Gökova Geometry / Topology Conferences 14 13 12 11 10 09 08 07 06 05 04 03 02 01 00 98 96 95 94 93 92

TWENTY-FIRST GÖKOVA
GEOMETRY / TOPOLOGY CONFERENCE

May 26 - May 31 (2014)
Gökova, Turkey

List of invited speakers/participants

B. Siebert       D. Zvonkine       M. Abouzaid
K. Shaw       E. Brugallé       I. Coskun
B. Chantraine       J. Risler       P. Rossi
P. Ghiggini       P. Melvin       Z. Wu
A. Wand       C. Karakurt       M. Kalafat
B. Ozbagci       A. Keating       E. Ionel
A. Petrunin       S. Finashin       B. Coskunuzer
      O. Viro      

Scientific Committee : D. Auroux, Y. Eliashberg, I. Itenberg, G. Mikhalkin, S. Akbulut

Organizing Commitee : T. Önder, S. Koçak, A. Degtyarev, Ö. Kişisel, Y. Ozan, T. Etgü, S. Salur

Supporting Organizations: NSF (National Science Foundation) and TUBITAK


Application

Application period has ended on February 25, 2014.


The participants of 21st Gökova Geometry - Topology Conference

Preliminary List of Talks
Erwan Brugalle,
Ilia Itenberg,
Grigory Mikhalkin,
Kristin Shaw
   Mini course: On basic notions in tropical geometry
The minicourse of 4 interconnected lectures is aimed to provide an elementary introduction to tropical geometry.
In Lecture 1 we introduce tropical curves and the way how they appear in tropical limits of Riemann surfaces.
In Lecture 2 tropical curves will be used as a tool in several classical enumerative problems as well as their modern refinements.
Lecture 3 will focus on higher-dimensional tropical varieties, in particular, tropical surfaces, as well as on tropical intersection theory.
Lecture 4 will review further recent applications of tropical techniques, particularly in symplectic geometry.
Dimitri Zvonkine    Cohomological relations on Mbar_{g,n} via 3-spin structures
We construct a family of relations between tautological cohomology classes on the moduli space Mbar_{g,n}. This family contains all relations known to this day and is expected to be complete and optimal. The construction uses the Frobenius manifold of the A_2 singularity, the 3-spin Witten class and the Givental-Teleman classification of semi-simple cohomological field theories (CohFTs) I will start with a short introduction into the cohomology of moduli spaces and give simplest examples of tautological relations. Then I will proceed to define Witten's r-spin class, explain why it is a CohFT and how Teleman's classification applies to it. In the end I will compute several cohomological relations using our method. This is a joint work with R. Pandharipande and A. Pixton.
Andy Wand    Tightness and Legendrian surgery
A well known result of Giroux tells us that isotopy classes of contact structures on a closed three manifold are in one to one correspondence with stabilization classes of open book decompositions of the manifold. We will introduce a characterization of tightness of a contact structure in terms of corresponding open book decompositions, and show how this can be used to resolve the question of whether tightness is preserved under Legendrian surgery.
Ailsa Keating    Lagrangian tori in four-dimensional Milnor fibres
The Milnor fibre of any isolated hypersurface singularity contains exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. I will explain how to construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four. Time allowing, I will use these to give examples of fibres whose Fukaya categories are not generated by vanishing cycles, and explain applications to mirror symmetry for those fibres.
Mohammed Abouzaid    Generators of Fukaya categories
In order to apply categorical ideas in symplectic topology, it is often necessary to have a concrete collection of Lagrangians which encode all Floer-theoretic information; such Lagrangians are called generators of the Fukaya category. I will survey known results about when collections of Lagrangian give generators of different flavours of the Fukaya category.
Eleny Ionel    Gopakumar-Vafa conjecture for symplectic manifolds
The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. In this talk, based on joint work with Tom Parker, we describe the proof of this conjecture, coming from a more general structure theorem for the Gromov-Witten invariants of symplectic 6-manifold.
Izzet Coskun    The birational geometry of the moduli spaces of sheaves on the plane
I will survey recent work with Arcara, Bertram, Huizenga and Woolf on the birational geometry of the moduli spaces of Gieseker semistable sheaves on the projective plane. I will mostly concentrate on the case of the Hilbert scheme of points and discuss important invariants such as the ample cone, the effective cone and the stable base locus decomposition. I will describe how to run the minimal model program and explain the relation to Bridgeland stability. The talk will be example based. I will try to illustrate the general theory in small examples.
Oleg Viro    Relations among reflections
Some classical Lie groups are generated by reflections, all the generators-reflections are conjugate to each other, and satisfy simple relations that completely define the group. In the talk examples of this phenomenon will be discussed.
Cagri Karakurt    Star Surgery of Symplectic 4-Manifolds
It is an interesting problem in 4-manifold topology to construct new symplectic manifolds out of given ones. One classical tool to do such constructions is Fintushel and Stern's rational blow-down operation which is useful for understanding not only symplectic 4-manifolds but also their exotic smooth structures. Star surgery is a variation of rational blow-down in which one removes a convex neighborhood of a union of symplectic spheres intersecting according to a star shaped plumbing, and replaces it with a smaller symplectic filling of the contact manifold on the boundary which is not necessarily a rational ball. This operation admits many nice properties which allows one to control Seiberg-Witten invariants, and produce exotic smooth structures. In this talk I'll give an example of star surgery, and use it to construct a minimal symplectic exotic copy of complex projective plane blown-up n=8 times. If time permits, I'll also give two more examples where n=6,7. This is a joint work with L. Starkston.
Zhongtao Wu    On Minimal Genus Problems
In joint work with Yi Ni, we obtained a lower bound for the rational genus of knots in a given homology class. We will discuss its connection to one-sided Heegaard splittings and complexity of manifolds, and speculate a slice genus version of the bound.
Paolo Ghiggini    Floer theory for Lagrangian cobordisms-1: holomorphic Cthulhus and cultist maps.
In this talk I will describe a version of Floer homology for certain Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville domain. A Lagrangian cobordism L from a Legendrian submanifold Λ- to a Legendrian submanifold Λ+ is an exact Lagrangian submanifold in the symplectisation of a contact manifold which, outside a compact set, coincides with the cylinder over the Legendrian submanifolds. The main obstruction to the existence of a Floer theory for cobordisms are the SFT-like degenerations of the holomorphic strips at the negative end. Thus the boundary will count holomorphic strips with tentacles converging to Reeb chord at the negative end (Cthulhus) and use augmentations at the negative end (when they exist) to cancel the SFT-like degenerations algebraically and obtain a chain complex. I will use this theory to prove an exact sequence relating the linearised contact cohomologies of the Λ+ and Λ- to the homology of L relative to Λ-. This exact sequence generalises previous results of Ekholm, who considered the case when Λ- is empty, and of Sabloff and Traynor, who considered the case when L is defined by a generating family. This is a joint work with B.Chantraine, R. Golovko and G. Rizell.
Baptiste Chantraine    Floer theory for Lagrangian cobordisms-2: applications.
We will use the results of Paolo Ghiggini's talk to deduce some rigidity properties for the topology of exact Lagrangian cobordisms between certain Legendrian submanifolds in contactisations. As an example we will show that in higher dimension any exact Lagrangian cobordisms from a Legendrian homotopy sphere to itself is topologically a cylinder is the Legendrian homotopy sphere admits an augmentation. In order prove this type of constraint we will the functoriality of some characteristic classes in Legendrian contact homology and a version of Floer theory with twisted coefficients. This is joint work with P. Ghiggini, R. Golovo et G. Rizell.
Paolo Rossi    Rational reductions of the 2d-Toda hierarchy and mirror symmetry of local P1 orbifolds
The 2d-Toda hierarchy of PDEs is, together with the KP hierarchy, one of the archetypical examples of integrable hamiltonian PDEs in two space dimensions. Its reductions have proven to be central in the Gromov-Witten theory of the projective line and its orbifold or local variants. Further, their relation with Hurwitz spaces have given some of the most explicitly computable examples of mirror symmetry. In a recent preprint with A. Brini, G. Carlet and S. Romano we introduced and studied a new family of reductions of the 2d-Toda hierarchy which contains and generalize many of the previously known examples (bigraded Toda, q-deformed Gelfand-Dickey, Ablowitz-Ladik) and provides the mirror to the quantum cohomology of local footballs, an infinite family of toric Calabi-Yau threefolds.
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Last updated: June 2014
Wed address: GokovaGT.org/2014