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
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Dan Knopf
Bruce Kleiner
John Lott | |
Mini lectures on Ricci Flow (5 talks)
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| 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.
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| Ko Honda | |
Reeb vector fields and open book decompositions (II)
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| 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.
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| 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.
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| 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.
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| 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.
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| 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.
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| Grigory Mikhalkin | |
Some 3-dimensional enumerative problems
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| 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.
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| 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.
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| Wilderisch Tuschmann | |
Manifolds with almost non-negative curvature
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| Selman Akbulut | |
G2 manifolds and mirror duality (Joint work with Sema Salur)
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| Alex Degtyarev | |
A decomposability inequality for real trigonal curves
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