Seminar on GeometrySpring/Fall 2024, Boğaziçi, İstanbulTime / Location: Fridays 4:00 / TB240 

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 2sphere 
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 510 
The 13th GTSS GeometryTopology 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 1719 
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 1722 
Youtube 
The 11th GTSS GeometryTopology Summer School
IMBM  Week 1 
Jul 2429 
Videos  Summer School
IMBM  Week 2 
Jul 31 Aug 4 
Conference  Workshop on Curvature and Global Shape
Münster 
Aug 711 
Conference  Analytic Methods in Complex Geometry
Münster 
Aug 18 Fri, 4:00 
Minimal Surfaces and the Bernstein Problem 15
Scalar curvature rigidity 2  
Aug 2026 
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 1116 
The 12th GTSS GeometryTopology
Summer School
Nesin Mathematics Village  Week 1  
Sep 1823 
Summer School
Week 2  
Sep 29 Fri, 4:00 
No Seminar
 
Oct 46 
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 
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:
BS1: Introduction, metrics on the 2sphere.
BS3: Spectrum of the complex projective space.
BS4: Jacobi Fields on Spheres.
BS5: Orthogonal, graded decomposition of the Berger Eigenspace.
BS6: Nontriviality of Eigenspaces.
BS7: Dimension counting for Eigenspaces.
BS8: Dimension counting continued.
BS9: First eigenvalue of the Laplacian on Berger Spheres.
We will be using the following resources.
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:
B1: Index of Spheres as totally geodesic minimal submanifolds.
B2: Jacobi fields on totally geodesic minimal spheres in higher dimensional spheres.
B3: Negative definite index form.
B4: Killing Fields on the sphere.
B5: Using Killing Fields on the sphere to find nullity.
B6: An extrinsic rigidity theorem.
B7: Rigidity theorem for higher codimension.
B8: Rigidity theorem for higher codimension 2.
B10: Proof of the rigidity theorem.
B11: Reformulation of the rigiditty theorem.
B12: Laplacian of the second fundamental form on a sphere.
B13: Laplacian of the second fundamental form on a hypersurface of a sphere.
B14: Estimates on the shape operator.
B14: Scalar curvature rigidity.
We will be following the classical beautiful paper.
The Plateau problem can be stated as follows:
Given an (n−1)manifold(surface) as a boundary in an (n+k)manifold,
find an nsurface 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 massminimizing 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:
Differential Geometry Seminar Archive
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