EIGHTEENTH GÖKOVA GEOMETRY / TOPOLOGY CONFERENCE
May 30  June 4 (2011)
Gökova, Turkey
List of invited speakers/participants
C. Leung  
K. Yasui  
D. Auroux 
G. Mikhalkin  
D. Gayet  
J. Greene 
S. Smirnov  
J. Johns  
M. Tange 
L. Katzarkov  
I. Cheltsov  
A. Petrunin 
V. Przyjalkowski  
I. Itenberg  
M. Yan 
S. Salur  
A.J. Todd  
W. Zhang 
N. Lebedeva  
P. Georgieva  
S. Finashin 
B. Ozbagci  
F. Arikan  
M. Bhupal 
Ö. Kisisel  
M. Kalafat  
M. Shapiro 
 
A. Akhmedov  

Scientific Committee : D. Auroux, Y. Eliashberg, G. Mikhalkin, R. Stern, S. Akbulut
Organizing Commitee : T. Önder, S. Koçak, A. Degtyarev, M. Korkmaz, Y. Ozan, T. Etgü, B.Coskunuzer
Supporting Organizations: NSF (National Science Foundation) and
TUBITAK.
The participants of 18^{th} Gökova Geometry  Topology Conference
List of Talks

S. Smirnov  
Conformal Field Theory and SchrammLoewner Evolutions
We will give an introductory talk about SLE  an elegant way to define random fractal curves in the plane, invented by Oded Schramm in 1999. It provides a new (and completely rigorous) way to look at CFT, and allows to prove many of its predictions about lattice models of statistical physics.


C. Leung  
SYZ mirror transformation
Mirror Symmetry is a duality between complex geometry and symplectic geometry. In these lectures, we will discuss the proposal by Strominger, Yau and Zaslow of explaining this mysterious duality as a geometric Fourier transformation.


D. Auroux  
Mirror symmetry in complex dimension 1
The goal of this mostly introductory talk will be to illustrate some key concepts in mirror symmetry, by considering two of the simplest examples: the cylinder and the 2sphere. These examples will be a pretext to discuss the StromingerYauZaslow picture of mirror symmetry (according to which mirror pairs carry dual torus fibrations), as well as Kontsevich's homological mirror symmetry conjecture. If time permits, at the end we will mention recent work with Abouzaid, Efimov, Katzarkov and Orlov on mirror symmetry for punctured Riemann surfaces.


K. Yasui  
Cork twisting exotic Stein 4manifolds
From any 4dimensional 2handlebody X with b_2>0, we construct arbitrary many compact Stein 4manifolds which are mutually homeomorphic but not diffeomorphic to each other, so that their topological invariants (their fundamental groups, homology groups, boundary homology groups, and intersection forms) coincide with those of X. If the time permits, we will discuss related work. This is a joint work with Selman Akbulut.


P. Georgieva  
Open GromovWitten disk invariants
In the presence of an antisymplectic involution on M, open GromovWitten disk invariants were defined by Cho and Solomon when the dimension of M is less than or equal to 6. I will describe a generalization to higher dimension under some technical conditions. Time permitting, I will discuss a connection to real algebraic geometry and an approach to relax the conditions.


W. Zhang  
NakaiMoishezon Theorem and Donaldson's question for almost complex four manifolds
There are two interesting questions for almost complex four manifolds.The classical NakaiMoishezon theorem (for surfaces) states the duality between ample divisor cone and curve cone for projective surfaces. DemaillyPaun, Buchdahl and Lamari generalized this duality to Kahler surfaces. It is natural to ask for such a duality between Jcompatible symplectic cone and curve cone for almost Kahler surfaces. Another interesting question is raised by Donaldson. He asked that, in dimension four, if there is a Jtamed symplectic form, do we have a Jcompatible symplectic form as well? I will discuss both questions and some recent developments (joint with TianJun Li and partly with Tedi Draghici).


J. Greene  
Conway Mutation and Alternating Links
Let D and D' denote connected, reduced, alternating diagrams for a pair of links, and Y and Y' their branched doublecovers. I'll discuss the proof and consequences of the following result: Y and Y' have isomorphic Heegaard Floer homology groups iff Y and Y' are homeomorphic iff D and D' are mutants.


J. Johns  
Some Lefschetz fibrations on cotangent bundles
I will describe a method for constructing a Lefschetz fibration on some cotangent bundles. The main virtue is an explicit model for the fiber and vanishing cycles. Morally, the Lefschetz fibration should be thought of as a complexification of a Morse function on the zero section (and this idea guides the construction). I will put emphasis on the cotangent bundle of a 2manifold since we are interested in 4manifolds at this conference.


D. Gayet  
Rarefaction of the real curves with many components
In a real algebraic surface X, among the real algebraic curves of degree N, the proportion of the curves with an almost maximal
number of connected components decreases to zero when N grows to the infinity. This is a joint work with JeanYves Welschinger.


I. Itenberg  
Real trigonal curves in ruled surfaces
The talk is devoted to real trigonal curves and elliptic surfaces (over a base of an arbitrary genus) and their equivariant deformations. The study of these objects via a real version of Grothendieck's dessins d'enfants leads to an explicit description of the deformation classes of real trigonal curves and elliptic surfaces in several "extremal" cases (for example, in the case of maximal or submaximal in the senseof the Smith inequality curves and surfaces, and in the case of maximally inflected trigonal curvesof type I in rational geometrically ruled surfaces).


S. Akbulut  
Visualizing 4manifolds
I will discuss handlebodies of some interesting 4manifold constructions, such as FintushelStern knot surgery operation, and the surface bundles over surfaces. I will show how to use them to describe some exotic manifolds and solve some 4manifold problems, such as showing that all the "Scharlemann manifolds" are standard, and some previously unknown families of homotopy S^4's and S^2 x S^2's are standard (a corollary is that the immersed 2sphere with one self intersection can knot inside of S^4 infinitely many different ways).


M. Yan  
Compact Aspherical Manifolds Whose Fundamental Groups Have Nontrivial Center
A theorem of Borel asserts that if a torus acts on an aspherical manifold, then the inclusion of the orbit of any point is injective on the fundamental group, and the image consists of central elements. Conversely, it was conjectured that, if the fundamental group has nontrivial center, then some nontrivial central element is represented by an orbit of a circle action. The conjecture can be thought of as a very strong form of the Borel conjecture and is true in dimension 3. In this talk, for dimensions at least 6, we construct aspherical manifolds with the infinite cyclic subgroup as the center of the fundamental group, yet the manifold has no effective circle actions.


L. Katzarkov  
Degenerations and wall crossings
In this talk we will introduce new categorical structures
from classical perspective of degenerations.


I. Cheltsov  
Subgroups of Cremona groups
The group of birational selfmaps of the projective
space of dimension N is called the Cremona group of rank N.
We will show how to apply algebraic L^{2}methods
(NadelShokurov vanishing, Kawamata subadjunction etc)
to study conjugacy classes of some finite
subgroups of the Cremona groups of high rank.
In particular, we give a partial answer to a question of Serre
on normalizers of finite simple subgroups in the Cremona of rank 3.
This is a joint work with Costya Shramov (Moscow).


