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 27 - June 1 (2013)
Gökova, Turkey

List of invited speakers/participants

A. Zelevinsky       S. Bauer       E. Eftekhary
A. Oancea       M. Can       I. Tyomkin
O. Ceyhan       A. Alekseev       T. Ekholm
I. Baykur       B. Ozbagci       M. Kalafat
A. Akhmedov       A. Levine       C. Karakurt
P. Rossi       F. Arikan       R. Kirby
A. Petrunin       S. Finashin       B. Coskunuzer

Scientific Committee : D. Auroux, Y. Eliashberg, I. Itenberg, G. Mikhalkin, S. Akbulut

Organizing Commitee : T. Önder, S. Koçak, A. Degtyarev, O. Kisisel, Y. Ozan, T. Etgü, S. Salur

Supporting Organizations: NSF (National Science Foundation) and TUBITAK.


Application period has ended on January 22, 2013.

The participants of 20th Gökova Geometry - Topology Conference

List of Talks
Tobias Ekholm    Topological strings and knot contact homology
After a brief description of some aspects of topological string theory and its relation to Chern-Simons theory, we discuss possible geometric explanations of the recently observed relation between knot contact homology and open topological strings. A key role is played by Lagrangian fillings of the Legendrian unit conormal lift of a link that we will use in several ways. For example, we discuss a certain class of non-exact fillings that when equipped with a flat U(1)-connection induce augmentations and thereby parametrize branches of the augmentation variety that agree with branches of the corresponding variety defined through topological string theory. The talk reports on joint work with Aganagic, Ng, and Vafa.
Ozgur Ceyhan    Feynman integrals as periods in configuration spaces
Mid 90's, Broadhurst and Kreimer observed that multiple zeta values persist to appear in Feynman integral computations. Following this observation, Kontsevich proposed a conceptual explanation, that is, the loci of divergence in these integrals must be very particular type of object in algebraic geometry; mixed Tate motives. In 2000, Belkale and Brosnan disproved this conjecture. In this talk, I will describe a way to `correct' Kontsevich's proposal and show that the regularized Feynman integrals in position space setting as well as their ambiguities are given in terms of periods of suitable configuration spaces, which are mixed Tate. Therefore, the Feynman integrals for massless scalar QFTs are indeed Q[(2 π i)-1]-linear combinations of multiple zeta values. Moreover, the regularized Feynman amplitude for a massive scalar Euclidean field have an asymptotic expansion given by a formal series where the terms are combinations of multiple zeta values with coefficients that are polynomials in Q[√(π/2),(2 π i)-1,m-1]. This is a joint work with Matilde Marcolli.
R. Inanc Baykur    Topological complexity of symplectic 4-manifolds and Stein fillings
Following the ground-breaking works of Donaldson and Giroux, Lefschetz pencils and open books have become central tools in the study of symplectic 4-manifolds and contact 3-manifolds. An open question at the heart of this relationship is whether or not there exists an a priori bound on the topological complexity of a symplectic 4-manifold, coming from the genus of a compatible Lefschetz pencil on it, and a similar question inquires if there is such a bound on any Stein filling of a fixed contact 3-manifold, coming from the genus of a compatible open book. We will present our solutions to both questions, making heroic use of positive factorizations in surface mapping class groups of various flavors. This is a joint work with J. Van Horn-Morris.
Cagri Karakurt    Heegaard Floer homology and numerical semigroups
Recently, Nemethi gave a combinatorial description of Heegaard Floer homology for a class of 3-manifolds containing all Seifert fibered spaces. I will talk about a reformulation of this description in terms of some numerical semigroups generated by Seifert invariants. This is a joint work with Mahir Bilen Can.
Paolo Rossi    Cohomological field theories with boundary
Cohomological field theories were introduced by Kontsevich and Manin to encode and axiomatize the algebraic structure of Gromov-Witten invariants. They consist in a system of cohomology classes on the Deligne-Mumford moduli space of curves satisfying some properties and are known to produce (and in some measure be classified) by Frobenius manifolds. There are various natural bordifications of the Deligne-Mumford moduli space of curves obtained by different (real oriented) blowups of the nodal divisors and their intersections. After reviewing Kontsevich and Manin's definition, we introduce a version of cohomological field theories on these bordifications and analyze the ensuing algebraic structure.
Anton Petrunin    Telescopic actions
A group action H on X is called "telescopic" if for any finitely presented group G, there exists a subgroup H' in H such that G is isomorphic to the fundamental group of X/H'. We construct examples of telescopic actions on some CAT[-1] spaces, in particular on 3 and 4-dimensional hyperbolic spaces. As applications we give new proofs of the following statements:
(1) Aitchison's theorem: Every finitely presented group G can appear as the fundamental group of M/J, where M is a compact 3-manifold and J is an involution which has only isolated fixed points;
(2) Taubes' theorem: Every finitely presented group G can appear as the fundamental group of a compact complex 3-manifold.
This is a joint work with Dmitri Panov.
Anar Akhmedov    Construction of new symplectic 4-manifolds via Luttinger surgery
Luttinger surgery is a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold. The surgery was introduced by Karl Murad Luttinger in 1995, who used it to study Lagrangian tori in R4. Luttinger's surgery has been a very effective tool recently for constructing exotic smooth structures on 4-manifolds. In this talk, using Luttinger surgery, I will present new constructions of Lefschetz fibration over the 2-sphere whose total space has arbitrary finitely presented group G as the fundamental group. I will also construct infinitely many exotic Stein fillings with the fundamental group G. This is a joint work with Burak Ozbagci.
Eaman Eftekhary    A refinement of the sutured Floer homology
I will discuss a joint work with Akram Alishahi. Associated with a weakly balanced sutured manifold (X,T), we construct an algebra AT, and the sutured Floer complex CF(X,T) as a chain complex with coefficients in AT. The "filtered" chain homotopy type of CF(X,T) is an invariant of the sutured manifold (X,T). This generalizes both the sutured Floer homology of Juhasz and some of the constructions of Ozsvath and Szabo (e.g., 3-manifold invariants, knot and link invariants).
Mahir Can    G2 and the geometry of its wonderful embedding
In the first part of our talk, we recall basic constructions about the exceptional Lie group G2, and review its representation theory. In the second half of our talk, building on our knowledge of the special representations of G2, we describe its equivariant embeddings, and review some known facts about the equivariant K-theory of a wonderful compactification. In particular, we work out in detail the (torus-equivariant) K-theory of the wonderful compactification of G2. This is a joint work with Soumya Banerjee.
Adam Levine    Embeddings of non-orientable surfaces in L(p,q)×I
We use the Heegaard Floer homology correction terms to solve the minimal genus problem for embeddings of closed, non-orientable surfaces in the product of a lens space and an interval. We show that if a non-orientable surface embeds essentially in L(p,q)×I, the genus and normal Euler number of the embedding are the same as those of a stabilization of a non-orientable surface embedded in L(p,q) itself. This is a joint work with Danny Ruberman and Saso Strle.
Ilya Tyomkin    The irreducibility problem for Severi varieties on toric surfaces
A classical theorem due to Joe Harris asserts that over the field of complex numbers the Severi varieties parameterizing irreducible reduced plane curves of given degree and geometric genus are irreducible. Little is known about the irreducibility problem on other surfaces. In my talk I'll discuss several cases when the Severi varieties are known to be irreducible, and will give examples of toric surfaces for which the irreducibility doesn't hold (in any characteristic). Although, in these examples one can prove reducibility algebraically, the examples can be understood (and, in fact, were discovered) tropically. The link to tropical geometry will be explained.
Anton Alekseev    The Horn problem and planar networks
The Horn problem is a classical problem of Linear Algebra which establishes a complete set of inequalities on the eigenvalues of a sum of two Hermitian matrices with given spectra. It was solved by Klyachko and Knutson-Tao in the end of 1990s. Surprisingly, it turns out that exactly the same set of inequalities governs a problem of maximal multi-paths in planar networks with Boltzmann weights on their edges. The link between the two problems is via a tropical limit. In order to control this limit, we are using the Liouville volume on the space of solutions of the Horn problem. The talk is based on a joint work with M. Podkopaeva and A. Szenes.
Alexandru Oancea    Quantum string topology
I will explain how to deform the Chas-Sullivan product on the free loop space of a symplectic manifold in the presence of holomorphic spheres. This is the first step towards a theory of quantum string topology for symplectic manifolds.
Firat Arikan    Legendrian Realization in Convex Lefschetz Fibrations
After a brief introduction, we will show that, up to a Liouville homotopy and a deformation of compact convex Lefschetz fibrations on W, any simply connected embedded Lagrangian submanifold of a page in a convex open book on ∂W can be assumed to be Legendrian in ∂W with the induced contact structure. This can be thought as the extension of Giroux's Legendrian realization (which holds for contact open books) for the case of convex open books. Also some corollaries will be given. This is a joint work with Selman Akbulut.
Paul Melvin    Spherical projections of 4-manifolds
Subtle information about the differential topology of a smooth 4-manifold can be gained from a study of its maps to the 2-sphere. As a framework for this study, I will describe the homotopy classification of such maps - recent joint work with Rob Kirby and Peter Teichner, and independently Larry Taylor - and illustrate this classification through examples.
Rob Kirby    Trisections of 4-manifolds: existence and uniqueness
A trisection of a 4-manifold is described in the arXiv preprint of Dave Gay and myself, with a proof of existence. There is now a uniqueness theorem which states that two trisections of a given 4-manifold are equivalent under stabilization, handle slides, and a move corresponding to how a 1-handle can slide over a 2-handle in a 2-parameter family of Morse functions.
Sema Salur    Mirror Duality and G2 manifolds
Calibrated submanifolds of Calabi-Yau and G2 manifolds are volume minimizing in their homology classes and their moduli spaces have many important applications in geometry, topology and physics. In particular, they are believed to play a crucial role in explaining the mysterious "mirror duality" between pairs of Calabi-Yau and G2 manifolds. In this talk we give an introduction to calibrated geometries and a report of recent research on the calibrations inside the manifolds with special holonomy. This is a joint work with Selman Akbulut.
Selman Akbulut    Algebraic topology of manifolds with G2 structure
We will discuss a joint work with Mustafa Kalafat on cohomology rings of various Grassmannians related to the manifolds with G2 structure. From these results we deduce existence of certain special 3 and 4 dimensional submanifolds of G2 manifolds with special properties, which were previously used with Sema Salur to study Mirror duality phenomena in G2 manifolds.
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Last updated: June 2013
Wed address: GokovaGT.org/2013