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The 3rd GTWS

GEOMETRY-TOPOLOGY WINTER SCHOOL

Nesin Mathematics Village, Şirince, İzmir
January 6-19, 2020



First Week

 

TIME              SPEAKER                  TITLE
Jan 6-10
9-10:30
Martin Svensson
Lectures on Harmonic Maps
Jan 6-11
10:30-12
Craig Van Coevering
Sasakian Geometry A
Jan 6-12
12-1:15
Eyüp Yalçınkaya     Spin Geometry and Seiberg-Witten Equations    
Lunch
Jan 6-12
3-4:30
Kurando Baba     Lectures on Calibrations and Generalized Duality    
Jan 6-12
4:30-6
Mustafa Kalafat
Representation Theory of the Lie Algebra of G_2
Jan 6-11
6-7:15
Mehmet Kılıç Metric Geometry
Dinner
Jan 6-12
9:00
Çağrı Hacıyusufoğlu     Introduction to Riemannian Geometry    

Seminar

 

TIME              SPEAKER                  TITLE
Jan 12
2:00
Mehmet Kılıç Geodesic Spaces, Hyperconvexity and Tight Span of a Metric Space


Second Week


 
TIME              SPEAKER                  TITLE
Jan 13-18
9-10:30
Craig Van Coevering
Sasakian Geometry B
Jan 13-18
10:30-12
Dennis The Symmetry, Tanaka Theory and Cartan Geometry
Jan 13-18
12-1:15
Çağrı Hacıyusufoğlu     Minimal Surfaces    
Lunch
Jan 13-18
3-4:30
Özgür Kelekçi Einstein-Maxwell Manifolds
Jan 13-18
4:30-6
Buket Can Bahadır
Introduction to Elliptic Partial Differential Equations
Jan 13-19
6-7:15
Murat Savaş Hyperbolic Geometry

Seminar

 

TIME              SPEAKER                  TITLE
Jan 13
3:00
Mustafa Kalafat On minimal submanifolds of the Page space


Scientific Commitee

 

Vicente Cortés University of Hamburg, Germany
Ljudmila Kamenova Stony Brook University, USA
Lei Ni University of California at San Diego, USA
Misha Verbitsky IMPA, Brasil


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Information

The 3rd GTWS will be held at the Nesin Mathematical Village in Şirince, İzmir.
There will be about 15 mini-courses of introductory nature, related to the Geometry-Topology research subjects.
Venue-Classroom: 2nd floor of the Library building.
In the middle of the week there is an excursion to the Ancient City of Ephesus .

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 under-represented minorities are especially encouraged to the winter school.
Partial financial support is available. Daily expenses including bed, breakfast, lunch, dinner is around 20€.

Please fill out the application form and send it to

Aslı Can Korkmaz: aslicankorkmaz@nesinvakfi.org, okelekci@gmail.com

to attend to the 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 Google-Hangouts 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.


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Abstracts



Sasakian Geometry

Abstract or a daily curriculum: This course will present some recent ideas in Sasakian geometry, with applications to Sasaki-Einstein manifolds, existence of constant scalar curvature and extremal metrics. In particular, we will discuss ideas involving complex geometry such as toric varieties, Hilbert series, and stability, with applications to the above special metrics. We will start with an introduction, which will only presuppose some knowledge of differential geometry. Then we will present recent topics involving the existence of special metrics, their obstructions, with applications to physics. And some recent topics involving complex geometry, Hilbert series, and K-stability.

Textbook or/and course webpage:

1. Boyer, Charles P.;Galicki, Krzysztof. Sasakian geometry, Oxford Mathematical Monographs. Oxford University Press, Oxford, 2008.

2. Martelli, Dario; Sparks, James, Yau; Shing-Tung. The geometric dual of a-maximisation for toric Sasaki-Einstein manifolds, Comm. Math. Phys. 268 (2006), no. 1, 39–65.

3. Martelli, Dario; Sparks, James; Yau, Shing-Tung. Sasaki-Einstein manifolds and volume minimisation. Comm. Math. Phys. 280 (2008), no. 3, 611–673.

4. Collins, Tristan C.; Székelyhidi, Gábor. K-semistability for irregular Sasakian manifolds. J. Differential Geom. 109 (2018), no. 1, 81–109.

5. Collins, Tristan C.; Székelyhidi, Gábor. Sasaki-Einstein metrics and K-stability. Geom. Topol. 23 (2019), no. 3, 1339–1413.

Prerequisites: Beginning graduate differential geometry, i.e. some knowledge of manifolds, Riemannian geometry, vector bundles, connections.

Level: Graduate

Lectures on Calibrations and Generalized Duality

In this lecture series we discuss calibrated geometry in the view point of Lie groups actions on symmetric spaces. Concrete examples of calibrations given by Harvey-Lawson are invariant under the action of the holonomy groups of Riemannian manifolds, which give Lie group actions on Grassmannian manifolds with geometrically nice properties. We also explain constructions of calibrated submanifolds. In particular, we give special Lagrangian submanifolds in Stenzel spaces by using moment map techniques. The calibrated geometry was generalized by Mealy in the pseudo-Riemannian geometry category. We give a correspondence between Harvey-Lawson’s calibrations and Mealy’s calibrations by a generalization of the duality in symmetric spaces.

Textbook and References:

1. M. Arai and K. Baba, Special Lagrangian submanifolds and cohomogeneity one actions on the projective spaces, Tokyo J. Math. 42 (2019), 255-284.

2. R. Harvey and H. B. Lawson, Jr., Calibrated geometries, Acta Math, 148 (1982), 47-157.

3. S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press (1979).

4. J. Mealy, Volume maximization in semi-Riemannian manifolds, Indiana Univ. Math. J. 40 (1991) 793-814.

Prerequisites: Linear Algebra, Riemannian Geometry (not a must but preferable)

Level: Graduate Advanced undergraduate

Lectures on Harmonic Maps

The area of harmonic maps includes a range of familiar topics from differential geometry such as harmonic functions, parametrized geodesics, holomorphic maps in Kähler geometry and minimal branched immersions. In this lecture series we will discuss harmonic maps in Riemannian geometry. After a general introduction we will focus on harmonic maps from Riemann surfaces to Lie groups and symmetric spaces. We will see a number of constructions of such maps, and along the way we will encounter twistor theory, loop groups and integrable systems in the context of harmonic maps.

Textbook, Reference or/and course webpage:

1. F.E. Burstall, Harmonic tori in spheres and complex projective spaces, J. Reine Angew. Math. 469 (1995), 149 – 177.

2. F.E. Burstall and M.A. Guest, Harmonic two-spheres in compact symmetric spaces, revisited, Math. Ann. 309 (1997), 541 – 572.

3. J. Davidov and A.G. Sergeev, Twistor spaces and harmonic maps (Russian), Uspekhi Math. Nauk 48 (1993), no. 3 (291), 3 – 96; translation in Russian Math. Surveys 48 (1993), no. 3, 1 – 91.

4. J. Dorfmeister, F. Pedit and H. Wu, Weierstrass type representations of harmonic maps into symmetric spaces, Comm. Anal. Geom. 6 (1998), no. 4, 633 – 668.

5. K. Uhlenbeck, Harmonic maps into Lie groups: classical solutions of the chiral model, J. Differential Geom. 30 (1989), no. 1, 1 – 50.

6. M. Svensson and J.C. Wood, Filtrations, factorizations and explicit formulae for harmonic maps, Comm. Math. Phys. 310 (2012), no. 1, 99 – 134. arXiv: 0909.5582 M. Svnesson and J.C. Wood, New constructions of twistor lifts for harmonic maps, Manuscripta Math. 144 (2014), no. 3 – 4, 457 – 502. arXiv: 1106.1832

Prerequisites: Linear Algebra, Differential Geometry, Riemannian Geometry (not a must but strongly recommended)

Level: Advanced undergraduate

Symmetry, Tanaka Theory and Cartan Geometry

In Klein’s Erlangen program, geometries were organized and studied according to their symmetry groups. Cartan geometry provides a vast generalization of Klein’s program and the goal of this lecture series is to motivate and use the notion of a Cartan geometry as a “curved version” of a “flat” homogeneous model. A student of Riemannian geometry is introduced to the curvature of a Riemannian metric, but the Cartan geometric framework provides a means for speaking about the notion of curvature for a much broader class of (finite-type) geometric structures. For example, what is the curvature of a 2nd order ODE, or a generic rank 2 distribution on a 5-manifold, i.e. a (2,3,5)-distribution? Two key algebraic tools will be introduced: Tanaka prolongation and Kostant’s theorem for Lie algebra cohomology. The former is used to identify the maximal symmetry dimension for a class of geometric structures, while the latter is used in particular to arrive at structural information about the fundamental curvature quantities obstructing flatness for so-called parabolic geometries (which include 2nd order ODE, (2,3,5)-geometry, conformal, projective, CR, and many geometries besides). I will conclude with an outline of how these tools were used for a wide class of parabolic geometries to solve their symmetry gap problem: namely, what is the submaximal (next realizable) symmetry dimension below the maximal one? e.g. for 2nd order ODE, max = 8 and submax = 3 (Tresse 1896); for (2,3,5)-distributions, max = 14 and submax = 7 (Cartan 1910).

References:

1. R.W. Sharpe, Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program, GTM 166, 1997.

2. A. Cap, J. Slovak, Parabolic Geometries I: Background and General Theory, Mathematical Surveys and Monographs, vol. 154.

3. B. Kruglikov, D. The, The gap phenomenon in parabolic geometries, Journal für die reine und angewandte Mathematik, 2017(723), 153–215, DOI: https://doi.org/10.1515/crelle-2014-0072.

Prerequisites: Basic Differential Geometry, Lie Groups / Lie Algebras. (Some exposure to semisimple structure theory, e.g. roots / weights, Weyl group, etc. would be helpful.)

Level: Graduate

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. Springer-Verlag, New York, 1991. xvi+551 pp. ISBN: 0-387-97527-6
Lectures 11,12,13,14 and 22.

Prerequisites:

Lie groups and Lie algebras.

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 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)

Spin Geometry and Seiberg-Witten Equations

The Spin Geometry has its roots in physics and the study of spinors. We will cover spin geometry around algebra, geometry and analysis. When combined with the Atiyah-Singer index theorem - one of the most remarkable results in twentieth century mathematics - it has far-reaching applications to geometry and topology. This course has some goals. The first goal is to understand the concept of Dirac operators. The second is to state, and prove, the Atiyah-Singer index theorem for Dirac operators. The last goal is to apply these concepts to topology: A remarkable number of topological results - including the Chern-Gauss-Bonnet theorem, the signature theorem and the Hirzebruch-Riemann-Roch theorem - can be understood just by computing the index of a Dirac operator. Finally, we focus on Seiberg-Witten invariants of four-manifolds.

Textbook or/and course webpage:

1. Lawson, H. B. and M. Michelsohn - Spin Geometry , Princeton University Press, Princeton, New Jersey, 1989.

2. Lounesto P.- Clifford Algebras and Spinors , Cambridge University Press, London Mathematical Society Lecture Note Series 239, 2001

Introduction to Riemannian Geometry and Minimal Surfaces

Firstly, I will start with a short introduction to Riemannian Geometry: connections, gedoesics, curvature tensors, Ricci and scalar curvatures and the second fundamental form will be defined. Hopf-Rinow and Hadamard’s theorems may be discussed. In the second part, I will define minimal submanifolds, calculate the second variation of area for minimal submanifolds, introduce stability and Morse index of minimal hypersurfaces. My aim is to study the proof of the following theorem [3]: the plane is the only complete stable oriented minimal surface in R^3. For this I will introduce the eigenvalue problem for the Schrodinger operator on a Riemannian manifold. If time permits I will also introduce Weierstrass Representation and Gauss Map of a minimal surface and present some examples of complete minimal surfaces in R^3. We will mainly follow the first seven chapters of [1] and the first chapter of [2].

Textbook, References or/and course webpage:

1. M. P. do Carmo, Francis Flaherty - Riemannian Geometry - Birkhauser(1992)

2. T. Colding, W. Minicozzi II - A course in minimal surfaces - AMS(2011)

3. Fischer-Colbrie, D., and Schoen, R., The structure of complete stable minimal surfaces in 3- manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980) 199–211.

Prerequisites: Differentiable manifolds (definitions of the tangent bundle and differential forms)

Level: Graduate/Advanced undergraduate

Introduction to Elliptic Partial Differential Equations

In this lecture series we want to examine uniformly elliptic second order partial differential equations. In order to solve these equations we will take advantage of two different methods from analysis, namely, from Sobolev Spaces and maximum principle techniques. The simplest non-trivial examples of elliptic PDEs are the Laplace Equation and the Poisson Equation. Any other elliptic PDE in two variables can be expressed as a generalization of one of these equations. The general theory of solutions to Laplace's equation is known as potential theory, which is, in fact, the study of harmonic functions. So it is only natural to look at techniques from analysis to solve these equations. Daily description is as follows.

Elliptic PDEs 1: Definitions and Motivation.

Elliptic PDEs 2: Existence of Weak Solutions and Existence Theorems.

Elliptic PDEs 3: Existence of Weak Solutions and Existence Theorems (cont).

Elliptic PDEs 4: Regularity.

Elliptic PDEs 5: Maximum Principles.

Elliptic PDEs 6: Eigenvalues and Eigenfunctions.

Textbook or/and course webpage:

1. Lawrence C. Evans - Partial Differential Equations: Second Edition (Graduate Studies in Mathematics), American Mathematical Society, 1998.

2. Thomas Ransford, Potential Theory in the Complex Plane, Cambridge University Press, 2010.

Prerequisites: Partial Differential Equations, Sobolev Spaces

Level: Graduate Advanced undergraduate

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 non-Euclidean systems. We generate useful volume and area formulas for tetrahedrons and triangles in low-dimensional 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 Gauss-Bonnet theorem.

Textbook or/and course webpage:

1. Anderson, James W. – Hyperbolic Geometry, Springer-Verlag, London, 2005.

2. Ratcliffe, John G., Foundations of Hyperbolic Manifolds, Graduate texts in Mathematics, 149, Springer, New York, 2006.

Prerequisites:

Linear Algebra

Algebraic Topology - Homology and Cohomology of 2-manifolds

In this lecture series we will define fundamental groups, cellular homology and cellular cohomology groups. We will especially focus on 2 dimensional spaces. Daily description is as follows.

Day 1: Cell Complexes, Homotopy

Day 2: Fundamental Groups, Homology Axioms

Day 3: Homology Groups, Calculation of Homology Groups of Cell Complexes

Day 4: Cohomology Axioms

Day 5: Cohomology Groups

Day 6: Calculation of Cohomology Groups of Cell Complexes

Textbook, References or/and course webpage:

1. Hajime, Sato - Algebraic Topology An Intuitive Approach (Ams, 1999)(T)(124S)

2. Greenberg M.,Harper J. - Algebraic Topology.. A First Course - Benjamin-Cummings 1981

Prerequisites: Basic Algebra, Point Set Topology

Level: Graduate Advanced undergraduate

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Contact: aslicankorkmaz@nesinvakfi.org, okelekci@gmail.com



Activities are supported by Nesin Mathematical Village and Turkish Mathematical Society