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The 3rd GTWS

GEOMETRY-TOPOLOGY WINTER SCHOOL

Nesin Mathematics Village, Şirince, İzmir
January 6-19, 2020

(Invitation process still continues. Let us know
if you have a research-mini-course proposal!)



First Week

 

TIME              SPEAKER                  TITLE
Jan 6-11*
8-10
Martin Svensson
Topics in Harmonic Maps*
Jan 6-11
10-12
Craig Van Coevering
Special Metrics in Sasakian Geometry*
Jan 6-12
2:30-4
Kurando Baba     Lectures on Calibrations and Generalized Duality    
Jan 6-12
4-5:30
Mustafa Kalafat
Representation Theory of the Lie Algebra of G_2
Jan 6-11
10-12
Mehmet Kılıç Metric Geometry
Jan 6-19
5:30-7
Eyüp Yalçınkaya     Spin Geometry and Seiberg-Witten Equations    

Second Week

 

TIME              SPEAKER                  TITLE
Jan 13-19
8-10
Andreas Savas-Halilaj Minimal Submanifolds, Mean Curvature Flow
and Isotopy Problems

Jan 13-18
10-12
Lorenzo Foscolo Gibbons-Hawking Ansatz of Hyperkähler Metrics
in Dimension 4
*
Jan 13-19
2:30-4
Murat Savaş Hyperbolic Geometry
Jan 13-18
4-5:30
Özgür Kelekçi Kähler Geometry (and Einstein Manifolds)
Jan 13-18
5:30-7
Buket Can Bahadır
Introduction to Elliptic Partial Differential Equations

Scientific Commitee

 

Vicente Cortés University of Hamburg, Germany
Ljudmila Kamenova Stony Brook University, USA
Lei Ni University of California at San Diego, USA
Misha Verbitsky IMPA, Brasil







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Information

The 3rd GTWS 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 .

Wifi info: Passwords are TermoS1! at the Library networks and zeytinlik for istasyon at the registration lobby.



Application

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

Please fill out the application form and send it to

Aslı Can Korkmaz: aslicankorkmaz@nesinvakfi.org, okelekci@gmail.com-->
to attend to the school.

Airport: İzmir Adnan Menderes Airport - ADB is the closest one.
Visas: Check whether you need a visa beforehand.


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Abstracts



Minimal Submanifolds, Mean Curvature Flow and Isotopy Problems

Many fundamental results in geometry and topology have been established through the development of minimal submanifold theory and geometric flow techniques. In this mini course, I will start by discussing minimal submanifolds and scalar/vectorial maximum principles for elliptic and parabolic PDEs. Then, I will use these tools to prove Bernstein type theorems for graphical minimal submanifolds. Finally, I will focus on the mean curvature flow in high codimensions and will demonstrate how to use this powerful method to derive topological results for maps between Riemannian manifolds.

Textbook or/and course webpage:

1. K. Smoczyk, Mean curvature flow in higher codimension: introduction and survey. Springer Proceedings in Mathematics, Vol 7, 231-274 (2012). Text also available on arXiv 1104.3222.

2. Y.-L. Xin, Minimal submanifolds and related topics, Nankai Tracts in Mathematics, Vol. 16 (2018).

Prerequisites:

Differential Geometry, Riemannian Geometry.

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.

Lectures on Calibrations and Generalized Duality

In this lecture series we discuss calibrated geometry in the view point of Lie groups actions on symmetric spaces. Concrete examples of calibrations given by Harvey-Lawson are invariant under the action of the holonomy groups of Riemannian manifolds, which give Lie group actions on Grassmannian manifolds with geometrically nice properties. We also explain constructions of calibrated submanifolds. In particular, we give special Lagrangian submanifolds in Stenzel spaces by using moment map techniques. The calibrated geometry was generalized by Mealy in the pseudo-Riemannian geometry category. We give a correspondence between Harvey-Lawson’s calibrations and Mealy’s calibrations by a generalization of the duality in symmetric spaces.

Textbook and References:

1. M. Arai and K. Baba, Special Lagrangian submanifolds and cohomogeneity one actions on the projective spaces, Tokyo J. Math. 42 (2019), 255-284.

2. R. Harvey and H. B. Lawson, Jr., Calibrated geometries, Acta Math, 148 (1982), 47-157.

3. S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press (1979).

4. J. Mealy, Volume maximization in semi-Riemannian manifolds, Indiana Univ. Math. J. 40 (1991) 793-814.

Prerequisites: Linear Algebra, Riemannian Geometry (not a must but preferable)

Level: Graduate Advanced undergraduate

Topics in Harmonic Maps



Gibbons-Hawking Ansatz of Hyperkähler Metrics in Dimension 4



Representation Theory of the Lie Algebra of G_2

In this lecture series we give an introduction to the representation theory of the Lie algebra of G_2. We review Lie groups, Lie algebras, Cartan subalgebra, Root systems. Then, starting from concrete examples we will work on sl(2,C), sl(3,C), sl(4,C) and finally sl(n,C) and their representations. We also go into their geometric interpretations. Also spend time on sp(2n,C) and so(m,C) if time permits.

Textbook or/and course webpage:

1. Fulton, William; Harris, Joe - Representation theory. A first course.
Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991. xvi+551 pp. ISBN: 0-387-97527-6

Prerequisites:

Lie groups and Lie algebras.

Kähler Geometry (and Einstein Manifolds)

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)

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

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



Activities are supported by Nesin Mathematical Village and Turkish Mathematical Society