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 | |
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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
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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.
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| 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.
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| 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.
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| 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.
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| 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.
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| 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.
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| 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.
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| 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.
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| 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.
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| 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.
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| 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.
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| 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.
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| 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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