The 12th GTSS
GEOMETRYTOPOLOGY SUMMER SCHOOL
Nesin Mathematics Village, Şirince, İzmir September
1123, 2023
First Week
Second Week
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 asurbanipal at the Library (Kutuphane), tahtakurusu for KUTUPHANE networks,
a1b2c3d4 for istasyon, and terasbahce for kisbahcesi at the registration lobby. a1b2c3d4 for Cahit Arf
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. Register for the corresponding weeks on the website of the Nesin Mathematics Village.
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
Hyperbolicity of NonKähler Compact Complex Manifolds
We will start by covering some basic material in complex analysis and
geometry: holomorphic functions of several variables, the notions of
complex structure and complex manifold (= a differentiable manifold with a
holomorphic atlas), bidegrees for differential forms and currents (= a
kind of forms with possibly singular coefficients), positivity notions for
differential forms, Hermitian metrics (= smooth, positive definite forms
of bidegree (1, 1)), complex vector bundles and the notions of connections
and curvature thereon, etc. We will then introduce the Kähler metrics and
will present their basic properties and point out several examples of
complex manifolds carrying such metrics.
Then if time permits we will first present the classical notions of
Kobayashi and Brody hyperbolicity and we will then go on to present the very recent
notions of balanced, divisorial and partial hyperbolicity that we introduced
jointly with S. Marouani and H. Kasuya.
References:
 Hisashi Kasuya, Dan Popovici. Partially Hyperbolic Compact Complex Manifolds
Available on the ArXiv at https://arxiv.org/abs/2304.01697
Contact Geometry and Some Physical Applications
Contact geometry was formally born in 1896 in the monumental work of Sophus Lie on Berührungstransformationen (contact transformations).
However, the key ideas of contact geometry can be traced back to as early as Huygens’ 1690 formulation of geometric optics.
Contact geometry has experienced a surge of interest from various fields of mathematics and physics research community in the last two decades.
In the first part of the lecture, we aim to introduce basics of contact geometry such as vector fields and distributions, contact form, contact hyperplane, contact diffeomorphisms and so on. After covering the basic ingredients we will study some physical applications of contact geometry including examples from thermodynamics,
Hamiltonian mechanics and electrodynamics.
We will be using the following resources.
References:
 A. McInerney. First Steps in Differential Geometry, Riemannian, Contact, Symplectic.
Springer 2013.
 Blair, David E. Riemannian geometry of contact and symplectic manifolds.
Progress in Mathematics, 203. Birkhäuser, Boston, MA, 2002. xii+260 pp. ISBN: 0817642617.
Prerequisites: Basic Differential Geometry (not a must but preferable)
Level: Graduate, advanced undergraduate
Sasakian Geometry and Lie groups
Sasakian geometry is an odddimensional counterpart of Kähler geometry. The main purpose of the lectures is to compare Sasakian geometry with K”ahler geometry observing Sasakian structures on Lie groups and homogeneous spaces.
In particular, we study the classification three dimensional Sasakian manifolds proved by Belgun and related topics.
References:
 Boyer, C.P., Galicki, K. Sasakian Geometry.
Oxford Mathematical Monographs. Oxford University Press, Oxford (2008).
 F. A. Belgun. Normal CR structures on S3.
Math. Z. 244 (2003), no. 1, p. 123–151.
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 2dimensional 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 areaminimizing.
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 threemanifold 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.
 Mean curvature vector field on a Riemannian submanifold.
 First variational formula for the volume functional.
 Second variation of energy for a minimally immersed submanifold.
 Stability of minimal submanifolds.
We will be using the following resources.
References:
Li, Peter. Geometric analysis. Cambridge University Press, 2012.
Index of spheres as minimal submanifolds
This is a continuation of the basic minimal submanifold theory lectures.
In particular minimal embeddings of spheres into higher dimensional spheres.
Using the spherical harmonic functions we understand the eigenvalues and eigenvectors of the Laplacian on the round 2sphere
also the general nsphere.
[MGM71] contains explicit descriptions of the eigenvalues and eigenvectors of the standard basic manifolds including the nsphere.
Of historical interest is the treatment in what is arguably the first textbook on physics by Tait and Thomson.
The latter (a.k.a. Lord Kelvin) used it to estimate the age of the sun. Inaccurately, but not due to errors in the mathematics,
thermonuclear reactions hadn’t yet been discovered [MO].
Topics to be covered are as follows.
S1: First and second variational formula for the volume functional for a minimally immersed submanifold.
Formulation of Simons [S68].
S2a: Jacobi Operator, its index and nullity [S68].
S2b: Resolution of index and nullity of totally geodesic pspheres in an nsphere [S68].
S3: Spectrum of the Riemannian Laplacian on the round pdimensional sphere [H].
a: Homogenous extension of functions on the sphere to the Euclidean space.
b: Homogenous polynomials on the Euclidean space. Its harmonic subset and their restrictions to the psphere.
i.e. solid and surface spherical harmonics.
We will be using the following resources.
References:
 Berger, Marcel; Gauduchon, Paul; Mazet, Edmond  Le spectre d'une variété riemannienne.
Lecture Notes in Mathematics, Vol. 194 SpringerVerlag, BerlinNew York 1971 vii+251 pp.
 Hajlasz, Piotr  Functional Analysis notes. Available from the authors website at
https://sites.pitt.edu/~hajlasz/Notatki/Functional%20Analysis2.pdf
 Simons, James  Minimal varieties in riemannian manifolds.
Ann. of Math. (2), 88:62–105, 1968.
Introduction to hyperbolic geometry
We begin with the Poincaré halfplane and disc models of hyperbolic geometry. We determine the isometry group, and the geodesics (the straight lines, if you will). We explain Poincaré’s geometric construction of discrete subgroups of the isometry group, allowing us to construct surfaces of infinitely many topological types, whose geometry is hyperbolic. This proves that hyperbolic geometry is richer than Euclidean geometry, since the only complete Euclidean surfaces are the plane, the cylinder, the Möbius strip, the torus, and the Klein bottle. We also prove that the space of all tori (or moduli space of elliptic curvesdon’t worry if you have never heard any of these words) has a natural geometry, and this geometry is hyperbolic, so in a sense hyperbolic geometry rules over Euclidean geometry.
Prerequisites: Complex numbers, linear algebra (2×2 matrices)
Level: 1st year undergraduate
An introduction to knot theory
Topology is a field of mathematics that studies geometrical objects by considering them made of rubber and thus instead of rigid measurements such as area, volume or angle, measurements subject to the rubber soul are the tools of topology. Topology roots back in the studies of the 19th century scientists such as Gauss, Tait, Ampere, Thomson. Gauss tried to understand earth’ s magnetic potential via linked curves in space, Thomson suggested that atoms were knotted vortices in aether. These studies aroused a great mathematical interest in nicely shaped curves in space so called knots and with the developments in topology in the beginning of the 20th century, the study of knots became a mathematical theory on its own.
Knot theory is still one of the active areas in mathematics with many striking applications in biology (studies in DNA structure and enzymology), physics (quantum physics, ChernSimons theory, Gauge theory), and chemistry (in molecule structure, synthesizing molecules). In this course, we will construct the fundamentals of knot theory, learn about mathematical tools for classifying knots, investigate the physical aspects of the theory
and will discuss basic notions of algebraic topology and lowdimensional topology like homotopy, surfaces, 3manifolds, and such.
Level: Graduate, advanced undergraduate, beginning undergraduate
Lowdimensional topology
The world we live in is a 4dimensional space if we also consider time. Therefore, studying low dimensional spaces leads to a better understanding of our universe. In this course, we will learn how to visualize universes of different dimensions, and how we might distinguish one universe from another. More specifically, we will discuss various ways to decompose a complicated space into simpler pieces. From such a decomposition, we can draw many conclusions. For instance, we can deduce that a space is very complex if we need a large number of basic pieces to build it.
Level: Graduate, advanced undergraduate, beginning undergraduate
Contact: cerenaydin@nesinkoyleri.org, okelekci@gmail.com
Activities are supported by Nesin Mathematical Village, Turkish Mathematical Society and University of Bonn
