The 8th GTSS

GEOMETRY-TOPOLOGY SUMMER SCHOOL

İstanbul Center for Mathematical Sciences - Online (Youtube Channel)
August 15-27, 2022

First Week


TIME	SPEAKER	TITLE
Aug 19 11-12:30	Craig van Coevering	Extremal Kähler metrics and the moment map S : 1
		Lunch Time
Aug 15-19 3-4:15	Eyüp Yalçınkaya	Spin(7) Geometry S : 1 , 2 , 3
Aug 15-19 4:30-6	Ernani Ribeiro Jr.	Four-dimensional gradient Ricci solitons S : 1 , 2 , 3 , 4

Second Week


TIME	SPEAKER	TITLE
Aug 22-26 11-12:30	Alberto Raffero	Lectures on closed G_2 structures S : 1 , 2 , 3 , 4
		Lunch Time
Aug 22-26 3-4:15	Mustafa Kalafat	Minimal Surfaces S : 1 , 2 , 3 , 4
Aug 25 4:30-6	Craig van Coevering	Extremal Kähler metrics and the moment map 2 S : 2 , 3

Scientific Commitee


Vicente Cortés	University of Hamburg, Germany
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

Registration Poster Participants Arrival

Information

The 8th GTSS will be held at the IMBM, İstanbul Center for Mathematical Sciences (Online)
which is established in the main (South) Campus of Boğaziçi University (Bosphorus).
The first photo on top of this webpage demonstrates the view from the venue.
There will be mini-courses of introductory nature, related to the Geometry-Topology research subjects.

Application

Graduate students, recent Ph.D.s and under-represented minorities are especially encouraged to the summer school.
Daily expenses including bed, breakfast, lunch, dinner is around 20€.
You may stay at cheap hostels around Taksim(Nightlife) square and take the subway to the Campus easily.
We are trying to arrange accomodation on Campus as well.

Please fill out the Registration form in order to attend the "Online" summer school. Registration is free but mandatory.

Airport: İstanbul Airport - IST is the closest one. Take a bus from the airport to the 4. Levent. Then take a taxi or subway to reach to the main/south campus.
Alternatively you can use Sabiha Gökçen Airport - SAW and take a bus from there to 4. Levent. Then take a taxi or subway to reach to the main/south campus.
Navigate the link above for more detailed arrival and venue information.

Visas: Check whether you need a visa beforehand.

Abstracts

Spin(7) Geometry

Spin structures have wide applications to mathematical physics, in particular to quantum field theory. In order to study the geometry of the special class Spin(7), there are different approaches. One of them is by holonomy groups. According to the Berger classification (1955), the group Spin(7) is one of the members of the list of holonomy classes. Firstly, we are going to present its properties [1]. After that, we will present normed algebras. Normed algebras are an important concept of this geometry to define metric and measure angle between vectors. Finally, we talk about Calabi-Yau manifolds which play an important role in string theory. Induced from the properties of Calabi-Yau manifolds, we will investigate Mirror Duality on Spin(7) manifolds [2] [3].

References

D. Joyce, Lectures on Calabi-Yau and special Lagrangian geometry, Arxiv, (2002).

S. Akbulut, S. Salur, Mirror Duality via G 2 and Spin(7) Manifolds, Arithmetic and Geometry Around Quantization, Progress in Mathematics, vol 279, (2010).

E. Yalcinkaya, Mirror Duality on Spin(7) manifolds, Preprint.

Four-dimensional gradient Ricci solitons
In this minicourse, we discuss the geometry of four-dimensional gradient shrinking Ricci solitons.

Lecture 1: We will show that gradient Ricci solitons are special (self-similar) solutions of the Ricci flow.

Lecture 2: Some basic results and examples of gradient shrinking Ricci solitons.

Lecture 3: Classification of four-dimensional gradient shrinking Ricci solitons.

Lecture 4: Open problems and their motivations.

Textbook and References

Chow-Li-Nu. Hamilton’s Ricci Flow.

Huai-Dong Cao. Recent Progress on Ricci Solitons.

Extremal Kähler metrics and the moment map
An extremal Kähler metric is a canonical Kähler metric, introduced by E. Calabi, which is somewhat more general than a constant scalar curvature Kähler metric. The existence of such a metric is an ongoing research subject and expected to be equivalent to some form of geometric stability of the underlying polarized complex manifold (M, J, [ω]) –the Yau-Tian-Donaldson conjecture. Thus it is no surprise that there is a moment map, the scalar curvature (A. Fujiki, S. Donaldson), and the problem can be described as an infinite dimensional version of the familiar finite dimensional G.I.T.

In this talk I will give an introduction to extremal metrics. Then I will describe how the moment map can be used to describe the local deformation problem of extremal metrics. Essentially, the local picture can be reduced to finite dimensional G.I.T. In particular, we can construct a course moduli space of extremal Kähler metrics with a fixed polarization [ω] ∈ H2(M, R), which is an Hausdorff complex analytic space.

Lectures on closed G_2 structures
These lectures include two introductory lectures on G2 geometry and two lectures concerning recent developments and open problems on closed G2 structures.

Lecture 1: Introduction to G2 geometry I

Lecture 2: Introduction to G2 geometry II

Lecture 3: Curvature properties and symmetries of closed G2-structures

Lecture 4: Special types of closed G2-structures (Laplacian solitons, extremally Ricci pinched, exact)

References

R. L. Bryant. Some remarks on G2-structures.
Proceedings of Gökova Geometry-Topology Conference, 75–109, 2006. Available at https://arxiv.org/abs/math/0305124

D. Joyce. Riemannian holonomy groups and calibrated geometry.
Oxford Graduate Texts in Mathematics, vol. 12, Oxford University Press, 2007.

S. Karigiannis, N. C. Leung, J. D. Lotay eds. Lectures and Surveys on G2-manifolds and Related Topics,
Fields Institute Communications, vol. 84, Springer US, 2020.

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.

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.

Contact: ilaydabariss@gmail.com, eyupyalcinkaya@gmail.com

Activities are supported by University of Bonn and Boğaziçi University