The 2nd GTSS
GEOMETRYTOPOLOGY SUMMER SCHOOL
Nesin Mathematics Village, Şirince, İzmir September
922, 2019
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Second Week
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Program
Poster
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Information
The 2nd 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.
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 less than 20$.
Please fill out the application form to attend to the summer school.
Airport: İzmir Adnan Menderes Airport  ADB is the closest one.
Visas: Check whether you need a visa beforehand.
Abstracts
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.
Lie groups, Lie algebras and representation theory
This minicourse is an introduction to Lie groups and Lie algebras. The topics covered include the structure theory and classification of complex semisimple Lie algebras using root systems, some elements of representation theory, with special emphasis on the complex semisimple case, weight theory. Towards the end, a more advanced topic might be discussed, time permitting, such as universal algebras, the PoincareBirkhoffWitt theorem, or another topic.
Textbook or/and course webpage:
1. 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, Algebra, Differentiable Topology (not a must)
Lectures on G2 Geometry
In this lecture series we present a combinatorial approach to the exceptional Lie group G2. We give a survey of various results about the algebraic structure. For the sake of completeness we decided to present them in a selfcontained way to be easily accessible for future usage. We also present some applications to geometry. Manifolds with G2 structures, Decomposition of the exterior algebra into irreducible G2 representations, Metric of a G2 structure, 16 classes of G2 structures, Deformations of G2 structures. Lie algebra of G2, roots and their spaces, order, Killing form.
Language: TR, EN
We will be following the Reference:
Karigiannis, Spiro  Deformations of G2 and Spin(7) structures.
Canad. J. Math. 57 (2005), no. 5, 1012–1055.
Author's Ph.D. Thesis also available on the Arxiv.org.
Bryant, Robert L. Some remarks on G2structures.
Proceedings of Gökova GeometryTopology Conference 2005, 75–109, Gökova Geometry/Topology Conference (GGT), Gökova, 2006.
Other useful references are:
Anthony W. Knapp. Lie groups beyond an introduction, second edition.
volume 140 of Progress in Mathematics. Birkhauser Boston, Inc., Boston, MA, 2002.
G2 Geometry 1: 4 categories of vector cross product structures.
G2 Geometry 2: Decomposition of A*(M) into irreducducible
G2 representations.
G2 Geometry 3: Metric of a G2 structure.
G2 Geometry 4: More into cross product identities.
G2 Geometry 5: 16 classes of G2 structures.
G2 Geometry 6: Deformations of G2 structures.
Prerequisites:
Lectures on Lie groups, Lie algebras and representation theory by Joseph Malkoun.
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
Kähler Geometry
Kähler geometry has been an important area of differential geometry and attracted significant interest from both mathematics and mathematical physics research community. In this lecture series our aim is to provide an introduction to different aspects of Kähler geometry. We will start with a review of Riemannian geometry. Then emphasis will be on complex and Hermitian geometry which form the basis for Kähler manifolds. We will study several aspects of Kähler manifolds such as
the CalabiYau theorem, Weitzenböck techniques, Calabi–Yau manifolds.
Textbook or/and course webpage:
A. Moroianu, “Lectures on Kähler Geometry”
Prerequisites:
Basic Differential Geometry (not a must but preferable)
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)
Spin Geometry and SeibergWitten Equations
In this lecture series we present spin geometry by using Clifford algebras. Then we focus on SeibergWitten equations.
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.
Metric Geometry
Prerequisites: Temel Analiz bilgisi ve metrik uzaylara aşinalık.
Level: Graduate, advanced undergraduate
Abstract: Bu derste, uzunluk uzayları ve jeodezik uzaylar tanıtılacaktır. Ascoli Teorem'i ifade ve ispat edilecek ve bunun sonucu olarak bir ''proper'' metrik uzayda, sonlu uzunluğa sahip bir yolla birleştirilebilen iki nokta arasında, minimal uzunluğa sahip bir yolun var olduğu gösterilecektir. Daha sonra metrik uzaylarda bazı konvekslik kavramları incelenecektir.
Language: TR
Textbook: Metric Spaces, Convexity and Nonpositive Curvature; Athanase Papadopoulos.
A Course in Metric Geometry; Dmitri Burago, Yuri Burago, Sergei Ivanov.
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
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