GEOMETRY / TOPOLOGY CONFERENCE
May 27 - June 1 (2013)
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
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
| 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.