Sekizinci Diferansiyel Geometri Yaz okulu kapsamında, İstanbul Matematiksel Bilimler Merkezinde (IMBM) çevrimiçi olarak, 15 - 27 Ağustos 2022 tarihleri arasında, aşağıdaki araştırma dersleri verilecektir.
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
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Lecturer: Alberto Raffero
Title: Lectures on closed G_2 structures
Abstract: Two introductory lectures on G2 geometry and two lectures concerning recent developments and open problems on closed G2 structures.
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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. Initially, we will show that gradient Ricci solitons are special (self-similar) solutions of the Ricci flow. Moreover, we will present some basic results and examples of gradient shrinking Ricci solitons. In the second part, we will discuss some important results on the classification of four-dimensional gradient shrinking Ricci solitons. To conclude, we will talk about some open problems and their motivations.
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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.
- Mean curvature vector field on a Riemannian submanifold.
- First variational formula for the volume functional.
- Second variation of energy for a minimally immersed submanifold.
- 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 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 an Hausdorff complex analytic space.
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