The 5th GTSS

GEOMETRY-TOPOLOGY SUMMER SCHOOL

İstanbul Center for Mathematical Sciences - Online (Youtube Channel)
July 6-18, 2020

First Week


TIME	SPEAKER	TITLE
Jul 6-11 9-10:30	Roger Nakad	Spin-c structures on manifolds and geometric applications
Jul 6-11 10:30-12	Vamsi Pritham Pingali	The Calabi Conjectures
Jul 6-10 2:30-4	Simone Calamai	Constant Scalar Curvature Kähler metric (CscK) problem: A priori estimates - Videos

Second Week


TIME	SPEAKER	TITLE
Jul 13-18 10:30-12	Özgür Kelekçi	Introduction to Einstein-Maxwell Manifolds
Jul 13-18 2:30-4	Buket Can Bahadır	Special Metrics in Complex Analysis
Jul 14-18 4:30-6	İzzet Coşkun	Birational Geometry of the Hilbert Schemes of points on the plane

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 mini-courses of introductory nature, related to the Geometry-Topology 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 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-c 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 3-dimensional 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 Kahler-Einstein 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 Levi-Civita 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

Einstein-Maxwell 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). Einstein-Maxwell manifolds, on the other hand, satisfy Einstein-Maxwell 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, Yang-Mills 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 Einstein-Maxwell 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, Kang-Tae; 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: 978-1-4614-7923-9.

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 Brill-Noether 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.