[Turkmath:5727] Diferansiyel Geometri Yaz Okulu

[Mustafa Kalafat]

Sevgili Geometriseverler...


Diferansiyel Geometri Yaz okulu kapsamında, İstanbul Matematiksel
Bilimler Merkezinde (IMBM) çevrimiçi olarak,

   "15 - 27 Ağustos 2022"

tarihleri arasında, aşağıdaki araştırma dersleri verilmektedir:


  1. Craig van Coevering - Extremal Kähler metrics and the moment map

  2. Alberto Raffero - Closed G_2 structures

  3. Ernani Ribeiro Jr. - 4-dimensional gradient Ricci solitons

  4. Eyüp Yalçınkaya - Spin(7) Geometry

  5. Mustafa Kalafat - Minimal Surfaces


Dersler ''Online'' olarak gerçekleşmekte, güncel araştırma konularında,
fakat elementer düzeyde olmaktadır. Katılım ücretsiz olup kayıt gereklidir.
Dersler Zoom yazılımı üzerinden verilmektedir. Etkinliğin Web Sayfası:

    https://gtnmk.droppages.com/2022a/

herkese iyi çalışmalar, iyi tatiller...

Doç. Dr. Mustafa Kalafat




 |Mustafa Kalafat -------------------------------------------------
 |    Rheinische Friedrich-Wilhelms-Universität Bonn   |
 |    Office: Endenicher Allee 60, Zimmer 1.036            |
 |    Email: kalafat at math.uni-bonn.de                          |
 |    Web  : http://kalafat.droppages.com/                     |
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     DERS İÇERİKLERİ
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Lecturer: Alberto Raffero

Title: Lectures on closed G_2 structures

Abstract: These lectures include two introductory lectures on G2 geometry and
two lectures concerning recent developments and open problems on
closed G2 structures.

Lecture 1: Introduction to G2 geometry I

Lecture 2: Introduction to G2 geometry II

Lecture 3: Curvature properties and symmetries of closed G2-structures

Lecture 4: Special types of closed G2-structures (Laplacian solitons,
extremally Ricci pinched, exact)

References

R. L. Bryant. Some remarks on G2-structures.
Proceedings of Gökova Geometry-Topology Conference, 75–109, 2006.
Available at https://arxiv.org/abs/math/0305124

D. Joyce. Riemannian holonomy groups and calibrated geometry.
Oxford Graduate Texts in Mathematics, vol. 12, Oxford University Press, 2007.

S. Karigiannis, N. C. Leung, J. D. Lotay eds. Lectures and Surveys on
G2-manifolds and Related Topics,
Fields Institute Communications, vol. 84, Springer US, 2020.
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Lecturer: Ernani Ribeiro Jr.

Title: Four-dimensional gradient Ricci solitons

Abstract: In this minicourse, we discuss the geometry of
four-dimensional gradient shrinking Ricci solitons.

Lecture 1: We will show that gradient Ricci solitons are special
(self-similar) solutions of the Ricci flow.

Lecture 2: Some basic results and examples of gradient shrinking Ricci solitons.

Lecture 3: Classification of four-dimensional gradient shrinking Ricci solitons.

Lecture 4: Open problems and their motivations.

Textbook and References

Chow-Li-Nu. Hamilton’s Ricci Flow.

Huai-Dong Cao. Recent Progress on Ricci Solitons.
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Lecturer: Eyüp Yalçınkaya

Title: Spin(7) Geometry

Abstract: Spin structures have wide applications to mathematical
physics, in particular to quantum field theory. In order to study the
geometry of the special class Spin(7), there are different approaches.
One of them is by holonomy groups. According to the Berger
classification (1955), the group Spin(7) is one of the members of the
list of holonomy classes. Firstly, we are going to present its
properties [1]. After that, we will present normed algebras. Normed
algebras are an important concept of this geometry to define metric
and measure angle between vectors. Finally, we talk about Calabi-Yau
manifolds which play an important role in string theory. Induced from
the properties of Calabi-Yau manifolds, we will investigate Mirror
Duality on Spin(7) manifolds [2] [3].

References

D. Joyce, Lectures on Calabi-Yau and special Lagrangian geometry, Arxiv, (2002).
S. Akbulut, S. Salur, Mirror Duality via G 2 and Spin(7) Manifolds,
Arithmetic and Geometry Around Quantization, Progress in Mathematics,
vol 279, (2010).
E. Yalcinkaya, Mirror Duality on Spin(7) manifolds, Preprint.
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Lecturer: Mustafa Kalafat

Title: Minimal Surfaces

Abstract: A minimal surface is a surface that locally minimizes its
area. This is equivalent to having zero mean curvature. They are
2-dimensional analog to geodesics, which are analogously defined as
critical points of the length functional.

Currently, the theory of minimal surfaces has diversified to minimal
submanifolds in other ambient geometries, becoming relevant to
mathematical physics (e.g. the positive mass conjecture, the Penrose
conjecture) and three-manifold geometry (e.g. the Smith conjecture,
the Poincaré conjecture, the Thurston Geometrization Conjecture).

In this lecture series, we will give an introduction to some topics in
minimal submanifold theory. The topics to be covered are as follows.

1. Mean curvature vector field on a Riemannian submanifold.

2. First variational formula for the volume functional.

3. Second variation of energy for a minimally immersed submanifold.

4. Stability of minimal submanifolds.

We will be using the following resources.

References:

Li, Peter. Geometric analysis. Cambridge University Press, 2012.
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Lecturer: Craig van Coevering

Title: Extremal Kähler metrics and the moment map

Abstract: 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 is 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.

In this talk I will give an introduction to extremal metrics. Then I
will describe how the moment map can be used to describe the local
deformation problem of extremal metrics. 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 a Hausdorff complex
analytic space.
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