Seminar on Geometry 

Schedule of talks

Abstracts

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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