The 10th GTWS
GEOMETRYTOPOLOGY WINTER SCHOOL
Nesin Mathematics Village, Şirince, İzmir January
1621, 2023
Schedule of talks
Scientific Commitee
Vicente Cortés 
University of Hamburg, Germany

İzzet Coşkun 
University of Illinois at Chicago, USA

Anna Fino 
University of Torino, Italy 
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

Organizing Commitee
Craig van Coevering 
Bosphorus University

İzzet Coşkun 
University of Illinois at Chicago, USA

Mustafa Kalafat 
Rheinische FriedrichWilhelmsUniversität Bonn

RegisterTR
Poster
Participants
Arrival
Information
The 10th GTSS will be held at the
Nesin Mathematical Village
in Şirince, İzmir.
There will be minicourses of introductory nature, related to the GeometryTopology research subjects.
The venue is close to the Ancient City of Ephesus
and/or Kuşadası Beach.
Wifi info: Passwords are asurbanibal at the Library (Kutuphane) networks, bakkalnmk for istasyon and terasbahce
for kisbahcesi at the registration lobby.
Application
Graduate students, recent Ph.D.s and underrepresented minorities are
especially encouraged to the research 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 research school.
Airport: İzmir Adnan Menderes Airport  ADB is the closest one.
For pick up, faculty is advised to get out of the terminal building from the domestic gate and wait there.
Also let us know before hand, your whatsapp number and/or GoogleHangouts address in case of a problem.
Small amounts of Turkish Lira may be needed in the village
which you can withdraw from ATM machines at the
airport or tell the driver along the way.
In case you are coming to the Village independently, from the Airport you may
take the regional trains to Selçuk, possibly with a break at Tepeköy.
Then from Selçuk take a taxi which costs about 10€.
Taxi Drivers: +90 232 892 3125 (Gar Taxi), +90 553 243 0835 (Volkan), +90 532 603 1518 (Ata).
Visas: Check whether you need a visa beforehand.
Abstracts
Variation formulae for submanifolds in Kähler and G2 geometry
In any Riemannian manifold it is interesting to detect which submanifolds are minimal,
i.e. critical points of the volume functional, and to study their further properties.
We shall discuss the first and second variation formulae in this general context,
then specialize them to the case of Lagrangian/coassociative submanifolds in special Riemannian situations:
Kähler/G2 manifolds. We shall develop the necessary concepts as we go along, starting with the simplest case of geodesics.
An outline of these topics is provided in the following paper and related lecture notes.
References:
 Pacini, Tommaso. Variation formulae for the volume of coassociative submanifolds.
Available on the ArXiv at https://arxiv.org/abs/2207.13956
 Pacini, Tommaso. Lecture Notes.
Nesin Mathematics Village 2023.
Lectures on Locally Conformally Kähler Manifolds
In these lectures we will make an introduction to the locally conformally Kähler (LCK) geometry.
A LCK metric is a structure on a complex manifold which falls somewhere between a Hermitian metric and a Kähler metric.
Ingredients of the individual seminars are as follows:
LCK 0: Example of a nonKähler compact complex manifold.
LCK 1: Introduction. Lee form. Torsion 1form.
LCK 2: Weyl connection.
LCK 3: Relation with conformal structures:
HermiteWeyl structures with vanishing distance curvature. (Complex dimension ≥ 3)
LCK 4: Globally conformally Kähler (GCK) manifolds.
LCK 5: Vaisman's conjectures.
We will be using the following resources.
References:
 S. Dragomir, L. Ornea 
Locally conformal Kähler geometry.
Progress in Mathematics, 155. Birkhäuser Boston, Inc., Boston, MA, 1998.
 Vaisman, Izu. Some curvature properties of complex surfaces.
Ann. Mat. Pura Appl. (4) 132 (1982), 1–18 (1983).
 Vaisman, Izu. On locally and globally conformal Kähler manifolds.
Trans. Amer. Math. Soc. 262 (1980), no. 2, 533–542.
 Gauduchon, Paul. La 1forme de torsion d'une variété hermitienne compacte.
[Torsion 1forms of compact Hermitian manifolds] Math. Ann. 267 (1984), no. 4, 495–518.
 Ornea L., Verbitsky M.  Principles of Locally Conformally Kähler Geometry.
Available on the ArXiv at https://arxiv.org/abs/2208.07188. 772pp. v3 2023.
Gauge Theory and 4manifolds
We will begin, in this minicourse, by discussing the
classification of closed, simply connected topological 4manifolds
given in the theorems of Whitehead and Freedman. We will then move on
to the question of the existence and classification of smooth
structures on 4manifolds. A powerful tool for distinguishing smooth
structures on 4manifolds is given by the SeibergWitten invariants.
These will be introduced without proofs and some computations on
elliptic surfaces will be given.
Some textbook and references for the subject are as follows.
References:
 R. E. Gompf and A. I. Stipsicz. 4Manifolds and Kirby Calculus.
Graduate Studies in Mathematics, Vol. 20, Amer. Math. Soc., RI, 1999.
 A. Scorpan. The Wild World of 4Manifolds.
Amer. Math. Soc. Providence, RI, 2005.
 R. Fintushel and R. Stern. Six lectures on four 4manifolds.
Low dimensional topology, 265–315, IAS/Park City Math. Ser., 15, Amer.
Math. Soc., Providence, RI, 2009.
Prerequisites: Basic Topology
Introduction to Elliptic Partial Differential Equations
In this lecture series we want to examine uniformly elliptic second order partial differential equations. In order to solve these
equations we will take advantage of two different methods from analysis, namely, from Sobolev Spaces and maximum principle
techniques. The simplest nontrivial examples of elliptic PDEs are the Laplace Equation and the Poisson Equation. Any other elliptic PDE in
two variables can be expressed as a generalization of one of these equations. The general theory of solutions to Laplace’s equation is
known as potential theory, which is, in fact, the study of harmonic functions. So it is only natural to look at techniques from analysis
to solve these equations.
Daily description is as follows.
Elliptic PDEs 1: Definitions and Motivation.
Elliptic PDEs 2: Existence of Weak Solutions and Existence Theorems.
Elliptic PDEs 3: Existence of Weak Solutions and Existence Theorems (cont).
Elliptic PDEs 4: Regularity.
Elliptic PDEs 5: Maximum Principles.
Elliptic PDEs 6: Eigenvalues and Eigenfunctions.
The following resources would be helpful as a reference.
Textbook and References:
 Lawrence C. Evans. Partial Differential Equations: Second Edition.
Graduate Studies in Mathematics, American Mathematical Society, 1998.
Prerequisites: Partial Differential Equations, Sobolev Spaces
Contact: cerenaydin@nesinkoyleri.org, okelekci@gmail.com
Activities are supported by Nesin Mathematical Village, Turkish Mathematical Society and University of Bonn
