The 7th GTSS
GEOMETRYTOPOLOGY SUMMER SCHOOL
Nesin Mathematics Village, Şirince, İzmir September
720, 2020
(Invitation process is started.
Let us know
if you have a researchminicourse proposal!)
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

RegisterTR
Poster
Participants
Arrival
Information
The 7th GTSS will be held at the
Nesin Mathematical Village
in Şirince, İzmir.
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 to the Ancient City of Ephesus
and/or Kuşadası Beach.
Wifi info: Passwords are TermoS1! at the Library (Kutuphanegenel) networks, zeytinlik for istasyon and kurugolet
for kisbahcesi at the registration lobby.
Application
Graduate students, recent Ph.D.s and underrepresented minorities are
especially encouraged to the summer 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 summer school.
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 GoogleHangouts 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.
Abstracts
Special relativity from a mathematician’s point of view
This will be a gentle introduction to special relativity, from the mathematician’s point of view. So we shall understand special relativity as a geometry, that is, as a mathematical space together with a group of transformations acting on that space.
1. Principles of special relativity
2. The Lorentz group
3. Effects in space and time
4. The metric, and causality
5. Aberration
6. The celestial sphere, and SL(2,C)
Prerequisites: Groups of transformations, elementary linear algebra, elementary geometry.
Level: Advanced undergraduate
Twistor Geometry
This will be a gentle introduction to twistor geometry. We will describe the Klein correspondence between Minkowski space and twistor space, and we will give an elementary account of the Penrose transform.
1. Compactification of Minkowski space
2. Definition of Twistor space
3. The Klein correspondence
4. Causal structure
5. Real points
6. Functions on Twistor space
7. A double fibration
Prerequisites: Special relativity, complex analysis, projective geometry
Level: Graduate
Character Theory of Finite Groups
Foundation of the theory of linear representations and characters of finite groups. Basic tools: induction and restriction, Clifford theory. Applications to group theory: theorems of Burnside and Frobenius. Character degrees and groups structure: theorems of Ito and Thompson.
Textbook: I.M. Isaacs, Character Theory of Finite Groups
Prerequisites: Some finite group theory and linear algebra.
Level: Advanced undergraduate
Representation Theory of Complex Lie Algebras
In this lecture series we give an introduction to the representation theory of some complex Lie algebras. We review Lie groups, Lie algebras, Cartan subalgebra, Root systems. Then, starting from concrete examples we will work on sl(2,C), sl(3,C), sl(4,C) and finally sl(n,C) and their representations. We also go into theri geometric interpretations. Also spend time on sp(2n,C) and so(m,C) if time permits.
Textbook or/and course webpage:
1. Fulton, William; Harris, Joe  Representation theory. A first course.
Graduate Texts in Mathematics, 129. Readings in Mathematics. SpringerVerlag, New York, 1991. xvi+551 pp. ISBN: 0387975276
Lectures 11,12,13,14 and 22.
2. Anthony W. Knapp. Lie groups beyond an introduction,
volume 140 of Progress in Mathematics. Birkhauser Boston, Inc., Boston, MA, second edition, 2002.
Prerequisites:
Linear Algebra. Lie groups and Lie algebras.
Representation Theory of the Lie Algebra of G_2
In this lecture series we give an introduction to the representation theory of the Lie algebra of G_2.
We review Lie groups, Lie algebras, Cartan subalgebra, Root systems. Then, starting from concrete examples we will work on sl(2,C), sl(3,C), sl(4,C) and finally sl(n,C) and their representations. We also go into their geometric interpretations. Also spend time on sp(2n,C) and so(m,C) if time permits.
Textbook or/and course webpage:
1. Fulton, William; Harris, Joe  Representation theory. A first course.
Graduate Texts in Mathematics, 129. Readings in Mathematics. SpringerVerlag, New York, 1991. xvi+551 pp. ISBN: 0387975276
Lectures 11,12,13,14 and 22.
Prerequisites:
Lie groups and Lie algebras.
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. 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 or/and course webpage:
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).
Level: Graduate, advanced undergraduate
Prerequisites: Basic Differential Geometry (not a must but preferable)
EinsteinMaxwell Kähler metrics(*)
Locally Conformally Kähler (lcK) Manifolds(*)
Special metrics in Sasakian geometry
This course will begin with an introduction to Sasakian geometry, a type of metric contact structure which is an odd dimensional analogue of a Kähler structure on a complex manifold. We will then consider the problem of finding a special Sasakian metrics, such as Einstein, and more generally constant scalar curvature and extremal metrics.
Textbook or/and course webpage:
1. Boyer, Charles P.; Galicki, Krzysztof. Sasakian geometry.
Oxford Mathematical Monographs.
Oxford University Press, Oxford, 2008. xii+613 pp.
2. Futaki, Akito; Ono, Hajime; Wang, Guofang
Transverse Kähler geometry of Sasaki manifolds and toric SasakiEinstein manifolds.
J. Differential Geom. 83 (2009), no. 3, 585–635.
3. Collins, Tristan C.; Székelyhidi, Gábor. Ksemistability for irregular Sasakian manifolds. J. Differential Geom. 109 (2018), no. 1, 81–109.
Prerequisites:
A first year graduated course in differential geometry should be sufficient.
Calibrated submanifolds in G2 manifolds
In this lecture, we first give a survey of calibrated submanifolds in G2 manifolds. Then we present their deformation theories.
Textbook or/and course webpage:
1. K. Kawai, Deformations of homogeneous associative submanifolds in nearly parallel G2manifolds, Asian J. Math. 21 (2017), 429462.
2. K. Kawai, Secondorder deformations of associative submanifolds in nearly parallel G2manifolds, Q. J. Math. 69 (2018), 241270.
3. J. D. Lotay, Associative Submanifolds of the 7Sphere,
Proc. Lond. Math. Soc. (3) 105 (2012), 11831214.
4. R. C. McLean, Deformations of Calibrated Submanifolds,
Comm. Anal.Geom. 6 (1998), 705747.
Prerequisites:
Linear Algebra, Riemannian Geometry
Complex geometric analysis
I shall give a quick course on the complex/Kaehler geometry with emphasis of PDE methods.
Textbook or/and course webpage:
I shall distribute some notes. The students can start with books such as
Complex Manifolds, by J. Morrow and K. Kodaira, AMS.
Principles of of Algebraic Geometry, by GriffithsHarris, John Wiley and Sons.
Differential Analysis on Complex Manifolds, by Wells. Springer.
Complex differential manifolds, by Zheng, AMS/IP.
Complex geometry, by Huybrechts, Springer.
Prerequisites:
Linear Algebra, Partial Differential Equations/Complex Analysis (not a must but preferable)
Differential Harnack estimate on manifolds
In this lecture series we present “LiYauHamilton“ type differential Harnack estimate on Riemannian manifolds and Kähler manifolds.
Textbook or/and course webpage:
1. Bennett Chow, Peng Lu and Lei Ni, Hamilton’s Ricci flow, Graduate Studies in Mathematics, 77. American Mathematical Society, Providence, RI; Science Press Beijing, New York, 2006.
2. B.Wilking, A Lie algebraic approach to Ricci ﬂow invariant curvature condition and Harnack inequalities, J. reine angew. Math. (Crelle), 679 (2013), 223–247.
3. L. Ni and Y. Y. Niu, Sharp diﬀerential estimates of LiYauHamilton type for positive (p, p)forms on
Kähler manifolds. Comm. Pure Appl. Math., 64 (2011), 920–974.
Prerequisites:
Riemannian manifold, Kähler manifold (not a must but preferable)
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
Partial Differential Equations (Sobolev and Hölder Spaces)
Partial Differential Equations, Functional Analysis, Riemannian Geometry (not a must but preferable).
Level: Graduate, advanced undergraduate
Abstract: In this lecture series, our aim is to introduce Sobolev Spaces, and to present techniques and ideas from functional analysis to develop the theory. Since the solutions of partial differential equations are naturally found in Sobolev spaces, this theory proves to be a useful tool for several applications in partial differential equations, as well as giving us an opportunity for a different aproach to classical problems using the methods of functional analysis. We also would like to make an introduction to how these concepts can be aplied in the set up of Riemannian Manifolds., and give examples, such as Yamabe problem, where this aprooach played an important role.
Textbook or/and course webpage:
Evans, Lawrence C.  Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.
Hyperbolic Geometry
In this lecture series we give an introduction to hyperbolic geometry with some discussion of other nonEuclidean systems. We generate useful volume and area formulas for tetrahedrons and triangles in lowdimensional hyperbolic space. Daily sections of the lecture are given below.
Description of the hyperbolic geometry models and the connection between them.
Description of the general Mobius group.
Characterization of the isometries of hyperbolic space.
Geometry of hyperbolic triangles.
Hyperbolic trigonometry.
Hyperbolic area and the GaussBonnet theorem.
Textbook or/and course webpage:
1. Anderson, James W. – Hyperbolic Geometry, SpringerVerlag, London, 2005.
2. Ratcliffe, John G., Foundations of Hyperbolic Manifolds, Graduate texts in Mathematics, 149, Springer, New York, 2006.
Prerequisites:
Linear Algebra
Generalization of Einstein metrics
In this lecture series we review some generalizations of Einstein metrics. These are Ricci solitons, static spaces and warpedproduct Eınstein metrics. As a tool to approach these strucures, we start with some study on the socalled Codazzi tensors and then talk about harmonic curvature and harmonic Weyl curvature. After these, we apply the Codazzi tensor approach to study each of the above generalizations of Einstein metrics.
Textbook or/and course webpage:
1. A.L. Besse  Einstein manifolds, Chapter 16, Ergebnisse der Mathematik, 3 Folge, Band 10, SpringerVerlag, 1987.
2. G. Catino, C. Mantegazza, L. Mazzieri, A note on Codazzi tensors, Math. Ann. 362 (2015), No. 12, 629638.
3. J. Kim. On a classification of 4d gradient Ricci solitons with harmonic Weyl curvature,
J. Geom. Anal. 27, (2017), no. 2, 9861012.
Prerequisites:
Riemannian Geometry
Hyperkähler Manifolds
Hyperkähler manifolds are manifolds with 3 complex structures satisfying quaternionic relations,
and a torsionfree orthogonal connection preserving these complex structures.
A number of objects arising in hyperkähler geometry are equipped with hyperkähler structure,
such as moduli of vector bundles, deformation spaces, subvarieties and so on.
This allows one to speak of “hyperkähler geometry”, similar to the usual complex algebraic geometry.
I would explain basic properties of hyperkähler manifolds,
give some examples and prove results about subvarieties.
Textbook or/and course webpage:
Besse, “Einstein manifolds”
Prerequisites:
Differential geometry, topology
Kalafat:
We show that a compact complex surface which admits a
conformally Kahler metric g of positive orthogonal holomorphic
bisectional curvature is biholomorphic to the complex projective
plane. In addition, if g is a Hermitian metric which is Einstein, then
the biholomorphism can be chosen to be an isometry via which g becomes
a multiple of the FubiniStudy metric. This is a joint work with
C.Koca.
Reference:
Kalafat, Mustafa; Koca, Caner 
Conformally Kähler surfaces and orthogonal holomorphic bisectional curvature.
Geom. Dedicata 174 (2015), 401–408.
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Contact: aslicankorkmaz@nesinvakfi.org, okelekci@gmail.com
Activities are supported by Nesin Mathematical Village and Turkish Mathematical Society
