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

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 13th 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 xi 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 xi 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 3-manifold.
Ko Honda    Reeb vector fields and open book decompositions (II)
Baris Coskunuzer    Non-uniqueness of the solutions to the asymptotic Plateau problem
We show that there exist examples of codimension-1 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 3-space.
Ilia Itenberg    A real analog of the Caporaso-Harris 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 Welschinger-type invariants of real toric surfaces. In the Del Pezzo case, these invariants can be seen as real tropical analogs of relative Gromov-Witten invariants, and are subject to a recursive formula similar to the Caporaso-Harris one. As application we obtain new results concerning Welschinger invariants of real toric Del Pezzo surfaces.
Bruce Kleiner    Bi-Lipschitz embedding in Banach spaces, Rademacher-type theorems, and functions of bounded variation (Joint work with Jeff Cheeger)
A mapping between metric spaces is L-bi-Lipschitz 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 bi-Lipschitz embedded in another, and if so, to estimate the optimal bi-Lipschitz 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 bi-Lipschitz embedded (with controlled bi-Lipschitz constant) in a nice space, such as L2 or L1. The lecture will discuss several new existence and non-existence results for bi-Lipschitz embeddings in Banach spaces. One approach to non-existence theorems is based on generalized differentiation theorems in the spirit of Rademacher's theorem on the almost everywhere differentiability of Lipschitz functions on Rn. We first show that earlier differentiation based results of Pansu and Cheeger, which proved non-existence of embeddings into Rk, generalize to many Banach space targets, such as Lp, for 1 < p < ∞. We then focus on the case when the target is L1, 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 Carnot-Caratheodory metric, we show that a modified form of differentiation still holds for Lipschitz maps into L1, 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    Long-time behavior of type-III Ricci flow solutions
A type-III 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 long-time behavior of type-III Ricci flowsolutions, especially in three dimensions.
Burak Ozbagci    Sections of elliptic Lefschetz fibrations (Joint work with Mustafa Korkmaz)
We find a new relation among right-handed Dehn twists in the mapping class group of a k-holed torus for 4 ≤ k ≤ 9. This relation induces an elliptic Lefschetz pencil structure on the 4-manifold CP2 # (9-k) \bar{CP2} 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)= CP2 # (9-k) \bar{CP2} → S2 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 n-th power of our relation gives an explicit description for k disjoint sections of the induced elliptic fibration on the complex elliptic surface E(n) → S2 for n = 1.
Grigory Mikhalkin    Some 3-dimensional 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    Kähler-Ricci flow on surfaces
We study the Kähler-Ricci 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 Kähler-Einstein 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 Kähler-Ricci 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 non-negative curvature
Selman Akbulut    G2 manifolds and mirror duality (Joint work with Sema Salur)
Alex Degtyarev    A decomposability inequality for real trigonal curves
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Last updated: August 2006
Wed address: GokovaGT.org/2006