The 10th GTWS

GEOMETRY-TOPOLOGY WINTER SCHOOL

Nesin Mathematics Village, Şirince, İzmir
January 16-21, 2023

Schedule of talks


TIME	SPEAKER	TITLE
Jan 16-20 9-10:30	Tommaso Pacini	Variation formulae for submanifolds in Kähler and G2 geometry Lecture Notes
Jan 16-21 10:30-12	Mohan Bhupal	Gauge theory and 4-manifolds
		Lunch Time
Jan 16-21 14-15:30	Mustafa Kalafat	Locally Conformally Kähler (LCK) Geometry
Jan 16-21 16-17:30	Buket Can Bahadır	Introduction to Elliptic Partial Differential Equations

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 Friedrich-Wilhelms-Universität Bonn

Information

The 10th GTSS will be held at the Nesin Mathematical Village in Şirince, İzmir.
There will be mini-courses of introductory nature, related to the Geometry-Topology 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 under-represented 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 Google-Hangouts 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.

Simply Easy Learning

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 non-Kähler compact complex manifold.

LCK 1: Introduction. Lee form. Torsion 1-form.

LCK 2: Weyl connection.

LCK 3: Relation with conformal structures: Hermite-Weyl 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 1-forme de torsion d'une variété hermitienne compacte.
[Torsion 1-forms 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 4-manifolds
We will begin, in this minicourse, by discussing the classification of closed, simply connected topological 4-manifolds 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 4-manifolds. A powerful tool for distinguishing smooth structures on 4-manifolds is given by the Seiberg-Witten 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. 4-Manifolds and Kirby Calculus.
Graduate Studies in Mathematics, Vol. 20, Amer. Math. Soc., RI, 1999.

A. Scorpan. The Wild World of 4-Manifolds.
Amer. Math. Soc. Providence, RI, 2005.

R. Fintushel and R. Stern. Six lectures on four 4-manifolds.
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 non-trivial 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: