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


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 17th 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 Arf-Kervaire invariant problem in algebraic topology I and II
Mike Hill, Mike Hopkins and I recently solved the 50 year old Arf-Kervaire 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 Heegaard-Floer homology
The goal of this talk will be to outline an interpretation of the work of Lipshitz-Ozsvath-Thurston on Heegaard-Floer homology for 3-manifolds with boundary in terms of Fukaya categories of symmetric products of Riemann surfaces. One ingredient is recent work of Lekili and Perutz recasting Heegaard-Floer 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 Reshetikhin-Turaev 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 RT-TQFT'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 ΩΣ of a sphere. The dimension of this sphere is the index of the linearized Fredholm map. The Seiberg-Witten map is an element in such a space. The result is not unexpected - it is a technical improvement of the refined SW-invariants. There are no new applications to 4-manifolds.
Y. Lekili    Quilted Floer homology of 3-manifolds
We introduce quilted Floer homology (QFH), a new invariant of 3-manifolds equipped with a circle valued Morse function. This is a natural extension of Perutz's 4-manifold 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 spinc 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 Nu,v in the disk that correspond to the choice of a pair (u, v) of Coxeter elements in the symmetric group Sn and the corresponding networks Ñu,v in the annulus. Boundary measurements for Nu,v represent elements of the Coxeter double Bruhat cell Gu,v GLn. The Cartan subgroup H acts on Gu,v by conjugation. The standard Poisson structure on the space of weights of Nu,v induces a Poisson structure on Gu,v, and hence on the quotient Gu,v/H, which makes the latter into the phase space for an appropriate Coxeter–Toda lattice. The boundary measurement for Ñu,v is a rational function that coincides up to a nonzero factor with the Weyl function for the boundary measurement for Nu,v. The corresponding Poisson bracket on the space of weights of Ñu,v induces a Poisson bracket on the certain space Rn of rational functions, which appeared previously in the context of Toda flows. We introduce a cluster algebra A on Rn 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 = (z1,z2,z3) with radial degree dr and polar degree dp. It is a linear combination of the monomials zν zμ where zν = z1ν1z2ν2z3ν3 and zμ = z1μ1z2μ2z3μ3 satisfying |ν|+|μ| = dr, and |ν|-|μ| = dp. It defines an real algebraic curve in the complex projective space P2. 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 1-parameter 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 1-parametric 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 group-related 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 4-manifolds
The symplectic geography problem, originally posed by R. Gompf, ask which ordered pairs of nonnegative integeres are realized as (χ(X), c12(X)) for some symplectic 4-manifold X. In this talk we address the geography problem of simply connected spin and non-spin symplectic 4-manifolds in the regions with small Euler characteristic, with nonnegative signature or near the Bogomolov-Miyaoka-Yau line c12(X) = 9χ(X).
S. Akbulut    Cappell and Shaneson homotopy 4-spheres
In 1976 Cappell and Shaneson constructed an infinite family of smooth homotopy 4-spheres which they offered as possible counterexamples to the 4-dimensional (smooth) Poincare conjecture (because some of them double cover of an exotic projective 4-space). We will discuss the recent solution of this problem.
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Last updated: June 2010
Wed address: GokovaGT.org/2010