Winter Workshop 2017 (Dec 25 - Dec 29, 2017)


The purpose of this event is to expose graduate students some of the hot topics in Geometry/Topology. There will be four mini lectures given by S. Akbulut, B. Özbağcı, F. Arıkan, Ç. Karakurt. Additional research talks will be given by graduate students.


To register this event please contact Cagri Karakurt: cagri.karakurt@boun.edu.tr


List of Participants

Selman Akbulut

Fırat Arıkan

Çağrı Karakurt

Burak Özbağcı


Selman Akbulut

A discussion on high dimensional PALFs and open books and their relation to exotic structures on 4-manifolds

Fırat Arıkan

Legendrian contact homology

This lecture series will be an elementary introduction to Legendrian contact homology. Lectures will basicly focus on the construction of Legendrian contact homology on (2n+1)-dimensional euclidean space and its extension to more general open manifolds.

Çağrı Karakurt

Knot Floer homology and its applications

These lecture series will serve as an introduction to Knot Floer homology, a set of prominent invariants of knots in three manifolds, and an exposition of its applications on several problems in low dimensional topology such as knot concordance, homology cobordism, and minimal genus. The emphasis will be on computational techniques.

Burak Özbağcı

A few results in low-dimensional topology

Lecture 1-2. Fillings of unit cotangent bundles of nonorientable surfaces.
We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed connected smooth surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism. This is a joint work with Youlin Li.

Lecture 3-4. Trisections of 4-manifolds via Lefschetz fibrations.
We develop a technique for gluing relative trisection diagrams of 4-manifolds with nonempty connected boundary to obtain trisection diagrams for closed 4-manifolds. As an application, we describe a trisection of any closed 4-manifold which admits a Lefschetz fibration over S2 equipped with a section of square -1, by an explicit diagram determined by the vanishing cycles of the Lefschetz fibration. In particular, we obtain a trisection diagram for some simply connected minimal complex surface of general type. As a consequence, we obtain explicit trisection diagrams for a pair of closed 4-manifolds which are homeomorphic but not diffeomorphic. Moreover, we describe a trisection for any oriented S2-bundle over any closed surface and in particular we draw the corresponding diagrams for T2 × S2 and T2 S2 using our gluing technique. Furthermore, we provide an alternate proof of a recent result of Gay and Kirby which says that every closed 4-manifold admits a trisection. The key feature of our proof is that Cerf theory takes a back seat to contact geometry. This is a joint work with Nick Castro.