Seminar on Geometry 

Fall 2015, ODTÜ, Ankara

Time / Location: Fridays 2:15 / M-215

Schedule of talks

 

TIME              SPEAKER                  TITLE
Sep 30
Wed, 3:00
Hyunjoo Cho
(Hacettepe)
Manifolds with G2-structures and (almost) contact structures
Hacettepe, Yaşar Ataman Seminer Salonu
Oct 1
Thu, 2:00
Characteristic Classes of Bundles 11
Oct 2
Fri, 2:00
Mustafa Kalafat Einstein-Hermitian yüzeylerin eğriliği
Gazi University
Oct 7
Wed, 3:40
Bernhard Hanke
(Augsburg)
Manifolds of positive scalar curvature
Bilkent math seminar room
Oct 9
Fri, 2:15

Page Space 1
Oct 14
Wed, 2:30

Page Space 2
Oct 15
Thu, 3:40
Gökhan Benli
(ODTÜ)
How Groups Grow
General Seminar at İkeda Room
Oct 16
Fri, 2:15

Page Space 3
Oct 17
Sat, 3:45
Bayram Tekin
(ODTÜ)
Introduction to Abelian and Non-Abelian Gauge Theories
AG-NT meeting at İkeda Room

5:00
Özer Öztürk
(MSGSÜ)
Cohomology of Flag Varieties
Oct 21
Wed, 3:15
Eyüp Yalçınkaya Spectral sequences 1
Oct 23
Fri, 2:15
Page Space 4
Oct 28
Wed, 3:30

Conformal structures on tori 1
Oct 30
Fri, 2:15

Characteristic Classes of Bundles 1
Yer: Kızılay ofis: Bayındır 2.sk Hulkibey apt. 62/12

3:30

Characteristic Classes of Bundles 2
Nov 4
Wed, 3:30

Spectral sequences 2
Nov 5
Thu, 3:40
İbrahim Ünal
(ODTÜ)
Calibrated Geometries and φ-free Submanifolds
General Seminar at İkeda Room
Nov 6
Fri, 2:15

Conformal structures on tori 2

3:30

Conformal structures on tori 3
Nov 12
Thu, 3:30
Note an unusual day
Spectral sequences 3
Nov 13
Fri, 2:15

Characteristic Classes of Bundles 3
Nov 20
Fri, 2:15

Characteristic Classes of Bundles 4
Nov 25
Wed, 3:00
İbrahim Ünal
Calibrated Geometries and φ-free Submanifolds
Hacettepe Üniversitesi Yaşar Ataman Seminer Salonu
Nov 26
Thu, 3:30

Spectral sequences 4
Nov 27
Fri, 2:15
Characteristic Classes of Bundles 5
Nov 30
Mon, 3:40
İbrahim Ünal
Potential Theory on Calibrated Manifolds
GT Seminar at İkeda Room
Dec 2
Wed, 3:30

Characteristic Classes of Bundles 6
Dec 4
Fri, 2:15

Spectral sequences 5
Dec 7
Mon, 1:40

Algebraic topology of and coassociative-free immersions into G_2 holonomy Riemannian 7-manifolds.
Bilkent Topology Seminar

3:40
İbrahim Ünal
Potential Theory on Calibrated Manifolds, Part 2
Topology Seminar, ODTÜ
Dec 9
Wed, 1:30

Einstein-Maxwell Equations 2
Seminar at Hacettepe University

3:00
Zhe Sun
(Yau Center, Tsinghua-Beijing)
Mirzakhani's topological recursion and higher Teichmüller space
Seminar at Hacettepe University
Dec 10
Thu, 11:40
Mehmet Kılıç
(Eskişehir,Anadolu)
İçsel Metrik Uzaylar (Intrinsic Metric Spaces)
İkeda Room

1:40
İnjektif uzaylar, Hiperkonveks uzaylar ve bir metrik uzayın 'tight span'i (sıkı germesi)
İkeda Room
Dec 11
Fri, 2:15
Aleksandra Borówka Introduction to Lie Algebroids
Dec 18
Fri, 2:15

Characteristic Classes of Bundles 7

3:30
Eyüp Yalçınkaya Spectral sequences 6
Dec 23
Wed, 3:30

Characteristic Classes of Bundles 8
Dec 24
Thu, 3:40
Hyunjoo Cho
(Hacettepe)

Dec 25
Fri, 2:15
Mobbingle mücadele derneğindeyiz:
İlkiz sk No:22/3 Sıhhıye-Ankara (Metro durağı yanından giriş yapılan sokak)
Jan 1
Fri, 3:00
Özgür Kelekçi
Einstein-Maxwell Equations 3
Adres: Bahçelievler 9 sk 6-6

4:00
Eyüp Yalçınkaya Spectral sequences 7
Jan 5
Tue, 1:30
Andrei Vesnin
(Sobolev Institute of Math, Novosibirsk)
Around right-angled hyperbolic polyhedra
Gazi Univ. Room D-227
Jan 8
Fri, 2:15
Complexification

3:30
Spectral sequences 8
Jan 13
Wed, 3:30
Characteristic Classes of Bundles 9
Jan 15
Fri, 2:15
Einstein-Maxwell Equations 4

3:30
Spectral sequences 9
Jan 25-31
Fri, 2:15
23rd Southern California Geometric Analysis Seminar and winter school
Feb 6-7
Sat-Sun, 9:00

Mobbing Temel Analiz Uzmanlık Eğitimi Konferansı

Abstracts/Notlar


Lectures on the Page space

In this seminar series we will be introducing the Page metric. Also introduce an efficient coordinate system on it. Then analyse some submanifolds which make these coordinates partially constant. An introduction to the subject can be found at:

Reference: M. Kalafat, C. Koca - On the curvature of Einstein-Hermitian surfaces.

Page 1: Introduction. Topology of the space. Euler coordinates. Orthonormal frames(Vierbein).

Subpage 2: Special surfaces of constant coordinates.

Subpage 3: Computing the connection 1-forms.

Subpage 4: Curvature 2-forms. Submanifolds.



Lectures on Conformal Structures on Riemann Surfaces

In this seminar series we will give elementary talks on moduli space of conformal classes on elliptic curves. Reference:

J.Jost - Compact Riemann Surfaces. Springer-Verlag.

Tori 1: Teichmüller space and Moduli space of conformal structures. Meromorphic functions.

Tori 2-3: Weierstrass p-function. Embedding into CP_2. Elliptic integrals.




Characteristic Classes of Bundles

In this seminar series we will be working on the chapter 4 of the following book:

Reference: Bott, Raoul; Tu, Loring W. -
Differential forms in algebraic topology.
Graduate Texts in Mathematics, 82. Springer-Verlag, New York-Berlin, 1982. xiv+331 pp. ISBN: 0-387-90613-4.

Bundles 1: Chern classes of wedge products, symmetric products and tensor products. Split manifold (from iterated projectivization).

Bundles 2: Equivalence of Split manifold and Flag manifold.

Bundles 3: Cohomology of Flag varieties (from projectivized bundles).

Bundles 4: Realization and complexification.

Bundles 5: Pontrjagin classes.

Bundles 6: Free cohomology of complex Grassmannians.

Bundles 7: Pontrjagin classes.



Lectures on Spectral Sequences

In this seminar series we will give elementary talks on Serre spectral sequence. Reference:

Hyunjoo: We prove that any seven-dimensional manifold with G2-structures admits almost contact structures on it. In this talk, I give a brief introduction on G2-structures and (almost) contact structures, and consider the relationships between contact structures and G2-structures on it.

Kalafat: Başlık: Einstein-Hermitian yüzeylerin eğriliği Hülasa: C.Koca ile ortak olan bu çalışmada, Page uzayının matematiksel açıklamasını yapı onun üzerinde etkili bir koordinat sistemi kuracağız. Bu sitem vasıtasiyle uzayın bazı sabit eğrilikli ve total geodezik altyüzeylerini bulacağız. Bu kısımlar ise R. Sarı ile ortak çalışmadır. Vakit elverdiği ölçüde ise pozitif holomorfik bikesitsel eğriliğin olmadığı durumları görmek ve Einstein-Hermityen bir manifold pozitif dik bikesitsel eğriliği haiz ise karmaşık projektif uzayın Fubini-Study metriğinden başka olamayacağını görmeye çalışacağız. Bu sonuç Berger'in bir teoremindeki Kähler şartını relaks eder.

Hanke: Curvature is a basic notion in differential geometry, because it measures the deviation from flat, Euclidean space. This talk deals with scalar curvature, which can be defined by the volume growth of small balls in Riemannian manifolds. We give an overview of some old and new results concerning Riemannian metrics with everywhere positive scalar curvature, combining techniques from global analysis, topology, and general relativity. Previous Talk on Monday: Equivariant bordism of elementary abelian groups: Time: October 5, 2015 Monday 13:40-14:30 Place: Math Department Seminar Room Abstract: The computation of bordism groups of manifolds equipped with group actions is a classical topic in geometric topology. We recall the basic definitions and present some new computations for elementary abelian groups.

Benli: The growth function of a finitely generated group measures how fast the balls in its Cayley graphs grow. This asymptotic invariant attracted a lot of attention starting from M. Gromov's celebrated theorem about groups which grow polynomially. This talk will be a survey of history, open problems and recent developments around this notion. The main focus will be on groups whose growth is intermediate between polynomial and exponential constructed by R. Grigorchuk.

Eyüp: Spectral sequences are the powerful tool to solve (co)homology of space. In this lecture we will introduce the spectral sequences and give an example. Exact couples.

Ünal: Calibrated geometries, introduced by Harvey and Lawson in 1982, are the geometries of calibrated submanifolds, a distinguished type of minimal submanifolds determined by a fixed, closed differential form $\phi$ called a calibration on a Riemannian manifold $M$. A K\"ahler form $\omega$ in complex geometry provides the first rich example of a calibration and calibrated geometries can be viewed as the generalization of K\"ahler manifolds as they have many similar properties. Recently, the introduction of plurisubharmonic functions on calibrated manifolds, which provides us doing analysis on them very similar to the one on complex manifolds, has made another type of submanifolds very important. These submanifolds are called as $\phi$-free and are the generalization of totally real submanifolds of complex manifolds. In this talk, I will start with an introduction to calibrated manifolds, and give the most well-known examples coming from special holonomy. Then, I will talk about the geometry and topology of $\phi$-free submanifolds.

Tekin : I will give an introduction to the the basics of gauge theories in physics such as the Maxwell's theory, Yang-Mills theory, Seiberg-Witten and Gravity.

Ünal 2: Plurisubharmonic functions in calibrated geometries are introduced by Harvey and Lawson almost 30 years after their foundational paper on calibrated manifolds. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties. Moreover, quite a few results proved in complex geometry via plurisubharmonic functions can be extended to calibrated manifolds. One important example of these results is a notion of pseudo-convexity defined in the context of calibrated manifolds, similar to the one in complex analysis. In this talk, after a quick introduction to calibrated manifolds, I will try to explain these new concepts and results as well as showing similarities between complex and calibrated geometries. In addition to these, I will talk about geometric constructions, known as -free submanifolds, to give examples of strongly pseudo-convex domains in calibrated manifolds.

Kalafat : In this talk, we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group G_2. Some of these results are new. One often encounters these spaces when studying submanifolds of manifolds with calibrated geometries. As an application we deduce existence of certain special 3 and 4 dimensional submanifolds of G_2 holonomy Riemannian manifolds with special properties. These are called Harvey-Lawson(HL) pairs. Which appeared first in the work of Akbulut & Salur about G_2 dualities. Another application is to the coassociative-free embeddings. We show that if there is a coassociative-free embedding of a 4-manifold into the Euclidean 7-space then the signature vanishes along with the Euler characteristic. The converse of this theorem is proved in the more general sense by İ.Ünal using h-principle techniques. We will talk about this direction if time permits. Joint work with S.Akbulut and İ.Ünal.

Sun: Maryam Mirzakhani reproved Witten's conjecture (Kontsevich's theorem) via her topological recursion by integrating over her version of McShane's identity relevant to Weil-Petersson form on DM moduli space and its compactification. In this talk, I will represent Mirzakhani's work and also Francois Labourie's work on higher Teichmüller theory, which leads to conjecturally the existence of higher moduli space, its compactification and higher generalized Witten's conjecture.

Kılıç 1 : İçsel, kesin içsel metrik uzaylar ve tek jeodezik uzayları tanıtılacak ve iki tek jeodezik uzayın çarpım uzayının hangi metriklerle tek jeodezik uzay olacağı tartışılacaktır. Ş. Koçak'la müşterek çalışmadır.
Referans: Kılıç, M. , Koçak, Ş. - On products of uniquely geodesic spaces. Accepted by Amer. Math. Monthly 2015.

Kılıç 2 : İnjektif uzaylar, hiperkonveks uzaylar ve 'tight span' kavramları tanıtılacak, aralarındaki ilişkiler incelenecek ve sonrasında maksimum metriğiyle donatılmış düzlemin hangi alt kümelerinin hiperkonveks olacağı tespit edilecek ve bu sayede düzlemin boştan farklı herhangi alt kümesinin sıkı germesinin nasıl bir kümeye izometrik olacağı belirlenecektir. Ş. Koçak'la müşterek çalışmadır.
Referans: Mehmet Kılıç, Şahin Koçak -
Tight Span of Subsets of The Plane With The Maximum Metric. arXiv:1506.05982.

Borówka : Lie algeborids are becoming a modern way to describe some geometrical objects. In this talk I will discuss their basic properties.

Vesnin : Right-angled polyhedra in hyperbolic spaces are serving as useful building blocks for constructing hyperbolic manifolds and orbifolds with interesting properties. We will look at dimension three, where we describe existence conditions and the volume set structure. Then we will present a way to construct closed hyperbolic 3-manifolds related to color-epimorphisms of right-angled Coxeter groups and find 2-fold branched coverings of the 3-sphere. As a particular case, we will discuss the first example of a closed orientable hyperbolic 3-manifold constructed by F. Loebell in 1931.



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