Geometry Main 20b 20a 19 18 15b 15a 14

The 4th CAGWS


Nesin Mathematics Village, Şirince, İzmir
January 20-February 2, 2020

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

First Week


TIME              SPEAKER                  TITLE
Jan 20-24
Yoshinori Hashimoto   Introduction to Geometric Invariant Theory  
Jan 20-25
Tommaso Pacini Conformal Invariants and Quasi-conformal Maps
Jan 20-26
Mutsuo Oka
Introduction on Mixed Hypersurface Singularity
Jan 20-25
Turgay Bayraktar Dynamics in One Complex Variable
Jan 20-26
Buket Can Bahadır Several Complex Variables

Second Week


TIME              SPEAKER                  TITLE
Jan 27-Feb 2
Gregory Sankaran* Moduli of Abelian Varieties
Jan 27-Feb 1
Shin-ichi Matsumura Geometry of Holomorphic Sectional Curvature
and Rational Curves
Jan 27-Feb 2
Dmitry Kerner
Singular Algebraic Curves and Varieties*
Jan 27-Feb 2
Taro Sano Deformations of Fano and Calabi-Yau Varieties
Jan 27-Feb 2
Mustafa Kalafat
Representation Theory of Complex Lie Algebras
Jan 27-Feb 2
Kürşat Aker* Higgs Bundles

Scientific Commitee


İzzet Coşkun University of Illinois at Chicago, USA
Ljudmila Kamenova Stony Brook University, USA
Lei Ni University of California at San Diego, USA
Misha Verbitsky IMPA, Brasil

Register-TR       Poster       Participants       Arrival


The 4th CAGWS 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.


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 and send it to

Aslı Can Korkmaz:,
to attend to the winter school.

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

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Introduction on Mixed Hypersurface Singularity

In this lecture series, I introduce the basic of mixed hypersurface singularity. A mixed polynomial is a polynomial which contains z_1,...,z_n and also their conjugates z_1,...,z_n. Hypersurace defined by {f=0} involves much wider class of geometry. We introduce the notion of Newton boundary, non-degeneracy and local Milnor fibration.

Textbook or/and course webpage:

1. M.Oka, Non-degenerate mixed functions, Kodai J. Math. 33 (1), 1-62,2010

2. J. Milnor, Singular points of complex hypersurfaces, Annals of Math. Studies 61, Princeton , N.J., 1968


Linear Algebra, holomorphic function of one variable (not a must but preferable)

Singular Algebraic Curves and Varieties

Introduction to Geometric Invariant Theory

Suppose that a reductive algebraic group G acts on a projective variety X. Geometric Invariant Theory (GIT), initiated by Mumford, is a method of taking a quotient X/G in the category of varieties. It has many important applications in the theory of moduli and is also related to problems in differential geometry. This minicourse aims to be a rapid introduction to this topic; it will be mostly about the construction of the GIT quotient and the Hilbert—Mumford criterion, but related differential-geometric topics will also be mentioned.

Textbook or/and course webpage:

1. Dolgachev, I. Lectures on invariant theory. London Mathematical Society Lecture Note Series, 296. Cambridge University Press, Cambridge, 2003. xvi+220 pp. ISBN: 0-521-52548-9

2. Mukai, S. An introduction to invariants and moduli. Translated from the 1998 and 2000 Japanese editions by W. M. Oxbury. Cambridge Studies in Advanced Mathematics, 81. Cambridge University Press, Cambridge, 2003. xx+503 pp. ISBN: 0-521-80906-1

3. Huybrechts, D.; Lehn, M. The geometry of moduli spaces of sheaves. Second edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2010. xviii+325 pp. ISBN: 978-0-521-13420-0

4. Chriss, N.; Ginzburg, V. Representation theory and complex geometry. Reprint of the 1997 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2010. x+495 pp. ISBN: 978-0-8176-4937-1

5. Thomas, R. P. Notes on GIT and symplectic reduction for bundles and varieties. Surveys in differential geometry. Vol. X, 221--273, Surv. Differ. Geom., 10, Int. Press, Somerville, MA, 2006.

6. Donaldson, S. K.; Kronheimer, P. B. The geometry of four-manifolds. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1990. x+440 pp. ISBN: 0-19-853553-8

7. Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34. Springer-Verlag, Berlin, 1994. xiv+292 pp. ISBN: 3-540-56963-4

Prerequisites: Basic complex algebraic geometry

On the Geometry of Holomorphic Sectional Curvature and Rational Curves

In this lecture, we discuss the notation of holomorphic sectional curvatures after we review hermitian metrics, Chern curvatures, and their properties of holomorohic vector bundle over complex manifolds. In particular, I explain a realtion between positivity of holomorphic sectional curvatures and the geometry of rationtal curves (that is, 1-dim projective space embedded in manifolds). The goal of this talk is to give a structure theorem for rationally connected fibrations of projective manifolds with non-negative holomorphic sectional curvature.

Textbook or/and course webpage:

1. Foundations of Differential Geometry, written by Shoshichi Kobayashi and Katsumi Nomizu, Wiley Classics Library.

2. RC-positivity, rational connectedness and Yau's conjecture, written by Xiaokui Yang, Camb. J. Math. 6 (2018), no. 2, 183–212

3. On projective manifolds with semi-positive holomorphic sectional curvature,

4. On morphisms of compact Kähler manifolds with semi-positive holomorphic sectional curvature written Shin-ichi Matsumura, available at arXiv.

Prerequisites: Complex manifolds , (holomorphic) vector bundles(not a must but preferable)

Level: Graduate and Advanced undergraduate

Language: EN

Deformations of Fano and Calabi-Yau Varieties

In this lecture series we present an algebraic approach to deformations of Fano and Calabi-Yau varieties. Fano varieties and Calabi-Yau varieties are important objects in the classification of algebraic varieties. In the classification of vareties, it is fundamental to study their deformations. Starting from basic notions, I'll explain how to see that they have unobstructed deformations. I'll also explain the generalization to log CY varieties and normal crossing CY varieties.

Daily description is as follows.

1: Basics on complex manifolds (analytic spaces) and sheaves
2: Preliminaries on Deformation theory
3: Deformations of CY varieties (Bogomolov-Tian-Todorov theorem)
4: Deformations of log CY and normal crossing CY varieties

Textbook and References:

1. Greuel, G.-M.; Lossen, C.; Shustin, E. Introduction to singularities and deformations. Springer Monographs in Mathematics. Springer, Berlin, 2007. xii+471 pp.

2. Huybrechts, Complex geometry, An introduction. Universitext. Springer-Verlag, Berlin, 2005. xii+309 pp.

3.Namikawa, Yoshinori Calabi-Yau threefolds and deformation theory [translation of Sūgaku 48 (1996), no. 4, 337–357; MR1614448]. Suguku Expositions. Sugaku Expositions 15 (2002), no. 1, 1–29.

4. Sernesi, Edoardo Deformations of algebraic schemes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 334. Springer-Verlag, Berlin, 2006. xii+339 pp.

Prerequisites: Basic complex algebraic geometry (not a must but preferable)

Level: Graduate Advanced undergraduate

Representation Theory of Complex Lie Algebras

In this lecture series we give an introduction to the representation theory of some complex Lie algebras. 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 theri 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

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


Linerar Algebra. Lie groups and Lie algebras.

Moduli of Abelian Varieties

Conformal Invariants and Quasi-conformal Maps

Several Complex Variables

In this lecture series, our aim is to introduce Several Complex Variables theory which is used in many applications in Geometry. The course gives an introduction to the most central aspects and methods in the theory of holomorphic functions of several complex variables, and applications of these to approximation- and mapping problems.The contents are as follows. Holomorphic functions and mappings. Hartog's extension phenomenon, domains of holomorphy and holomorphic convexity. Plurisubharmonic functions. The d-bar equation and the Levi problem. Oka's approximation theorem, Runge pairs and the Cousin problems. Polynomial convexity and applications.

Textbook or/and course webpage:

1. S. Krantz. A Crash Course in Several Complex Variables

2. Raghavan Narasimhan, "Several complex variables", Chicago lectures in mathematics.

Level: Graduate, advanced undergraduate


Complex Analysis

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Activities are supported by Nesin Mathematical Village and Turkish Mathematical Society