Seminar on Geometry
Spring/Fall 2018
|
Schedule of talks
TIME |
SPEAKER |
TITLE |
Mar 19
Mon, 3:40 |
Mustafa Kalafat
|
On special submanifolds of the Page space
Talk at ODTÜ İkeda Room |
May 28
Jun 2 |
|
25th Gökova Geometry-Topology Conference |
Jul 23
Mon, 4:00 |
|
Lectures on G2 Geometry 1 |
Jul 24
Tue, 4:00 |
|
Lectures on G2 Geometry 2 |
Jul 25
Wed, 4:00 |
|
Lectures on G2 Geometry 3 |
Jul 26
Thu |
No Seminar
|
Excursion |
Jul 27
Fri, 4:00 |
|
Lectures on G2 Geometry 4 |
Jul 28
Sat, 4:00 |
|
Lectures on G2 Geometry 5 |
Jul 29
Sun, 4:00 |
|
Lectures on G2 Geometry 6 |
Oct 3
Wed, 3:40 |
Mehmet Kılıç
|
Geodesically convex and hyperconvex subsets of the plane with
the maximum metric
Talk at Atılım University - FEF 404 |
Oct 5
Fri, 3:40 |
|
Meeting on SW Theory |
5:00 |
|
Lectures on Grassmannians |
Nov 14
Wed, 3:40 |
Tülin Altunöz
|
The number of singular fibers in hyperelliptic Lefschetz fibrations
Talk at Atılım University - FEF 404 |
Nov 15
Thu, 3:40 |
Mustafa Kalafat
|
Algebraic Topology of G_2 manifolds and applications to Geometry
General Seminar at ODTÜ İkeda Room |
Nov 16
Fri, 1:30 |
|
Lie algebra meeting
Representations |
3:40 |
Emre Coşkun
|
Serre's GAGA II
AG Seminar at İkeda Room |
Dec 28
Fri, 3:00 |
E. Yalçınkaya
|
Some Problems on the Geometry of Calibrated Manifolds
Ph.D. Thesis Defence - M-214 |
Abstracts
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 self-contained 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 G2-structures.
Proceedings of Gökova Geometry-Topology 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.
Kalafat : Page manifold is the underlying differentiable manifold of
the complex surface, obtained out of the process of blowing up the
complex projective plane, only once. This space is decorated with a
natural Einstein metric, first studied by D.Page in 1978.
In this talk, we study some classes of submanifolds of codimension one
and two in the Page space. These submanifolds are totally geodesic. We
also compute their curvature and show that some of them are constant
curvature spaces. Finally we give information on how the Page space is
related to some other metrics on the same underlying smooth manifold.
This talk is based on a joint work with R.Sarı.
Related paper may be accessed from arXiv:1608.03252.
Despite working on basic submanifolds, we introduce a variety of
mutually-independent techniques, like graphic illustrations,
physicist computations, teichmüller space, 3-manifold topology,
ODEsystems etc. So that should not be confused with dry,
computational diff.geo. papers, involving only symbolic manipulations,
meaningless garbage of equalities followed by equalities.
We always consider the global topology of the submanifold for example,
and deal primarily with compact examples.
Kılıç : A metric space (X, d) is called a geodesic space if for
any p, q ∈ X, there exists a path α : [0, s] → X between p and
q, for which d(p, q) = L(α) holds, where L(α) is the length of the
path α. L(α) is defined as sup P
(Xn i=1 d(α(ti−1), α(ti)))
over all partitions P = {t0 = 0, t1, · · · , tn = s} of [0, s]. Such
a path can be reparametrized as γ : [0, d(p, q)] → X satisfying
d(γ(t), γ(t0)) = t 0 − t for 0 ≤ t ≤ t
0 ≤ d(p, q). A path parametrized
in this way is called a geodesic between p and q. To give a few
examples, consider the unit circle S1 in the plane with the metric
induced from the standard metric of the plane R^2 and choose two
antipodal points A and B on it. The distance between the points
A and B is 2, but there is no path on S1 with length realizing
this distance. So S1 with the induced metric is not a geodesic
space. If we put however the so-called “shorter arc-length metric”
on S1, then the distance between A and B becomes π and there
is a path between A and B with length π (for example, a suitably
parametrized half circle). This holds for any two points and S1
with the shorter arc-length metric becomes a geodesic space. Note
that there are two geodesics between the points A and B.
In this talk, geodesics in R^2 with the maximum (of the sides) metric
d∞ will be determined. A subspace X ⊆ (R
2, d∞) is called geodesically
convex if it is a geodesic space with the induced metric i.e. for any
two points p, q ∈ X, there exists a geodesic in (R^2, d∞) which is
contained in X. We will also introduce the concept of hyperconvexity and related
concepts of injectivity and tight span.
Finally, we will sketch of proof of the following theorem: A
nonempty closed and geodesically convex subset of the l∞ plane
R^2 ∞ is hyperconvex and we characterize the tight spans of arbitrary
subsets of R^2
∞ via this property: Given any nonempty X ⊆ R
2 ∞, a closed, geodesically convex and minimal subset Y ⊆ R
2 ∞ containing X is isometric to the tight span T(X) of X.
Altunöz :
For the past two decades the category of 4-dimensional has experienced explosive
growth. A number of surprising results, which exhibits rich and complicated rela-
tionships between different categories of manifold which are unique to dimension 4,
has been discovered associated with this growth. These developments have made
symplectic 4-manifolds that are another category of 4-dimensional manifolds natu-
ral candidates to be the building blocks of all smooth 4-manifolds having infinitely
differentiable local identification maps.
Donaldson and Gompf results ([1], [2], [3] and [4]) give the relation between sym-
plectic 4-manifolds and Lefschetz fibrations, which are a fibering of a 4-manifold by
surfaces, with a finite number of singularities of a prescribed type. Their results
say that symplectic 4-manifolds (after perhaps blowing up) admit the structure of
a Lefschetz fibration and a genus-g Lefschetz fibration with a fiber genus g ≥ 2 over
the Riemann surface admits a symplectic structure. Hence, Lefschetz fibrations
provide a combinatorial way to study symplectic 4-manifolds. The results on the
minimal number of vanishing cycles of a Lefschetz fibration provide some results on
symplectic 4- manifolds, which also gives a connection between symplectic topology
and geometric group theory.
The information about the number of singular fibers in a Lefschetz fibration provides
us important information about the topological invariants of its total space such as
σ(X), e(X), c
2
1
(X) and so on. In addition, it has been known that the number of
singular fibers in a Lefschetz fibration can not be arbitrary. So it makes sense to
ask what the minimal number of singular fibers in a nontrivial relatively minimal
genus-g Lefschetz fibration over the oriented surface of genus h.
In this talk, after giving some results about the minimal number of singular fibers in
a nontrivial relatively minimal genus-g Lefschetz fibration over the surface of genus
h, we will state our results about it for hyperelliptic Lefschetz fibrations which are
a special kind of Lefschetz fibrations.
References
[1] S.K.Donaldson, “ Lefschetz fibrations is symplectic geometry”, In Proceedings
of the International Congress of Mathematicians (Berlin,1998), Vol.II, Doc.
Math., Extra Vol. ICM Berlin, 1998, 309-314. 272.
1
[2] S.K.Donaldson, “Lefschetz pencils on symplectic manifolds”, J. Differential
Geom. 53 (1999), 205-236. 272.
[3] R.E. Gompf,“The topology of symplectic manifolds”, Turkish J. Math. 25
(2001), 43-59. 272,278.
[4] R.E. Gompf and A.I.Stipsicz,“4-manifolds and Kirby calculus”,Grad. Stud.
Math. 20, Amer. Math. Soc., Providence; R.I., 1999. 272, 278, 290.
Kalafat2 : 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 the 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 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. As a more recent application, we exhibit a family of
complex manifolds, which has a member at each odd complex dimension
and which has the same cohomology groups as the complex projective
space at that dimension, but not homotopy equivalent to it. We also
compute various cohomology rings. (Joint work with S.Akbulut et al.)
Kişisel :
See references:
1. Complex Semisimple Lie Algebras by Serre, Jean-Pierre
Coşkun :
Serre's famous theorem known as "GAGA" (Géometrie Algébrique et Géométrie Analytique
- Algebraic Geometry and Analytic Geometry)
is a fundamental result in algebraic geometry.
It basically says that the theory of complex analytic subvarieties of projective space
and the theory of algebraic subvarieties of projective space coincide.
In this series of lectures, we shall start with the fundamentals of complex analytic geometry
and then move toward the proof of GAGA.
Yalçınkaya :
We study three problems on the geometry of calibrated manifolds, which
are Riemannian manifolds equipped with a special closed differential form called a
calibration. Firstly, we compute the homology of Grasmannian manifold of oriented
3-planes in R6, namely G+3
(R6), and its special submanifold called SLAG, the set
of 3-planes in G+3
(R6) determined by the special Lagrangian calibration on Calabi-
Yau 3-fold C3 =
R6. We make an immediate application of these computations.
Secondly, we investigate a related problem on the embedding of oriented closed manifolds
into Cn as special Lagrangian-free (sLag-free). Finally, we study the geography
of symplectic 8-dimensional manifolds and obtain certain results on the existence of
symplectic 8-manifolds with Spin(7)-structure.
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