The 15th GTSS

GEOMETRY-TOPOLOGY SUMMER SCHOOL

Nesin Mathematics Village, Şirince, İzmir
September 9-21, 2024

First Week


TIME	SPEAKER	TITLE
Sep 9-15 10:30-12	Daniel Massart	Translation surfaces and their moduli space
		Lunch Time

Second Week


TIME	SPEAKER	TITLE
Sep 16-21 9-10:30	Alberto Rodríguez Vázquez	Totally geodesic submanifolds and curvature Lecture Notes
Sep 16-21 11-12:30	Özgür Kelekçi	Laplacian spectrum in Riemannian Geometry
		Lunch Time
Sep 16-21 4-5:30	Mustafa Kalafat	Berger Spheres in Differential Geometry
Sep 16-21 6-7:30	Buket Can Bahadır	Spectrum of the Laplacian Operator on Riemannian Manifolds

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


Özgür Kelekçi	Turkish Aeronautical Association University
Mustafa Kalafat	Rheinische Friedrich-Wilhelms-Universität Bonn

Information

The 15th 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 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 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. 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 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

Introduction to translation surfaces and their moduli space
We review the different notions about translation surfaces which are necessary to understand McMullen’s classification of GL+ 2 (R)-orbit closures in genus two.

In Section 2 we recall the different definitions of a translation surface, in increasing order of abstraction, starting with cutting and pasting plane polygons, ending with Abelian differentials.

In Section 3 we define the moduli space of translation surfaces and explain its stratification by the type of zeroes of the Abelian differential, the local coordinates given by the relative periods, its relationship with the moduli space of complex structures and the Teichműller geodesic flow.

In Part II we introduce the GL+ 2 (R)-action, and define the related notions of Veech group, Teichműller disk, and Veech surface.

In Section 8 we explain how McMullen classifies GL+ 2 (R)-orbit closures in genus 2: you have orbit closures of dimension 1 (Veech surfaces, of which a complete list is given), 2 (Hilbert modular surfaces, of which again a complete list is given), and 3 (the whole moduli space of complex structures).

In the last section we review some recent progress in higher genus.

Totally geodesic submanifolds and curvature
The geometric objects that can be perceived by means of our senses are curves and surfaces. Submanifolds provide the natural generalization for higher dimensions of these objects and totally geodesic submanifolds are those with the simplest geometry. Intuitively, a submanifold is totally geodesic if it curves as the ambient space where it lives. The most basic examples are affine subspaces of Euclidean spaces R^n, or great subspheres of round spheres S^n. As we will see, totally geodesic submanifolds are intimately linked with curvature, and one of the purposes of this course will be to explore these links. Moreover, the existence of totally geodesic submanifolds is usually related to the abundance of isometries. Thus, the theory of totally geodesic submanifolds is particularly rich in spaces with a big isometry group such as homogeneous and symmetric spaces. Indeed, in certain cases we will be able to use Lie theory to reduce the study of totally geodesic submanifolds to a problem of algebraic nature.

Tentative list of contents.

1. Basics of submanifold geometry. Totally geodesic submanifolds. Some basic characterizations. Examples. Fixed point components of isometries.

2. Existence and uniqueness of totally geodesic submanifolds.

3. Positive curvature. Frankel theorem. Critical points of sectional curvature.

4. Totally geodesic submanifolds in homogeneous spaces.

5. Totally geodesic submanifolds in symmetric spaces.

References:

Berndt, Jürgen; Console, Sergio; Olmos, Carlos. Submanifolds and holonomy.
Chapman & Hall/CRC Research Notes in Mathematics, 434. Chapman & Hall/CRC, Boca Raton, FL, 2003.

Prerequisites: Basic differential geometry

Level: Advanced undergraduate and graduate

Berger Spheres in Differential Geometry
Berger spheres are introduced by Marcel Berger. These are odd dimensional spheres, which are squashed in a direction. The shrinking direction can be provided by the circle action on the complex Euclidean space. In other words, it corresponds to shrinking or expanding the Hopf circles in an odd dimensonal sphere. Consequently, the resulting Riemannian metric is different than the round metric. In these seminar series, we understand the eigenvalues of the Laplacian on Berger spheres.

Ingredients of the individual lectures are as follows:

BS-1: Introduction, metrics on the 2-sphere.

BS-2: Berger sphere metrics.

BS-3: Spectrum of the complex projective space.

BS-4: Jacobi Fields on Spheres.

BS-5: Orthogonal, graded decomposition of the Berger Eigenspace.

BS-6: Nontriviality of Eigenspaces.

BS-7: Dimension counting for Eigenspaces.

BS-8: Dimension counting continued.

BS-9: First eigenvalue of the Laplacian on Berger Spheres.

We will be using the following resources.

References:

Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine. Riemannian Geometry.
Springer. 2004.

Robbin, Joel W.; Salamon, Dietmar A. Introduction to differential geometry
Springer Spektrum Stud. Math. Master, Wiesbaden, 2022, xiii+418 pp.

Tanno, Shûkichi. The topology of contact Riemannian manifolds.
Illinois J. Math. 12 (1968), 700–717.

Tanno, Shûkichi. The first eigenvalue of the Laplacian on spheres.
Tohoku Math. J. (2) 31 (1979), no. 2, 179–185.

Prerequisites: Multivariable Calculus, Linear Algebra, Algebraic Curves, Riemannian Geometry (not a must but preferable)

Level: Advanced undergraduate and graduate

Riemann Surfaces and Algebraic Curves: Concepts, Classifications, and Elliptic Curves(Cancelled)
Riemann surfaces are obtained by gluing together patches of the complex plane by holomorphic maps, whereas algebraic curves are one-dimensional shapes defined by polynomial equations. (Compact) Riemann surfaces and (complex smooth projective) algebraic curves are equivalent in a precise sense. This means we can study the same objects using both complex analysis and algebra. The course starts with an introduction to the concept of Riemann surfaces, followed by illustrative examples. It then covers the classification of Riemann surfaces using covering space theory and uniformization, based on the fundamental group. Finally, we will explore compact Riemann surfaces of genus 1, specifically focusing on elliptic curves and their details.

We will be using the following resources.

References:

E. Girondo and G. Gonzalez-Diez. Introduction to compact Riemann surfaces and dessin d’enfants.
LMS Students texts 79, Cambridge University Press, 2012.

R. Miranda. Algebraic curves and Riemann surfaces.
Graduate Studies in Mathematics. American Mathematical Society.

H. M. Farkas and I. Kra. Riemann surfaces.
Graduate Texts in Mathematics. Springer.

S. Donaldson. , Riemann Surfaces.
Oxford Graduate Texts in Mathematics.

Prerequisites: Complex Analysis and Set Point Topology, basic notion about algebraic curves.

Level: Graduate, advanced undergraduate.

Laplacian spectrum in Riemannian Geometry
In these lectures, we’ll begin by introducing fundamental concepts in Riemannian geometry such as differentiable manifolds, Riemannian metrics, connection, curvature etc. We’ll then focus on some applications such as Laplace operator and spectral theory.

We will be using the following resources.

References:

Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine. Riemannian Geometry.
Springer. 2004.

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

P. Petersen. Riemannian geometry 2nd edition.
Springer. 2006.

Robbin, Joel W.; Salamon, Dietmar A. Introduction to differential geometry
Springer Spektrum Stud. Math. Master, Wiesbaden, 2022, xiii+418 pp.

Prerequisites: Basic Differential Geometry (not a must but preferable).

Level: Graduate, advanced undergraduate.

Spectrum of the Laplacian Operator on Riemannian Manifolds
In this lecture series we want to explore how we can employ theoretical tools from functional analysis for Laplace Operator on compact Riemannian manifolds, how (and if) we can use these techniques to compute spectra of torus, sphere, real and complex projective spaces. Along the way we would like to discuss the relationship between the geometry and the spectrum of a given manifold, mainly “Given the geometry, what can we say about the spectrum of a manifold?” and “If we know its spectrum, can we say anything about the geometry of a manifold?”

Ingredients of the individual lectures are as follows:

SLO-1: Introduction and a (very) short summary of theoretical background and necessary tools.

SLO-2: The Laplacian on a compact Riemannian manifold.

SLO-3: How to calculate the spectrum: Flat Tori.

SLO-4: How to calculate the spectrum: Sphere.

SLO-5: How to calculate the spectrum: Projective Space.

SLO-6: Can one hear the holes of a drum? Inverse Problems.