Dec 29
Fri, 4:00
Plateau problem for minimal surfaces in differential geometry 4
Mean curvature and minimal varieties

Abstracts/Notlar


Eigenvalues of the Laplacian on Berger spheres


Berger spheres are introduced by Marcel Berger. These are odd dimensional spheres, which are squashed in a direction. The shrinking direction can be provided by the circle action on the complex Euclidean space. In other words, it corresponds to shrinking or expanding the Hopf circles in an odd dimensonal sphere. Consequently, the resulting Riemannian metric is different than the round metric. In these seminar series, we understand the eigenvalues of the Laplacian on Berger spheres.

Ingredients of the individual seminars are as follows:

BS-1: Introduction, metrics on the 2-sphere.

BS-2: Berger sphere metrics.

BS-3: Spectrum of the complex projective space.

BS-4: Jacobi Fields on Spheres.

BS-5: Orthogonal, graded decomposition of the Berger Eigenspace.

BS-6: Nontriviality of Eigenspaces.

BS-7: Dimension counting for Eigenspaces.

BS-8: Dimension counting continued.

BS-9: First eigenvalue of the Laplacian on Berger Spheres.

BS-10:

We will be using the following resources.

References:
  1. Tanno, Shûkichi. The topology of contact Riemannian manifolds.
    Illinois J. Math. 12 (1968), 700–717.

  2. Tanno, Shûkichi. The first eigenvalue of the Laplacian on spheres.
    Tohoku Math. J. (2) 31 (1979), no. 2, 179–185.




Minimal Surfaces and the Bernstein Problem


In this learning seminar series, we will give an introduction to minimal submanifolds of the higher dimensional spheres. In particular we give estimates on the index of the Jacobi operator. Talk about applications on the Plateau's problem and Bernstein conjecture.

Ingredients of the individual seminars are as follows:

B-1: Index of Spheres as totally geodesic minimal submanifolds.

B-2: Jacobi fields on totally geodesic minimal spheres in higher dimensional spheres.

B-3: Negative definite index form.

B-4: Killing Fields on the sphere.

B-5: Using Killing Fields on the sphere to find nullity.

B-6: An extrinsic rigidity theorem.

B-7: Rigidity theorem for higher codimension.

B-8: Rigidity theorem for higher codimension 2.

B-9: Sphere rigidity theorem.

B-10: Proof of the rigidity theorem.

B-11: Reformulation of the rigiditty theorem.

B-12: Laplacian of the second fundamental form on a sphere.

B-13: Laplacian of the second fundamental form on a hypersurface of a sphere.

B-14: Estimates on the shape operator.

B-14: Scalar curvature rigidity.

We will be following the classical beautiful paper.

References:
  1. James Simons. - Minimal varieties in riemannian manifolds.
    Ann. of Math. (2), 88:62–105, 1968.

Plateau problem for minimal surfaces in differential geometry


The Plateau problem can be stated as follows: Given an (n−1)-manifold(surface) as a boundary in an (n+k)-manifold, find an n-surface that is bounded by that boundary and has minimal area. The problem was first posed by Lagrange in 1760, and named after the Belgian Physicist Joseph Plateau, who studied soap films and observed several laws of their geometric properties.

Depending on the conditions we impose on the boundary and enclosing surface, the ambient manifold M, the codimension k and the interpretation of ”bounded by Γ”, we have variants of the Plateau problem. In this talk we are mainly focused on the oriented codimension one Plateau problem: Given a closed oriented immersed (n−1)-surface in the Euclidean (n+1)-space, find a oriented bounding surface which has minimal area among other candidates. To better understand the bounding condition, consider the following example. Take two parallel circles in R3 that are closed to each other. The oriented solution will be a catenoid if the two components are equipped with different orientations, and two disks if the two components are given the same orientation. Also, we know from the example that the oriented solution may not be minimizing among all surfaces that span.

To solve the Plateau problem, one wants to take a minimizing sequence of surfaces Σi, and hope that Σi converges to some minimal surface Σ. However, in general we do not have convergence as the area bound is not strong enough to control the surface. In the same spirit as the weak solution of a PDE, we want to find a space of ”weak manifolds” in which a notion of ”mass” is defined, and has nice functional analysis properties:

1. The space has good compactness property, so for a mass-minimizing sequence Σi, we can find a convergent subsequence.

2. The mass functional is lower semicontinuous, so the limit is minimizing.

3. The ”weak solution” generated above is actually regularity, thus a ”classical solution”.

In 1960, Federer and Fleming came up with a very powerful setting, called integral currents, which is suitable for the discussion of the oriented Plateau problem.

We will be using the following resources.

References:

  1. Jenny Harrison and Harrison Pugh. - Plateau’s Problem: What’s Next.
    In: arXiv:1509.03797v2 2016.

  2. Leon Simon. - Geometric Measure Theory.
    Stanford Univ. lecture notes.

  3. James Simons. - Minimal varieties in riemannian manifolds.
    Ann. of Math. (2), 88:62–105, 1968.




Seminars

Craig : An extremal Kähler metric is a canonical Kähler metric, introduced by E. Calabi, which is somewhat more general than a constant scalar curvature Kähler metric. The existence of such a metric is an ongoing research subject and expected to be equivalent to some form of geometric stability of the underlying polarized complex manifold (M,J,[ω]) –the Yau-Tian-Donaldson conjecture. Thus it is no surprise that there is a moment map, the scalar curvature (A. Fujiki, S. Donaldson), and the problem can be described as an infinite dimensional version of the familiar finite dimensional G.I.T.

I will describe how the moment map can be used to describe the local space of extremal metrics on a symplectic manifold. Essentially, the local picture can be reduced to finite dimensional G.I.T. In particular, we can construct a course moduli space of extremal Kähler metrics with a fixed polarization [ω]∈H2(M,ℝ) , which is an Hausdorff complex analytic space



Differential Geometry Seminar Archive


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Activities are supported by University of Bonn and Boğaziçi University