Geometry and Topology Seminar 

Schedule of talks

Abstracts

Kalafat : We construct handlebody diagrams of families of non-simply connected Locally Conformally Flat(LCF) 4-manifolds realizing rich topological types, which are obtained from conformal compactifications of the 3-manifolds, that are built from the Panelled Web Groups. These manifolds have strictly negative scalar curvature and the underlying topological 4-manifolds do not admit any Einstein metrics. This is a joint work with S. Akbulut.

Selma : (Oct21) Let (S,0) be a germ of a normal complex analytic surface. Consider a resolution (X,E) --> (S,0) of it where E is the exceptional divisor whose irreducible components are denoted by E1...En. The set of positive divisors Y supported on this exceptional divisor E satisfying the inequality Y Ei <= 0 for all i, forms a semigroup, called the semigroup of Lipman. The smallest element of the semigroup characterizes rational singularities. It can be calculated by Laufer's algorithm. All other elements can also be computed by the algorithms given by Lipman, Tosun, Altinok-Bhupal. The 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.

Selma : It is known that oriented contact structures on a closed 3-manifold M are related to open book decompositions of M by the fundamental theorem of Grioux. Therefore, this allows one to ask what either the minimal page genus or the minimal sum of page-genus and the number of connected components of the binding of an open book supporting a given contact sturture on M is. Let (X,x)be a germ of a normal complex anlytic surface and f be a germ of a holomorphic function on it. Then one can define an open book decomposition on the link M_X of (X,x) supported by the canonical contact structure on M_X. These open book decompoistions are called “Milnor open books” studied by C. Caubel, A. Nemethi and P. Papescu-Pampu. Here we solve the page-genus minimization problem among Milnor open books for the link of rational singularities.

Alex2 : The Alexander module of an algebraic curve is a certain purely algebraic invariant of the fundamental group of (the complement of) the curve. Introduced by Zariski and developed by Libgober, it is still a subject of intensive research. We will describe the Alexander modules and Alexander polynomials (both over Q and over finite fields F_p ) of a special class of curves, the so called generalized trigonal curves. The rational case is closed completely; in the case of characteristic p>0, a few points remain open. (Conjecturally, all polynomials that can appear are indeed listed.) Unlike most known divisibility theorems, which rely upon the degree and the types of the singularities of the curve, our bounds are universal: essentially, the Alexander module of a trigonal curve can take but a finitely many values.

Mohan : We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weight.

Beyaz : The main purpose of this talk is to give a mathematical definition of ``mirror symmetry'' for Calabi-Yau and G_2 manifolds. More specifically, we explain how to assign a G_2 manifold (M,\phi,\Lambda), with the calibration 3-form \phi and an oriented 2-plane field \Lambda, a pair of parametrized tangent bundle valued 2 and 3-forms of M. These forms can then be used to define various different complex and symplectic structures on certain 6-dimensional subbundles of T(M). When these bundles are integrated they give mirror CY manifolds. In a similar way, one can define mirror dual G_2 manifolds inside of a Spin(7) manifold (N^8, \Psi). In case N^8 admits an oriented 3-plane field, by iterating this process we obtain Calabi-Yau submanifold pairs in N whose complex and symplectic structures determine each other via the calibration form of the ambient G_2 (or Spin(7)) manifold.

Mehmetçik : In this talk we are going to show how one can use the group of homotopy self-equivalences of a 4-manifold together with the modified surgery theory of Matthias Kreck to give an s-cabordism classification of topological 4-manifolds. We will work with certain funda- mental groups and give s- cobordism classification in terms of standard invariants.

Morgan : Poincaré launched the subject of 3-dimensional topology in 1904. At the end of a long treatise on 3-manifolds he asked what became known as the Poincaré Conjecture: Is every simply connected 3-manifold homeomorphic to the 3-sphere. This problem sparked a century of work on manifolds of dimensions 3 and higher, work that made topology one of the most dynamic and exciting areas of mathematics during the 20th century. But in spite of all this work, at the end of the 20th century the Poincaré Conjecture still stood unresolved. Then in 2002 and 2003, Grigory Perelman put a series of 3 preprints on the archive that completely resolved this conjecture and the more general conjecture, due to Thurston, about the structure of all 3-manifolds. His approach was to use work of Richard Hamilton concerning what is called the Ricci flow. This is a parabolic evolution equation for a Riemannian metric on a manifold. In this talk we will review the motivating questions and the Ricci flow. After giving this background we will then sketch Perelman's method of solution.

Sema : A 7-dimensional Riemannian manifold (M,g) is called a G_2 manifold if the holonomy group of its Levi-Civita connection of g lies inside G_2. In this talk, I will first give brief introductions to G_2 manifolds, and then discuss relations between G_2 and contact structures. This is a joint work with Hyunjoo Cho and Firat Arikan.

Mahir : We describe the variety of fixed points of a unipotent operator acting on the space of symmetric matrices or, equivalently, the corresponding space of quadrics. We compute the determinant and the rank of a generic symmetric matrix in the fixed variety, yielding information about the generic singular locus of the corresponding quadrics.

Akbulut : We show here that an infinite sequence of homotopy 4-spheres constructed by Cappell-Shaneson are all diffeomorphic to S^4. This generalizes previous results of Akbulut-Kirby and Gompf. Reference: Annals of Mathematics (2) 171 (2010), no. 3.

Arzu : A cubic surface can be represented by blowing up of projective plane at six points. The six exceptional divisors yield a configuration of six skew lines out of 27 lines on a cubic surface. I will discuss how configurations of six generic points in the projective plane are related to the corresponding configurations of six skew lines in the three dimensional projective space.

Fιrat : Loi-Piergallini and Akbulut-Ozbagci showed that every compact Stein surface admits a Lefschetz fibration over the 2-disk with bounded fibers. In this talk, we describe an algorithm which gives an alternative proof of this result. This is a joint work with Selman Akbulut.

Çağrι : The knot Floer homology is a very powerful Floer theoretic invariant of knots in 3-manifolds which was recently introduced by Ozsvath and Szabo. It detects the genus of a knot and whether a given knot is fibred. It also provides lower bounds for the four ball genera of knots in the three-sphere. The purpose of this talk is to discuss yet another application of the knot Floer homology in low-dimensional topology. Using the underlying filtered chain complex of the knot Floer homology, we will calculate some invariants contact manifolds that are obtained by surgery on a Legendrian knot.

This page is maintained by  Mustafa Kalafat
Thanks to Hülya Argüz for handling the hardcopy posters.
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