Gökova Geometry Topology Institute

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===== 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]]

**Supporting Organization**: TMD (Turkish Mathematical Society)

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=== List of participants ===

| Selman Akbulut | Fırat Arıkan | Çağrı Karakurt |
| Burak Özbağcı | İsmail Aydoğdu | Deniz Genlik |
| Özgür Karabayır | Cana Pehlivan | İrem Özge Saraç |
| Oğuz Şavk | Merve Seçgin | Kürşat Sözer |
| Fulya Taştan | Yasemin Yıldırım | Eda Yıldız |
| Eylem Zeliha Yıldız | Kürşat Yılmaz | Yağmur Yılmaz |


=== List of Talks ===
^ Speaker ^ Title and Abstract ^
| 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 | **Symplectic fillings of contact 3-manifolds**\\ We will give an introduction to different types of fillings of contact manifolds. We will provide a survey of known results in the literature for the case of 3-dimensional contact manifolds. The ultimate goal of the lectures is to describe a proof of the following result obtained jointly with Youlin Li: 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. |
| Deniz Genlik | **Monotone Lagrangian Tori and Lagrangian Surgeries of Whitney Sphere in $\mathbb C^2$**\\ In the first part of the talk, we will define Maslov index, monotone Lagrangian submanifold of a symplectic manifold and we will give three examples of monotone Lagrangian tori: Clifford torus, Chekanov torus and monotone Lagrangian torus described by Eliashberg-Polterovich in $\mathbb C^2$. In the second part of the talk, we will show that tori defined by Chekanov and Eliashberg-Polterovich are Hamiltonian isotopic and they are not Hamiltonian isotopic to Clifford torus. Lastly, we will introduce Lagrangian surgery and talk about two Lagrangian surgeries of Whitney sphere which are Clifford and Chekanov tori.|
| Merve Seçgin | **Computing Ozsvath-Szabo Contact Invariant From Surgery Exact Triangle**\\ In this talk, we briefly introduce one of versions of Heegaard Floer Homology of a 3-manifold $Y$, $\widehat{HF}(Y)$, and contact invariant of a contact structure in $Y$, which is an element of $\widehat{HF}(Y)$. Tightness of a contact structure might be detected if its contact invariant is nonzero. We shall give some examples of contact 3-manifolds with nonzero contact invariant which is found by using surgery exact triangles. |
| Kürşat Sözer | **Topological Quantum Field Theories**\\ In this talk we will see Atiyah's definition of Topological Quantum Field Theories(TQFT) and several examples of \((0+1)\) and \((1+1)\) TQFTs.  We will compute the corresponding invariants of closed 1 and 2-manifolds. We will also briefly motivate the extended field theories and why they have higher categories as their targets.|
| Fulya Taştan | **Computation of Grid Homology**\\ The main aim is to introduce a combinatorial approach to compute Knot Floer Homology for a given knot (or a link) in \(S^3\), called grid homology. We will define the bigrading structure on the chain complexes, and the differential map. The difference and relation between the variants of grid homology will be investigated. We will show, by a simple example, the invariance of simply blocked grid homology \(\widehat {GH}( \mathbb{G})\) under stabilization move.|
| Eylem Zeliha Yıldız | **From knot concordances to exotic $4$-manifolds**\\ I will discuss PL and smooth knot concordances in $3$-manifolds and an application to constructing exotic $4$-manifolds. The application part of this talk is joint work with Selman Akbulut.|
| Kürşat Yılmaz | **Tight Contact Structures on The Brieskorn Homology Spheres \(\sum (2,3,6m+1)\)**\\ It is well known that ever 3-manifold admits an overtwisted contact structure. However, it is still an open promlem for many 3-manifolds that whether they admit a tight contactstructure or not and if it does how many non-isotopic tight contact structure exist. In this talk,  using results of Mark-Tosun and Honda’s bypass technique, we will find an upper bound and hence all of the tight contact structures on this specific family.|
| Yağmur Yılmaz | **Introduction to Persistent Homology**\\ In this talk, we will explain some basic definitions of persistent homology, which are filtration, birth and death of homology classes, and persistence diagrams. Also, we will mention the differences between persistent homology and ordinary homology in algebraic topology. Moreover, we will discuss how to construct comlex structures on a given finite data set and how to obtain filtration from this complex structure to analyse the data by using barcodes. At the end, we will speak of some applications based on persistent homology.|
Gökova Geometry Topology Institute

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