Konusma Özetleri

Alex Degtyarev

Slopes of Colored Links (Joint work in progress with Vincent Florens and Ana G. Lecuona)

 

Abstract. This work is motivated by our previous study of the behavior of the signature of colored links under the splice operation. The signature is mainly additive, away from a certain "singular locus", which is the subject of our current work. To describe the extra correction term (arising as a certain Maslov index in Wall's non-additivity theorem), we introduce a collection of polynomial invariants of colored links, called slopes. It turns out that the slope can be represented as the ratio of two sign-refined Alexander polynomials (or rather derivatives thereof), whenever this ratio makes sense. However, experiments with the link tables show that, when both polynomials in question vanish, the rational function obtained is independent of the higher Alexander polynomials, thus providing a new link invariant. This invariant distinguishes some of the links in the tables. Even the isolated common zeroes of the two polynomials sometimes lead to surprises.

Should time permit, we will study the further properties of these invariants and discuss various ways to compute them.

 

 

Bahar Acu

Symplectic Mapping Class Group Relations Generalizing the Chain Relation

 

Abstract. In this talk, we will introduce an understanding of symplectomorphisms of higher dimensional symplectic manifolds by using fibered Dehn twists introduced by Biran and Giroux. We show that the Weinstein domain W2n = { f(z0 , … , zn ) = δ } B2n+2 ,  given by a homogeneous polynomial f, has a Boothby-Wang type boundary and admits a fibered Dehn twist along W which can be written as a product of k(k-1)n right-handed Dehn twists along Lagrangian spheres. Moreover, this identification turns out to be a generalization of the classical chain relation of surfaces. This is a joint work with Russell Avdek.

 

 

Elif Medetoğulları 

On a Certain Relation in Mapping Class Group and Lefschetz Fibrations

 

Abstract. There is an important connection between Mapping Class Group of a closed oriented surface and Lefschetz fibrations. In this talk, first we briefly explain this connection and mention about a certain relation known as Matsumoto relation in the mapping class group of a genus 2 surface.  This relation, which has some important applications in terms of 4-manifolds, is generalized to surfaces of arbitrary genera by Mustafa Korkmaz.  We give an alternative proof of the generalized version of the mentioned relation which uses only very elementary relations in the mapping class group. This is a joint work with Elif Dalyan and Mehmetcik Pamuk.

 

 

Hatice Çoban

CR Structures on Three Dimensional Contact Manifolds

 

Abstract. In this talk, we give a brief introduction of Cauchy-Riemann (CR) structures. Then we prove that any orientable closed contact 3-manifold admits a compatible CR structure by showing that any contact structure ξ; is isotopic to another contact structure which is induced by a CR structure.

 

 

İbrahim Ünal

Similarities between Complex and Calibrated Geometries

 

Abstract. Calibrated geometries, introduced by Harvey and Lawson in 1982, are the geometries of distinguished submanifolds determined by a fixed, closed differential form  called a calibration on a Riemannian manifold M. A Kähler form ω  in complex geometry provides the first rich example of a calibration and calibrated geometries can be viewed as the generalization of Kähler 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, show that they enjoy more common properties.

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 similarities between complex and calibrated geometries resulting from the newly defined plurisubharmonic functions.

 

 

Muazzez Şimşir

Information Geometry from the Viewpoint of Affine Differential Geometry

 

Abstract. One may describe information geometry as applying the techniques of differential geometry to the probability  theory. The set of probability distributions constitute a statistical model as a manifold. At this point the question of  "Why differential geometry is useful for statistics?" naturally arises. By means of this model,  the relationship between the geometric structure of the manifold and statistical estimation can be analyzed. The Fisher information metric  and dually flat connections play an important role in information geometry. On the other hand, The pair (M,g) where M is a flat affine manifold and g is a Kaehler affine metric is called a Kaehler affine manifold. A connection, which is also flat, dual to flat affine connection is  defined by the Kaehler affine metric. In addition, one can form a family of dual connections that are not flat. In this talk, information geometric structures will be discussed  from the point of view of Kaehler affine geometry.

Note that, this talk is a part of the TUBITAK 1001  Project 113F296 called ``Harmonic maps, affine manifolds and their applications to  information geometry".

 

 

Selma Altınok

Positive Cycles Supported on a Weighted Graph

 

Abstract. Let Γ be a weighted graph whose incidence matrix is negative definite. Lipman studied a certain set E+  of positive cycles supported on Γ. We call this the Lipman semigroup. Given a singularity (X, 0) there are analytic and topological invariants. The main question is to determine the analytic invariants such as the local ring of X at a singular point 0, the minimal resolution X→ X, the set Ean  of analytic cycles, the maximal ideal cycle Zmax ∈ Ean  and the geometric genus pg from topological invariants, for example, the link LX, the dual resolution graph ΓX, the minimal cycle Zmin ∈ E+, the arithmetic genus pa , etc. In case of a rational singularity the smallest element Zmin of the semigroup E+  characterizes the singularity. It can be calculated by Laufer’s algorithm. This makes it interesting to study E+ . A natural question to ask is how to determine the explicit Z-generators for the semigroup. To answer this question, we develop a method inspired by toric geometry to compute the minimal generating set for the semigroup of Lipman.

 

 

Semra Pamuk

Periodic Maps on Simply Connected 4-Manifolds

 

Abstract. It is  known that simply connected, closed, 4-manifolds are classified in terms of intersection pairing  and Kirby-Siebenman invariant. In the non-simply connected case, since the fundamental group acts freely on the universal cover, classification of these concern the topological classification of free periodic maps on simply connected 4-manifolds. In general, a periodic map is allowed to have fixed points or periodic points of period smaller than the map period. When this happens the quotient space has singularities, which is an orbifold. In the case of isolated singularities one can still classify periodic maps by some invariants associated to this orbifold. The aim of this talk is to give some overview of the homotopy types of pseudofree actions and to present the above mentioned invariants in the orbit category setting. This is an ongoing work with M.Pamuk.

 

 

Sergey Finashin

Projective Typical 7-configurations and Cayley Octads: Real Deformation Classification

 

Abstract. Real deformational classification of planar typical 7-configurations (in Ph.D thesis of A.Zabun) via Gale duality leads to a similar classification of special 7-configurations, which is related to classification of maximal real Cayley octads (8-configurations being the zero locus of a net of quadric surfaces). I will describe the 8 deformation classes of regular octads relating them to the associated even theta-characteristics on the spectral quartic and the corresponding 14 real deformation classes of 7-configurations.

 

 

Tolga Etgü

Koszul Duality Patterns in Floer Theory

 

Abstract. We study symplectic invariants of the open symplectic manifolds X obtained by plumbing cotangent bundles of spheres according to a plumbing tree. We prove that certain models for the Fukaya category of closed exact Lagrangians in X and the wrapped Fukaya category are related by Koszul duality. As an application, we give explicit computations of symplectic cohomology of the symplectic manifolds X for all trees but the exceptional cases E6, E7 and E8. This is joint work with Yanki Lekili.

 

 

Turgut Önder

Equivariant Almost Complex Substructures on Spheres

 

Abstract. Let G be a finite group and M be a unitary representation of G. We will give an up-to -date survey of sufficient and necessary conditions for the tangent bundle T(S(M)) of the unit sphere S(M) to admit  an orthogonal splitting into two G-equivariant subbundles one of which admitting  a G-equivariant complex structure. The emphasis will be on the recent results.