THIRTEENTH GÖKOVA GEOMETRY / TOPOLOGY CONFERENCE
May 29  June 03 (2006)
Gökova, Turkey
List of invited speakers and
participants
J.Lott  
B.Kleiner  
B.Parker  
R.Bryant  
D.Knopf  
J.Song 
V.Colin  
G.Mikhalkin  
B.Ozbagci 
B.Coskunuzer  
K.Honda  
A.Degtyarev  
I.Itenberg  
T.Etgu  
S.Salur  
Scientific Committee : G. Tian, R. Stern, C. Vafa, R. Kirby,
Y. Eliashberg, S. Akbulut
Organizing Commitee : T. Onder, T. Dereli, S. Kocak, S. Finashin, M.Korkmaz, Y.Ozan
This conference is sponsored by
TUBITAK (The Scientific and Technological
Research Council of Turkey) and NSF (National Science
Foundation)
The participants of 13^{th} Gökova Geometry  Topology Conference
List of Talks

Dan Knopf Bruce Kleiner John Lott  
Mini lectures on Ricci Flow (5 talks)


Vincent Colin  
Reeb vector fields and open book decompositions: The periodic case (I)
In collaboration with Ko Honda, we prove that every contact structure x_{i} in dimension 3 which is supported by an open book whose monodromy is isotopic to a periodic diffeomorphism satifies the Weinstein conjecture (every Reeb vector field associated to x_{i} has a periodic orbit). This result comes from a study of holomorphic curves in the symplectization. It also allows us to study the topology of the underlying 3manifold.


Ko Honda  
Reeb vector fields and open book decompositions (II)


Baris Coskunuzer  
Nonuniqueness of the solutions to the asymptotic Plateau problem
We show that there exist examples of codimension1 closed submanifolds of sphere at infinity of hyperbolic (n+1)space, which bounds more than one absolutely area minimizing hypersurface in hyperbolic (n+1)space. We also show that the same is true for area minimizing planes in hyperbolic 3space.


Ilia Itenberg  
A real analog of the CaporasoHarris formula
The Welschinger invariant is designed to bound from below the number of real rational curves which pass through a given generic collection of real points on a real rational surface. In some cases (for example, in the case of toric Del Pezzo surfaces) this invariant can be calculated using Mikhalkin's approach which deals with a corresponding count of tropical curves. We define a series of relative tropical Welschingertype invariants of real toric
surfaces. In the Del Pezzo case, these invariants can be seen as real tropical analogs of relative GromovWitten invariants, and are subject to a recursive formula similar to the CaporasoHarris one. As application we obtain new results concerning Welschinger invariants of real toric Del Pezzo surfaces.


Bruce Kleiner  
BiLipschitz embedding in Banach spaces, Rademachertype theorems, and functions of bounded variation (Joint work with Jeff Cheeger)
A mapping between metric spaces is LbiLipschitz if it stretches distances by a factor of at most L, and compresses them by a factor no worse than 1/L. A basic problem in geometric analysis is to determine when one metric space can be biLipschitz embedded in another, and if so, to estimate the optimal biLipschitz constant. In recent years, this question has generated great interest in computer science, since many data sets can be represented as metric spaces, and associated algorithms can be simplified, improved, or estimated, provided one knows that the metric space in question can be biLipschitz embedded (with controlled biLipschitz constant) in a nice space, such as L^{2} or L^{1}. The lecture will discuss several new existence and nonexistence results for biLipschitz embeddings in Banach spaces. One approach to nonexistence theorems is based on generalized differentiation theorems in the spirit of Rademacher's theorem on the almost everywhere differentiability of Lipschitz functions on R^{n}. We first show that earlier differentiation based results of Pansu and Cheeger, which proved nonexistence of embeddings into R^{k}, generalize to many Banach space targets, such as L^{p}, for 1 < p < ∞. We then focus on the case when the target is L^{1}, where differentiation theory is known to fail, and the embedding questions are of particular interest in computer science. When the domain is the Heisenberg group with its CarnotCaratheodory metric, we show that a modified form of differentiation still holds for Lipschitz maps into L^{1}, by exploiting a new connection with functions of bounded variation, and some very recent advances in geometric measure theory. This leads to a proof of a conjecture of Assaf Naor.


John Lott  
Longtime behavior of typeIII Ricci flow solutions
A typeIII Ricci flow solution is one that exists for all positive time and whose sectional curvatures decay at least as fast as the inverse of the time. I will give some results on the longtime behavior of typeIII Ricci flowsolutions, especially in three dimensions.


Burak Ozbagci  
Sections of elliptic Lefschetz fibrations (Joint work with Mustafa Korkmaz)
We find a new relation among righthanded Dehn twists in the mapping class group of a
kholed torus for 4 ≤ k ≤ 9. This relation induces an elliptic Lefschetz pencil structure on the
4manifold CP^{2} # (9k) \bar{CP^{2}} with k base points and twelve singular fibers. By blowing up the base points we get an elliptic Lefschetz fibration on the complex elliptic surface
E(1)= CP^{2} # (9k) \bar{CP^{2}} → S^{2} with twelve singular fibers and k disjoint sections. More importantly we can locate these k sections in a Kirby diagram of the induced elliptic Lefschetz
fibration. The nth power of our relation gives an explicit description for k disjoint sections of the induced elliptic fibration on the complex elliptic surface E(n) → S^{2} for n = 1.


Grigory Mikhalkin  
Some 3dimensional enumerative problems


Brett Parker  
Exploded torus fibrations
A common technique for studying holomorphic curves in symplectic manifolds involves studying the behaviour of holomorphic curves under a degeneration of the (almost) complex structure in what might be considered an adiabatic limit. For example, one can consider a limit collapsing a Lagrangian torus fibration. The images of holomorphic curves in the base converge under this limit to `Tropical curves' which look like piecewise linear graphs with a conservation of momentum condition at vertices. The smooth category is inadequate for describing these limiting objects.The category of exploded fibrations extends the smooth category so that we can consider some degenerating families of complex structures to have limits which are complex exploded fibrations. I will concentrate on the special case of exploded torus fibrations, which have torus symmetry. An example of when these might arise is given by an adiabatic limit collapsing a singular Lagrangian torus fibration. In this case, the moduli space of holomorphic curves corresponds to the moduli space of exploded curves, which itself has the structure of an exploded torus fibration. I will explain the relationship of this to Tropical geometry.


Jian Song  
KählerRicci flow on surfaces
We study the KählerRicci flow on minimal surfaces of Kodaira dimension one and show that the flow collapses and converges to a unique canonical metric on its canonical model. Such a canonical metric is a generalized KählerEinstein metric. Combining the results of Cao, Tsuji, Tian and Zhang, we give a metric classification for Kähler surfaces with anumerically effective canonical line bundle by the KählerRicci flow. In general, we propose a program of finding canonical metrics on the canonical models of projective varieties of positive Kodaira dimension. This is a joint work with G. Tian.


Wilderisch Tuschmann  
Manifolds with almost nonnegative curvature


Selman Akbulut  
G_{2} manifolds and mirror duality (Joint work with Sema Salur)


Alex Degtyarev  
A decomposability inequality for real trigonal curves


