The 8th GTSS
GEOMETRY-TOPOLOGY SUMMER SCHOOL
İstanbul Center for Mathematical Sciences - Online (Youtube Channel)
August 15-27, 2022
||University of Hamburg, Germany
||University of Torino, Italy
||Stony Brook University, USA
||University of California at San Diego, USA
||University of Torino, Italy
||University of Bath, UK
|Craig van Coevering
||University of Illinois at Chicago, USA
||Rheinische Friedrich-Wilhelms-Universität Bonn
The 8th GTSS will be held at the
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.
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
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.
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 .
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  .
- 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)
- R. L. Bryant. Some remarks on G2-structures.
Proceedings of Gökova Geometry-Topology Conference, 75–109, 2006. Available at
- 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.
We will be using the following resources.
- 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.
Li, Peter. Geometric analysis. Cambridge University Press, 2012.
(*) To be confirmed.
(**) No talk on the Thursday of the first week.
(***) No talk on the Wednesday of the second week.
Contact: firstname.lastname@example.org, email@example.com