Geometry Main

The 9th GTSS


Nesin Mathematics Village, Şirince, İzmir
September 12-24, 2022

First Week


TIME              SPEAKER                  TITLE
Sep 12-16
Mustafa Kalafat
Introduction to Minimal Submanifolds
Lunch Time
Sep 12-16
Stephen Huggett Special relativity from a mathematician’s point of view
Lecture Notes

Second Week


TIME              SPEAKER                  TITLE
Sep 19-23
Mustafa Kalafat
Riemann Surfaces in Einstein-Hermitian Spaces
Lunch Time
Sep 19-23
Stephen Huggett Twistor Geometry
Lecture Notes

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

Register-TR       Poster       Participants       Arrival


The 9th 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.


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.
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


Special relativity from a mathematician’s point of view

This will be a gentle introduction to special relativity, from the mathematician’s point of view. So we shall understand special relativity as a geometry, that is, as a mathematical space together with a group of transformations acting on that space.

1. Principles of special relativity

2. The Lorentz group

3. Effects in space and time

4. The metric, and causality

5. Aberration

6. The celestial sphere, and SL(2,C)

Prerequisites: Groups of transformations, elementary linear algebra, elementary geometry.

Level: Advanced undergraduate

Twistor Geometry

This will be a gentle introduction to twistor geometry. We will describe the Klein correspondence between Minkowski space and twistor space, and we will give an elementary account of the Penrose transform.

1. Compactification of Minkowski space

2. Definition of Twistor space

3. The Klein correspondence

4. Causal structure

5. Real points

6. Functions on Twistor space

7. A double fibration

Prerequisites: Special relativity, complex analysis, projective geometry

Level: Graduate

Lectures on Stability Of Minimal Submanifolds

A minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature. They are 2-dimensional analogue to geodesics, which are analogously defined as critical points of the length functional.

Minimal surface theory originates with Lagrange who in 1762 considered the variational problem of finding the surface z = z(x, y) of least area stretched across a given closed contour. He derived the Euler–Lagrange equation for the solution He did not succeed in finding any solution beyond the plane. In 1776 Jean Baptiste Marie Meusnier discovered that the helicoid and catenoid satisfy the equation and that the differential expression corresponds to twice the mean curvature of the surface, concluding that surfaces with zero mean curvature are area-minimizing.

By expanding Lagrange's equation, Gaspard Monge and Legendre in 1795 derived representation formulas for the solution surfaces. While these were successfully used by Heinrich Scherk in 1830 to derive his surfaces, they were generally regarded as practically unusable. Catalan proved in 1842/43 that the helicoid is the only ruled minimal surface.

Progress had been fairly slow until the middle of the century when the Björling problem was solved using complex methods. The "first golden age" of minimal surfaces began. Schwarz found the solution of the Plateau problem for a regular quadrilateral in 1865 and for a general quadrilateral in 1867 using complex methods. Weierstrass and Enneper developed more useful representation formulas, firmly linking minimal surfaces to complex analysis and harmonic functions. Other important contributions came from Beltrami, Bonnet, Darboux, Lie, Riemann, Serret and Weingarten.

Between 1925 and 1950 minimal surface theory revived, now mainly aimed at nonparametric minimal surfaces. The complete solution of the Plateau problem by Jesse Douglas and Tibor Radó was a major milestone. Bernstein's problem and Robert Osserman's work on complete minimal surfaces of finite total curvature were also important.

Another revival began in the 1980s. One cause was the discovery in 1982 by Celso Costa of a surface that disproved the conjecture that the plane, the catenoid, and the helicoid are the only complete embedded minimal surfaces in R^3 of finite topological type. This not only stimulated new work on using the old parametric methods, but also demonstrated the importance of computer graphics to visualise the studied surfaces and numerical methods to solve the "period problem" (when using the conjugate surface method to determine surface patches that can be assembled into a larger symmetric surface, certain parameters need to be numerically matched to produce an embedded surface). Another cause was the verification by H. Karcher that the triply periodic minimal surfaces originally described empirically by Alan Schoen in 1970 actually exist. This has led to a rich menagerie of surface families and methods of deriving new surfaces from old, for example by adding handles or distorting them.

Currently the theory of minimal surfaces has diversified to minimal submanifolds in other ambient geometries, becoming relevant to mathematical physics (e.g. the positive mass conjecture, the Penrose conjecture) and three-manifold geometry (e.g. the Smith conjecture, the Poincaré conjecture, the Thurston Geometrization Conjecture).

In this lecture series we will give an introduction to some topics in minimal submanifold theory. The topics to be covered are as follows.
  1. Mean curvature vector field on a Riemannian submanifold.

  2. First variational formula for the volume functional.

  3. Second variation of energy for a minimally immersed submanifold.

  4. Stability of minimal submanifolds.
We will be using the following resources.


Li, Peter. Geometric analysis. Cambridge University Press, 2012.

Riemann Surfaces in Einstein-Hermitian Spaces

This is a continuation of the basic minimal submanifold theory lectures. In particular minimal embeddings of spheres into higher dimensional spheres especially as holomorphic curves. Topics to be covered are as follows.

RS 1: Jacobi Operator. Higher dimensional fundamental forms.

RS 2: Higher dimensional curvatures.

RS 3: Minimal immersions into higher spheres.

RS 4: A Hermitian structure on the normal bundle.

RS 5: An application of the Riemann-Roch formula to the minimal surfaces.

RS 6: Index of minimal immersions of a sphere into higher dimensional spheres.

RS 7: Holomorphic curves in the 6-dimensional sphere.

We will be using the following resources.

  1. N. Ejiri - The Index of Minimal Immersions of S^2 into S^{2n}.
    Mathematische Zeitschrift. (1983).

  2. Kühnel, Wolfgang - Differential geometry. Curves—surfaces—manifolds. Third edition.
    Translated from the 2013 German edition. American Mathematical Society. 2015.

  3. J. Madnick - The Second Variation of Null-Torsion Holomorphic Curves in the 6-Sphere.
    ArXiv:2101.09580 (Jan 2021) 35 pages.

  4. J. Madnick - Free-Boundary Problems for Holomorphic Curves in the 6-Sphere.
    Arxiv:2105.10562 (May 2021). 17 pages.

  5. S. Montiel and F. Urbano - Second Variation of Superminimal Surfaces into Self-Dual Einstein 4-Manifolds.
    Trans. AMS. (1997).

Simply Easy Learning


Activities are supported by Nesin Mathematical Village, Turkish Mathematical Society and University of Bonn