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The 6th GTSS

GEOMETRY-TOPOLOGY SUMMER SCHOOL

Nesin Mathematics Village, Şirince, İzmir
August 17-30, 2020

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



First Week

 

TIME              SPEAKER                  TITLE
Aug 17-23
4-6
Joseph Malkoun Lie groups, Lie algebras and representation theory
Aug 17-23
8-10
Damien Gayet
Topics in Calibrated Geometries*
Sep 9-15
8-10
Craig Van Coevering
Special Metrics in Sasakian Geometry A
Sep 9-13
10-12
Misha Verbitsky Hyperkähler Manifolds
Sep 9-14
2:30-4
Kotaro Kawai Calibrated submanifolds in G2 manifolds
Sep 9-15
10-12
Mehmet Kılıç Metric Geometry
Sep 9-15
4-5:30
Özgür Kelekçi Kähler Geometry A
Sep 9-15
5:30-7
Buket Can Bahadır Introduction to Several Complex Variables A
Sep 9-15
5:30-7
İlker Savaş Yüce Hyperbolic Geometry
Sep 9-15
5:30-7
Özgür İnce Algebraic Curves
Jul 6-12
8-9:30
Marisa Fernández*
Lectures on G2 Geometry
Jul 6-10
9:30-11
Shimpei Kobayashi Harmonic maps, constant mean curvature surfaces
and integrable systems
Jul 6-11
10-12
Lorenzo Foscolo Gibbons-Hawking Ansatz of Hyperkähler Metrics
in Dimension 4
*
Jul 6-12
4-5:30
Spiro Karigiannis* Moduli Space of G_2 Manifolds
Jul 6-12
10-12
Andreas Savas-Halilaj* Minimal Submanifolds, Mean Curvature Flow
and Isotopy Problems


Second Week

 

TIME              SPEAKER                  TITLE
Aug 24-30
8-10
Andrei Moroianu Locally Conformally Kähler (lcK) Manifolds*
Aug 24-30
10-12
Akito Futaki Einstein-Maxwell Kähler metrics*
Jul 13-18
8-9:30
Jihun Park Topics in Stability and Fano Varieties/del Pezzo Surfaces
Jul 13-18
5:30-7
Susumu Tanabé Monodromy and Periods of Algebraic Varieties
Jul 6-11
5:30-7
Subhojoy Gupta Bers embedding of Teichmüller spaces
Jul 13-18
11-12:30
Yoshinori Gongyo Topics in Fano Varieties and Rational Connectedness
Aug 24-30
10-12
Lorenzo Foscolo Gibbons-Hawking Ansatz of Hyperkähler Metrics
in Dimension 4
*
Aug 24-30
5:30-7
Siddhartha Gadgil Geometric group theory and Hyperbolic Geometry
Jul 13-18
9:30-11
Miles Reid Topics in Algebraic Geometry
Jul 13-18
10-12
Gerard van der Geer* Abelian Varieties
Jul 13-18
9-10:30
Emre Sertöz* Topics in Moduli of Curves*
Sep 16-18
8-10
Craig Van Coevering Special Metrics in Sasakian Geometry B
Sep 16-20
10-12
Lei Ni Complex geometric analysis
Lecture Notes , Seminar
Sep 16-20
2:30-4
Yanyan Niu Differential Harnack estimate on manifolds
Sep 16-22
4-5:30
Murat Savaş Hyperbolic Geometry
Sep 16-18
5:30-7
Özgür Kelekçi
Kähler Geometry B
Sep 16-22
7-8:30
Buket Can Bahadır
Introduction to Several Complex Variables B
Jan 13-18
12-1:15
Çağrı Hacıyusufoğlu     Minimal Surfaces    
Jan 20-26
4:30-6
Mohan Bhupal Riemann Surfaces and Fuchsian Groups
Jul 13-18
2:30-4
Dmitry Kerner
Singular Algebraic Curves and Varieties*

Scientific Commitee

 

Vicente Cortés University of Hamburg, Germany
İzzet Coşkun University of Illinois at Chicago, USA
Marisa Fernández Universidad del País Vasco, Bilbao, Spain
Ljudmila Kamenova Stony Brook University, USA
Lei Ni University of California at San Diego, USA
Tommaso Pacini University of Torino, Italy
Gregory Sankaran University of Bath, UK
Misha Verbitsky IMPA, Brasil







Register-TR       Poster       Participants       Arrival



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.

Wifi info: Passwords are TermoS1! at the Library (Kutuphanegenel) networks, zeytinlik for istasyon and kurugolet for kisbahcesi at the registration lobby.



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 around 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



Einstein-Maxwell Kähler metrics(*)



Locally Conformally Kähler (lcK) Manifolds(*)



Topics in Stability and Fano Varieties/del Pezzo Surfaces



Harmonic maps, constant mean curvature surfaces and integrable systems

In this lecture, we give an overview of homogeneous spaces, harmonic maps, integrable systems, constant mean curvature surfaces and their relations. First, we see how harmonic maps from a surface into a homogeneous space can be understood as integrable systems and they are related to constant mean curvature surfaces in space forms via Gauss maps (Ruh-Vilms theorem). Finally, we give a construction method of harmonic maps (and constant mean curvature surfaces) via integrable systems (the so-called DPW method).

Daily description is as follows.

1. Overview

2. Manifolds and homogeneous spaces

3. Integrable systems and harmonic maps

4. Constant mean curvature surfaces, Gauss maps and Ruh-Vilms theorem

5. Construction of harmonic maps via integrable systems

6. Advanced topics

Textbook, References or/and course webpage:

1.Shoichi Fujimori, Shimpei Kobayashi, Wayne Rossman, Loop Group Methods for Constant Mean Curvature Surfaces, Rokko Lectures in Mathematics, 17 (2005), v+118 pp. arXiv:math/0602570

2. Josef F. Dorfmeister, Franz Pedit, Hongyou Wu, Weierstrass type representation of harmonic maps into symmetric spaces. Comm. Anal. Geom. 6 (1998), no. 4, 633–668.

Prerequisites: Linear Algebra, Calculus

Level (erase some): Graduate Advanced undergraduate

Bers embedding of Teichmüller spaces

In these lectures I will introduce Teichmüller spaces from the point of view of Ahlfors-Bers theory, and describe how it acquires a complex structure via an embedding into the vector space of holomorphic quadratic differentials. Along the way, I shall talk about quasiconformal maps and their boundary, the Schwarzian derivative, amongst other basic topics. The image of the Bers embedding is still mysterious, and I plan to describe some open problems.

Textbook, References or/and course webpage:

1. Teichmüller theory and applications, Volume 1, by John H. Hubbard.  Published by Matrix editions, 2006.

2. Lectures on Quasiconformal Mappings, 2nd edition, by Lars V. Ahlfors. AMS University Lectures Series, Volume 38. (original 1966, reprinted 2006)

3. Univalent Functions and Teichmüller spaces, by Olli Lehto.  GTM series, Springer, 1987.

Prerequisites: Complex Analysis, some knowledge of hyperbolic geometry and Riemann surfaces would be preferable.

Level (erase some): Graduate Advanced undergraduate

Monodromy and Periods of Algebraic Varieties

We discuss various topics pertaining to the topology of complex algebraic varieties, relying on classical techniques in complex algebraic geometry and algebraic topology. We will begin by introducing the Milnor fibration and construct the monodromy action on the homology of the fibers of the Milnor fibration. We will discuss vanishing cycles in this concrete setting. We will then proceed to discuss applications of these concepts in classical algebraic geometry, such as the theory of Lefschetz pencils. In the second part of the week, we will focus on the cohomological aspects of our subject. The concept of the Gauss-Manin connection will be introduced. For motivational purposes, we will discuss the classical theory of periods, Picard-Fuchs equations and similar topics in the context of the Legendre family of elliptic curves. A basic review of local systems and vector bundles will be given. Time permitting, Hodge Structures and more advanced topics may be introduced. If the need arises, tools from basic differential topology and homological algebra may also be discussed.

Textbook or/and course webpage:

1. Singular Points of Complex Hypersurfaces, J. Milnor, 1968

2.Griffiths-Harris, Principles of Algebraic Geometry

3. Otto Forster, Lectures on Riemann Surfaces, 1981

4. A Scrapbook of Complex Curve Theory, H. Clemens, 1980

5. Complex Algebraic Geometry and Hodge Theory Claire Voisin, 2002

6. Pierre Deligne, SGA 7, vol II

Advanced Topics

5. Mixed Hodge Structures, C.Peters, J.H.M. Steenbrink, 2008

6. Sheaves in Topology A. Dimca, 2004

Also see the papers:

The Topology of Complex Projective Varieties after S. Lefschetz, K.Lamotke, 1979

P.A. Griffiths, Periods of integrals on algebraic Manifolds 1970

Prerequisites:

Complex Analysis, Algebraic Topology, Intro to Algebraic Curves (preferred)

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.

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)

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

Kalafat: We show that a compact complex surface which admits a conformally Kahler metric g of positive orthogonal holomorphic bisectional curvature is biholomorphic to the complex projective plane. In addition, if g is a Hermitian metric which is Einstein, then the biholomorphism can be chosen to be an isometry via which g becomes a multiple of the Fubini-Study metric. This is a joint work with C.Koca.

Reference:

Kalafat, Mustafa; Koca, Caner -
Conformally Kähler surfaces and orthogonal holomorphic bisectional curvature. Geom. Dedicata 174 (2015), 401–408.


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