SEVENTEENTH GÖKOVA GEOMETRY / TOPOLOGY CONFERENCE
May 31  June 5 (2010)
Gökova, Turkey
Invited speakers/participants
D. Gabai  
S. Fomin  
V. Fock 
D. Ravenel  
Weimin Chen  
S. Bauer 
I. Itenberg  
M. Maydanskiy  
M. Shapiro 
D. Auroux  
G. Mikhalkin  
J. Risler 
I. Zharkov  
L. Williams  
Y. Lekili 
A. Akhmedov  
J.E. Andersen  
F. Bourgeois 
M. Oka  
A. Petrunin  
A. Degtyarev 
P. Ghiggini  
E. Brugalle  
J. Van Horn Morris 
C. Karakurt  
B. Efe  
M. Kalafat 
O. Ceyhan  
I. Baykur  
S. Salur 
Scientific Committee : D. Auroux, Y. Eliashberg, G. Mikhalkin, R. Stern, S. Akbulut
Organizing Commitee : T. Önder, S. Koçak, S. Finashin, M. Korkmaz, Y. Ozan, T. Etgü, B.Coskunuzer
Supporting Organizations: NSF (National Science Foundation) and
TUBITAK (The Scientific and Technological Research Council of Turkey).
The participants of 17^{th} Gökova Geometry  Topology Conference
List of Talks

F. Bourgeois  
Lectures on contact and symplectic homology (Mini course)
We shall give the geometric definition of several holomorphic curves
invariants for symplectic and contact manifolds, such as contact
homology, symplectic homology and some of their variants. We shall then
explain the relationship between these theories, leading to a common
algebraic framework. These relations can also be used to translate
structural results for one of these invariants to the other ones. This
can be illustrated with the effect of some geometric operations on these
invariants, such as Legendrian surgery.


D. Ravenel  
The ArfKervaire invariant problem in algebraic topology I and II
Mike Hill, Mike Hopkins and I recently solved the 50 year old
ArfKervaire invariant problem in algebraic topology. The first talk
will describe the background and history of the problem and give a brief
overview of the proof of our main theorem. The second talk will
describe the new tools we developed to prove it.


S. Fomin  
Cluster algebras and triangulated surfaces
I will give a quick introduction to cluster algebras, with a focus
on combinatorial topology of marked surfaces with boundary.


J. Van Horn Morris  
Planar open books and symplectic fillings
Recently, Wendl showed that any planar open book extends to
a Lefschetz fibration on a symplectic filling of a compatible contact
structure. We use this to establish uniqueness of symplectic fillings
of certain Lens spaces, generalizing work of McDuff and Lisca, and to
obstruct fillings of certain contact structures on Seifert fibered
spaces. This is joint work with O. Plamenevskaya.


D. Auroux  
Fukaya categories of symmetric products and bordered
HeegaardFloer homology
The goal of this talk will be to outline an interpretation of
the work of LipshitzOzsvathThurston on HeegaardFloer homology for
3manifolds with boundary in terms of Fukaya categories of symmetric
products of Riemann surfaces. One ingredient is recent work of Lekili
and Perutz recasting HeegaardFloer homology in terms of Lagrangian
correspondences; the other one is a generation result for "partially
wrapped" Fukaya categories of symmetric products.


J. Andersen  
TQFT, Hitchin's connection and Toeplitz operators
In the talk we will review the geometric gauge theory construction of
the vector spaces the ReshetikhinTuraev TQFT associates to a closed
oriented surface. Hence we will build the Hitchin connection in the vector
bundles over Teichmüller space, which is obtained by applying
geometric quantization to the moduli space of flat connections on the
surface. This will be followed by a discussion of
the relation between the Toeplitz operator construction and the
Hitchin connection. The talk will end with
a discussion of various results about the RTTQFT's which we have
proved using these geometric constructions.


M. Maydanskiy  
A criterion for emptiness
We call an exact symplectic manifold E empty if it does not contain
any compact exact Lagrangian submanifold. If E is a total space of a
Lefschetz fibration, we show that unless the vanishing cycles of the
fibration are dependent in a particular manner in the Fukaya category
of the fiber, the wrapped Fukaya category of the total space vanishes,
which implies that E is empty. As an application, we construct empty 
and hence exotic  symplectic structures on cotangent bundles of
spheres and related spaces. This is joint work with Paul Seidel.


S. Bauer  
Nonlinear Fredholm maps and stable homotopy
The main result is a description of spaces of such nonlinear maps. It turns out that
such spaces have the weak homotopy type of Ω^{∞}Σ^{∞} of a sphere.
The dimension of this sphere is the index of the linearized Fredholm map.
The SeibergWitten map is an element in such a space.
The result is not unexpected  it is a technical improvement of the refined SWinvariants.
There are no new applications to 4manifolds.


Y. Lekili  
Quilted Floer homology of 3manifolds
We introduce quilted Floer homology (QFH), a new invariant of 3manifolds equipped
with a circle valued Morse function. This is a natural extension of Perutz's
4manifold invariants, making it a (3+1) theory. We relate Perutz's theory to Heegaard
Floer theory by showing an isomorphism between QFH and HF^{+} for extremal spin^{c}
structures with respect to the fibre of the Morse function. As applications, we give
new calculations of Heegaard Floer theory and a characterization of sutured Floer
homology.


M. Shapiro  
Planar networks from cluster perspective and Bäcklund–Darboux transformations
We describe Poisson properties of planar directed networks in the disk and in the annulus. We concentrate on
special networks N_{u,v} in the disk that correspond to the choice
of a pair (u, v) of Coxeter elements in the symmetric group S_{n} and the corresponding
networks Ñ_{u,v} in the annulus.
Boundary measurements for N_{u,v} represent elements of the Coxeter double
Bruhat cell G^{u,v} ⊂ GL_{n}.
The Cartan subgroup H acts on G^{u,v} by conjugation. The standard
Poisson structure on the space of weights of N_{u,v} induces a
Poisson structure on G^{u,v}, and hence on the quotient
G^{u,v}/H, which makes the latter into the phase space for an appropriate Coxeter–Toda
lattice. The boundary measurement for Ñ_{u,v} is a rational
function that coincides up to a nonzero factor with the Weyl function for the boundary measurement for
N_{u,v}. The corresponding
Poisson bracket on the space of weights of Ñ_{u,v} induces a
Poisson bracket on the certain space R_{n} of rational functions,
which appeared previously in the context of Toda flows.
We introduce a cluster algebra A on R_{n} compatible with the
obtained Poisson bracket. Generalized Bäcklund–Darboux
transformations map solutions of one Coxeter–Toda lattice to solutions
of another preserving
the corresponding Weyl function. Using network representation, we
construct generalized
Bäcklund–Darboux transformations as appropriate sequences of cluster
transformations in A.
This is a joint work with Michael Gekhtman and Alek Vainshtein


M. Oka  
Mixed projective curves and Thom inequality
Let f be a strongly polar homogeneous polynomial of
three variables z = (z_{1},z_{2},z_{3})
with radial degree d_{r} and polar degree d_{p}.
It is a linear combination of the monomials
z^{ν} z^{μ}
where z^{ν} = z_{1}^{ν1}z_{2}^{ν2}z_{3}^{ν3}
and z^{μ} = z_{1}^{μ1}z_{2}^{μ2}z_{3}^{μ3}
satisfying
ν+μ = d_{r}, and νμ = d_{p}.
It defines an real algebraic curve in the complex projective space P^{2}.
We study the basic properties of this curve.


D. Gabai  
Topology of Ending Laminations Space
If S is not the punctured torus or 3 or 4 holed sphere
then the ending lamination space of S is path connected and locally
path connected.


G. Mikhalkin  
Tropical (p,q)cycles
Real 1parameter families of complex manifolds
have limits collapsing to certain polyhedral complexes
called tropical varieties. Tropical structure of these
limits is responsible for the asymptotics of the collapse.
Tropical varieties are simple enough to visualize
easily. Yet their homology theories reflect the Hodge
structure of the collapsing family. These theories
agree with the classical mixed Hogde structures
in the case when the real 1parametric families
can be extended to the complex domain.
This is a joint work with Ilia Itenberg, Ludmil Katzarkov and
Ilia Zharkov.


V. Fock  
Geometric and combinatorial structures on the space of triples of flags
The space of triple of flags considered up to the diagonal group action is a building block to construct
many grouprelated varieties, like moduli of flat connections, higher Teichmueller spaces and many others.
Sections of line bundles on this space are invariants of triple tensor products of representations.
We shall describe several geometric structures arising on this space and its relatives: cluster coordinates,
symplectic groupoid structure, integrable system over the space of plane algebraic curves and hyperkaehler structure.


A. Akhmedov  
The geography of symplectic 4manifolds
The symplectic geography problem, originally posed by R. Gompf, ask which
ordered pairs of nonnegative integeres are realized as (χ(X), c_{1}^{2}(X))
for some symplectic 4manifold X. In this talk we address the geography
problem of simply connected spin and nonspin symplectic 4manifolds in the
regions with small Euler characteristic, with nonnegative signature or near
the BogomolovMiyaokaYau line c_{1}^{2}(X) = 9χ(X).


S. Akbulut  
Cappell and Shaneson homotopy 4spheres
In 1976 Cappell and Shaneson constructed an infinite family of smooth homotopy 4spheres which they offered as possible counterexamples to the 4dimensional (smooth) Poincare conjecture (because some of them double cover of an exotic projective 4space). We will discuss the recent solution of this problem.


