The 5th GTSS
GEOMETRYTOPOLOGY SUMMER SCHOOL
İstanbul Center for Mathematical Sciences  Online (Youtube Channel)
July 618, 2020
First Week
Second Week
Scientific Commitee
Vicente Cortés 
University of Hamburg, Germany

İzzet Coşkun 
University of Illinois at Chicago, USA

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

İlhan İkeda 
Bosphorus University

Mustafa Kalafat 
Nesin Mathematical Village

Registration
Poster
Participants
Arrival
Information
The 5th 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 about 15 minicourses of introductory nature, related to the GeometryTopology research subjects.
In the middle of the week there is an excursion on the Bosphorus.
Do not forget to get your swimsuit with yourself.
Application
Graduate students, recent Ph.D.s and underrepresented 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
Spinc structures on manifolds and geometric applications
These lecture series aim to give an elementary exposition on basic results about the first eigenvalue of the Dirac operator, on compact Riemannian Spin and Spin^c manifolds and their hypersurfaces. For this, we select some key ingredients which illustrate the basic objects and some of their properties as Clifford algebras, spin and spin^c groups, connections, covariant derivatives, Dirac and Twistor operators. We end by giving beautiful geometric applications: a Lawson type correspondence for constant mean curvature surfaces in some 3dimensional Thurston geometries, extrinsic hyperspheres in manifolds with special holonomy, Alexandrov type theorems…
Textbook or/and course webpage:
1. Th. Friedrich, Dirac operator’s in Riemannian Geometry, Graduate studies in mathematics, Volume 25, American Mathematical Society, 2000
2. H.B. Lawson and M.L. Michelson, Spin Geometry, Princeton University press, Princeton, New Jersey, 1989.
3. J.P. Bourguignon, O. Hijazi, J.L. Milhorat, A. Moroianu and S. Moroianu, A Spinorial Approach to Riemannian and Conformal Geometry, Monographs in Mathematics, European Mathematical Society (June 2015) 462 pages.
Prerequisites:
Linear Algebra, Riemannian Geometry
The Calabi Conjectures
We shall state, prove, and study applications of some of Calabi’s Conjectures.
Topics to be covered are as follows:
Definition of complex manifolds and holomorphic maps, Examples.
Kahler metrics and examples.
Holomorphic line bundles, the canonical bundle and the first Chern class.
Statement of Calabi’s volume conjecture, the representability of the first Chern class conjecture, and the KahlerEinstein conjecture.
Examples and applications.
If time permits, a gentle introduction to the ideas behind the proof (method of continuity, a priori estimates).
Textbook and/or References: 
Prerequisites:
Manifold theory including Riemannian metrics and the LeviCivita connection, complex analysis, and functional analysis. Some exposure to elliptic PDE and complex geometry will be immensely helpful but is not necessary.
A priori estimates on the Constant Scalar Curvature Kähler metric (CscK) problem
In this lecture series we present the 2018 breakthrough results by XiuXiong Chen
and JingRui Cheng on the analytic a priori estimates for the fourth order fully
nonlinear elliptic PDE that gives constant scalar curvature Kähler metrics.
Preliminarily we deliver some motivations and historical background on the
problem; we also present the correspondence between the properness of Kenergy
and the existence of constant scalar curvature Kähler metrics. Then we
move to the review of the a priori estimates, with special emphasis on the C^0
order ones.
Textbook, References or/and course webpage:
1. An Introduction to Extremal Kähler Metrics. Gabor Szekelyhidi. American Math
Soc.
2. On the constant scalar curvature Kähler metrics, apriori estimates. XiuXiong Chen and JingRui
Cheng, available on Arxiv.
3. On the constant scalar curvature Kähler metrics, existence results. XiuXiong Chen and JingRui
Cheng, available on Arxiv.
Prerequisites:
Riemannian Geometry, Kähler geometry
Level: Graduate
EinsteinMaxwell Manifolds
An Einstein manifold is a (pseudo)Riemannian manifold (M,g) (a spacetime) such that the Ricci tensor is proportional to the metric tensor. Einstein manifolds are the solutions of Einstein's field equations for pure gravity with cosmological constant Λ (Lambda). EinsteinMaxwell manifolds, on the other hand, satisfy EinsteinMaxwell equations consisting of gravity and electromagnetism. These manifolds are not only interesting for physics but also for pure geometry since they are related to many important topics of Riemannian geometry such as Riemannian submersions, homogeneous Riemannian spaces, Riemannian functionals and their critical points, YangMills theory, holonomy groups etc. In these lectures we aim to provide basics of Einstein manifolds and some parts of their classifications. We will also deal with EinsteinMaxwell equations and study on some explicit examples.
Textbook and/or References:
1. A. L. Besse, “Einstein Manifolds”, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 10. Springer, Berlin (1987).
2. C. LeBrun, “The Einstein–Maxwell equations, Kähler metrics, and Hermitian geometry”, J. Geom. Phys. 91, 163–171 (2015).
Prerequisites:
Basic Differential Geometry (not a must but preferable)
Special Metrics in Complex Analysis
1 Complex Analysis and Differential Geometry Background
2 The Poincare Metric and Schwarz Lemma
3 Curvature, Liouville’s Theorem and Other Applications
4 Normal Families and the Spherical Metric
5 The Caratheodory and the Kobayashi Metric
6 The Bergman Metric and Some Applications
7 A Word About Several Complex Variables
Level: Graduate, advanced undergraduate
Textbook or/and course webpage:
1. Complex Analysis  The Geometric Viewpoint by Steven G. Krantz
2. Greene, Robert E.; Kim, KangTae; Krantz, Steven G. The geometry of complex domains.
Progress in Mathematics, 291. Birkhäuser Boston, Ltd., Boston, MA, 2011.
3. Krantz, Steven G. Geometric analysis of the Bergman kernel and metric.
Graduate Texts in Mathematics, 268. Springer, New York, 2013. xiv+292 pp. ISBN: 9781461479239.
Prerequisites:
Complex Analysis
Birational Geometry of the Hilbert Schemes of points on the plane
In these 4 lectures, I will give an introduction to the birational geometry of Hilbert schemes of points in the plane.
In lecture 1, I will introduce the Hilbert scheme of points on surfaces and discuss some of its basic geometric properties.
In lecture 2, I will introduce some key notions of birational geometry, moduli spaces of higher rank vector bundles and BrillNoether divisors.
In lecture 3, I will introduce Bridgeland stability conditions and discuss the interaction between the chamber decomposition of the Bridgeland stability manifold and the chamber decomposition of the effective cone.
In lecture 4, I will tie everything together and discuss the effective cone of moduli spaces of sheaves on the plane.
The Hilbert scheme will serve as the main example.
Level:
The intended audience of these lectures are graduate students and postdoctoral researchers who are familiar with algebraic geometry at the level of Hartshorne or Griffiths and Harris.
Textbook and/or References:
These lectures will be based on joint work with Daniele Arcara, Aaron Bertram, Jack Huizenga and Matthew Woolf.
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Contact: ilaydabariss@gmail.com, berkanuze@gmail.com
Activities are supported by Nesin Mathematical Village and Turkish Mathematical Society
