The 6th GTSS
GEOMETRYTOPOLOGY SUMMER SCHOOL
Nesin Mathematics Village, Şirince, İzmir August
1730, 2020
(Invitation process is started.
Let us know
if you have a researchminicourse proposal!)
First Week
Second Week
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

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Poster
Participants
Arrival
Information
The 2nd GTSS will be held at the
Nesin Mathematical Village
in Şirince, İzmir.
There will be about 15 minicourses of introductory nature, related to the GeometryTopology 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 underrepresented 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.
Abstracts
EinsteinMaxwell 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 (RuhVilms theorem). Finally, we give a construction method of harmonic maps (and constant mean curvature surfaces) via integrable systems (the socalled 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 RuhVilms 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 AhlforsBers 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 GaussManin connection will be introduced. For motivational purposes, we will discuss the classical theory of periods, PicardFuchs 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.GriffithsHarris, 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, 231274 (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 SasakiEinstein manifolds.
J. Differential Geom. 83 (2009), no. 3, 585–635.
3. Collins, Tristan C.; Székelyhidi, Gábor. Ksemistability 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 G2manifolds, Asian J. Math. 21 (2017), 429462.
2. K. Kawai, Secondorder deformations of associative submanifolds in nearly parallel G2manifolds, Q. J. Math. 69 (2018), 241270.
3. J. D. Lotay, Associative Submanifolds of the 7Sphere,
Proc. Lond. Math. Soc. (3) 105 (2012), 11831214.
4. R. C. McLean, Deformations of Calibrated Submanifolds,
Comm. Anal.Geom. 6 (1998), 705747.
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 CalabiYau 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 GriffithsHarris, 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 “LiYauHamilton“ 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 LiYauHamilton 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 nonEuclidean systems. We generate useful volume and area formulas for tetrahedrons and triangles in lowdimensional 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 GaussBonnet theorem.
Textbook or/and course webpage:
1. Anderson, James W. – Hyperbolic Geometry, SpringerVerlag, 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 warpedproduct Eınstein metrics. As a tool to approach these strucures, we start with some study on the socalled 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, SpringerVerlag, 1987.
2. G. Catino, C. Mantegazza, L. Mazzieri, A note on Codazzi tensors, Math. Ann. 362 (2015), No. 12, 629638.
3. J. Kim. On a classification of 4d gradient Ricci solitons with harmonic Weyl curvature,
J. Geom. Anal. 27, (2017), no. 2, 9861012.
Prerequisites:
Riemannian Geometry
Hyperkähler Manifolds
Hyperkähler manifolds are manifolds with 3 complex structures satisfying quaternionic relations,
and a torsionfree 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 FubiniStudy 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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