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The 2nd GTSS

GEOMETRY-TOPOLOGY SUMMER SCHOOL

Nesin Mathematics Village, Şirince, İzmir
September 9-22, 2019



First Week

 

TIME              SPEAKER                  TITLE
Sep 9-15
8-10
Craig Van Coevering
Special Metrics in Sasakian Geometry A
Sep 9-15
10-12
Misha Verbitsky Hyperkähler Manifolds
Sep 9-14
2:30-4
Kotaro Kawai Calibrated submanifolds in G2 manifolds
Sep 9-13
4-6
Jongsu Kim Generalization of Einstein metrics
Sep 9-15
4-6
Joseph Malkoun Lie groups, Lie algebras and representation theory
Sep 9-15
8-10
Özgür Kelekçi Kähler Geometry A
Sep 9-15
2:30-4
Buket Can Bahadır Partial Differential Equations (Sobolev and Hölder Spaces) A
Sep 9-15
10-12
Mehmet Kılıç Metric Geometry

Second Week

 

TIME              SPEAKER                  TITLE
Sep 16-18
8-10
Craig Van Coevering Special Metrics in Sasakian Geometry B
Sep 16-20
4-6
Yanyan Niu Differential Harnack estimate on manifolds
Sep 16-20
10-12
Lei Ni Complex geometric analysis
Sep 16-22
8-10
Murat Savaş Hyperbolic Geometry
Sep 16-22
4-6
Mustafa Kalafat
G2 Geometry
Sep 16-22
10-12
Özgür Kelekçi
Kähler Geometry B
Sep 16-22
4-6
Buket Can Bahadır
Partial Differential Equations (Sobolev and Hölder Spaces) B
Sep 16-22
8-10
Eyüp Yalçınkaya Spin Geometry and Seiberg-Witten Equations







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Information

The 2nd GTSS will be held at the Nesin Mathematical Village in Şirince, İzmir. There will be about 15 mini-courses of introductory nature, related to the Geometry-Topology research subjects.
In the middle of the week there is an excursion to the Ancient City of Ephesus and/or Kuşadası Beach.



Application

Graduate students, recent Ph.D.s and under-represented minorities are especially encouraged to the summer school. Partial financial support is available. Daily expenses including bed, breakfast, lunch, dinner is less than 20$.

Please fill out the application form to attend to the summer school.
Airport: İzmir Adnan Menderes Airport - ADB is the closest one.
Visas: Check whether you need a visa beforehand.


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Abstracts



Special metrics in Sasakian geometry

This course will begin with an introduction to Sasakian geometry, a type of metric contact structure which is an odd dimensional analogue of a Kähler structure on a complex manifold. We will then consider the problem of finding a special Sasakian metrics, such as Einstein, and more generally constant scalar curvature and extremal metrics.

Textbook or/and course webpage:

1. Boyer, Charles P.; Galicki, Krzysztof. Sasakian geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2008. xii+613 pp.

2. Futaki, Akito; Ono, Hajime; Wang, Guofang Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds. J. Differential Geom. 83 (2009), no. 3, 585–635.

3. Collins, Tristan C.; Székelyhidi, Gábor. K-semistability for irregular Sasakian manifolds. J. Differential Geom. 109 (2018), no. 1, 81–109.

Prerequisites:

A first year graduated course in differential geometry should be sufficient.

Lie groups, Lie algebras and representation theory

This mini-course is an introduction to Lie groups and Lie algebras. The topics covered include the structure theory and classification of complex semisimple Lie algebras using root systems, some elements of representation theory, with special emphasis on the complex semisimple case, weight theory. Towards the end, a more advanced topic might be discussed, time permitting, such as universal algebras, the Poincare-Birkhoff-Witt theorem, or another topic.

Textbook or/and course webpage:

1. Anthony W. Knapp. Lie groups beyond an introduction, volume 140 of Progress in Mathematics. Birkhauser Boston, Inc., Boston, MA, second edition, 2002.

Prerequisites:

Linear Algebra, Algebra, Differentiable Topology (not a must)

Lectures on G2 Geometry

In this lecture series we present a combinatorial approach to the exceptional Lie group G2. We give a survey of various results about the algebraic structure. For the sake of completeness we decided to present them in a self-contained way to be easily accessible for future usage. We also present some applications to geometry. Manifolds with G2 structures, Decomposition of the exterior algebra into irreducible G2 representations, Metric of a G2 structure, 16 classes of G2 structures, Deformations of G2 structures. Lie algebra of G2, roots and their spaces, order, Killing form. Language: TR, EN

We will be following the Reference:

Karigiannis, Spiro - Deformations of G2 and Spin(7) structures.
Canad. J. Math. 57 (2005), no. 5, 1012–1055.
Author's Ph.D. Thesis also available on the Arxiv.org.


Bryant, Robert L. Some remarks on G2-structures.
Proceedings of Gökova Geometry-Topology Conference 2005, 75–109, Gökova Geometry/Topology Conference (GGT), Gökova, 2006.


Other useful references are:

Anthony W. Knapp. Lie groups beyond an introduction, second edition.
volume 140 of Progress in Mathematics. Birkhauser Boston, Inc., Boston, MA, 2002.

G2 Geometry 1: 4 categories of vector cross product structures.

G2 Geometry 2: Decomposition of A*(M) into irreducducible G2 representations.

G2 Geometry 3: Metric of a G2 structure.

G2 Geometry 4: More into cross product identities.

G2 Geometry 5: 16 classes of G2 structures.

G2 Geometry 6: Deformations of G2 structures.


Prerequisites:

Lectures on Lie groups, Lie algebras and representation theory by Joseph Malkoun.

Calibrated submanifolds in G2 manifolds

In this lecture, we first give a survey of calibrated submanifolds in G2 manifolds. Then we present their deformation theories.

Textbook or/and course webpage:

1. K. Kawai, Deformations of homogeneous associative submanifolds in nearly parallel G2-manifolds, Asian J. Math. 21 (2017), 429-462.

2. K. Kawai, Second-order deformations of associative submanifolds in nearly parallel G2-manifolds, Q. J. Math. 69 (2018), 241-270.

3. J. D. Lotay, Associative Submanifolds of the 7-Sphere, Proc. Lond. Math. Soc. (3) 105 (2012), 1183-1214.

4. R. C. McLean, Deformations of Calibrated Submanifolds, Comm. Anal.Geom. 6 (1998), 705-747.

Prerequisites:

Linear Algebra, Riemannian Geometry

Kähler Geometry

Kähler geometry has been an important area of differential geometry and attracted significant interest from both mathematics and mathematical physics research community. In this lecture series our aim is to provide an introduction to different aspects of Kähler geometry. We will start with a review of Riemannian geometry. Then emphasis will be on complex and Hermitian geometry which form the basis for Kähler manifolds. We will study several aspects of Kähler manifolds such as the Calabi-Yau theorem, Weitzenböck techniques, Calabi–Yau manifolds.

Textbook or/and course webpage:

A. Moroianu, “Lectures on Kähler Geometry”

Prerequisites:

Basic Differential Geometry (not a must but preferable)

Complex geometric analysis

I shall give a quick course on the complex/Kaehler geometry with emphasis of PDE methods.

Textbook or/and course webpage:

I shall distribute some notes. The students can start with books such as

Complex Manifolds, by J. Morrow and K. Kodaira, AMS.

Principles of of Algebraic Geometry, by Griffiths-Harris, John Wiley and Sons.

Differential Analysis on Complex Manifolds, by Wells. Springer.

Complex differential manifolds, by Zheng, AMS/IP.

Complex geometry, by Huybrechts, Springer.

Prerequisites:

Linear Algebra, Partial Differential Equations/Complex Analysis (not a must but preferable)

Differential Harnack estimate on manifolds

In this lecture series we present “Li-Yau-Hamilton“ type differential Harnack estimate on Riemannian manifolds and Kähler manifolds.

Textbook or/and course webpage:

1. Bennett Chow, Peng Lu and Lei Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics, 77. American Mathematical Society, Providence, RI; Science Press Beijing, New York, 2006.

2. B.Wilking, A Lie algebraic approach to Ricci flow invariant curvature condition and Harnack inequalities, J. reine angew. Math. (Crelle), 679 (2013), 223–247.

3. L. Ni and Y. Y. Niu, Sharp differential estimates of Li-Yau-Hamilton type for positive (p, p)-forms on Kähler manifolds. Comm. Pure Appl. Math., 64 (2011), 920–974.

Prerequisites:

Riemannian manifold, Kähler manifold (not a must but preferable)

Spin Geometry and Seiberg-Witten Equations

In this lecture series we present spin geometry by using Clifford algebras. Then we focus on Seiberg-Witten equations.

Partial Differential Equations (Sobolev and Hölder Spaces)

Partial Differential Equations, Functional Analysis, Riemannian Geometry (not a must but preferable).

Level: Graduate, advanced undergraduate

Abstract: In this lecture series, our aim is to introduce Sobolev Spaces, and to present techniques and ideas from functional analysis to develop the theory. Since the solutions of partial differential equations are naturally found in Sobolev spaces, this theory proves to be a useful tool for several applications in partial differential equations, as well as giving us an opportunity for a different aproach to classical problems using the methods of functional analysis. We also would like to make an introduction to how these concepts can be aplied in the set up of Riemannian Manifolds., and give examples, such as Yamabe problem, where this aprooach played an important role.

Textbook or/and course webpage:

Evans, Lawrence C. - Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.

Metric Geometry

Prerequisites: Temel Analiz bilgisi ve metrik uzaylara aşinalık.

Level: Graduate, advanced undergraduate

Abstract: Bu derste, uzunluk uzayları ve jeodezik uzaylar tanıtılacaktır. Ascoli Teorem'i ifade ve ispat edilecek ve bunun sonucu olarak bir ''proper'' metrik uzayda, sonlu uzunluğa sahip bir yolla birleştirilebilen iki nokta arasında, minimal uzunluğa sahip bir yolun var olduğu gösterilecektir. Daha sonra metrik uzaylarda bazı konvekslik kavramları incelenecektir. Language: TR

Textbook: Metric Spaces, Convexity and Nonpositive Curvature; Athanase Papadopoulos. A Course in Metric Geometry; Dmitri Burago, Yuri Burago, Sergei Ivanov.

Hyperbolic Geometry

In this lecture series we give an introduction to hyperbolic geometry with some discussion of other non-Euclidean systems. We generate useful volume and area formulas for tetrahedrons and triangles in low-dimensional hyperbolic space. Daily sections of the lecture are given below.

Description of the hyperbolic geometry models and the connection between them. Description of the general Mobius group. Characterization of the isometries of hyperbolic space. Geometry of hyperbolic triangles. Hyperbolic trigonometry. Hyperbolic area and the Gauss-Bonnet theorem.

Textbook or/and course webpage:

1. Anderson, James W. – Hyperbolic Geometry, Springer-Verlag, London, 2005.

2. Ratcliffe, John G., Foundations of Hyperbolic Manifolds, Graduate texts in Mathematics, 149, Springer, New York, 2006.

Prerequisites:

Linear Algebra

Generalization of Einstein metrics

In this lecture series we review some generalizations of Einstein metrics. These are Ricci solitons, static spaces and warped-product Eınstein metrics. As a tool to approach these strucures, we start with some study on the so-called Codazzi tensors and then talk about harmonic curvature and harmonic Weyl curvature. After these, we apply the Codazzi tensor approach to study each of the above generalizations of Einstein metrics.

Textbook or/and course webpage:

1. A.L. Besse - Einstein manifolds, Chapter 16, Ergebnisse der Mathematik, 3 Folge, Band 10, Springer-Verlag, 1987.

2. G. Catino, C. Mantegazza, L. Mazzieri, A note on Codazzi tensors, Math. Ann. 362 (2015), No. 1-2, 629-638.

3. J. Kim. On a classification of 4-d gradient Ricci solitons with harmonic Weyl curvature, J. Geom. Anal. 27, (2017), no. 2, 986-1012.

Prerequisites:

Riemannian Geometry

Hyperkähler Manifolds

Hyperkähler manifolds are manifolds with 3 complex structures satisfying quaternionic relations, and a torsion-free orthogonal connection preserving these complex structures. A number of objects arising in hyperkähler geometry are equipped with hyperkähler structure, such as moduli of vector bundles, deformation spaces, subvarieties and so on. This allows one to speak of “hyperkähler geometry”, similar to the usual complex algebraic geometry. I would explain basic properties of hyperkähler manifolds, give some examples and prove results about subvarieties.

Textbook or/and course webpage:

Besse, “Einstein manifolds”

Prerequisites:

Differential geometry, topology


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Contact: aslicankorkmaz@nesinvakfi.org, okelekci@gmail.com

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