TWENTYFOURTH GÖKOVA GEOMETRY / TOPOLOGY CONFERENCE
May 29  June 3 (2017)
Gökova, Turkey
List of invited speakers/participants (tentative)
T. Ekholm  
R. Cavalieri  
P. Massot 
B. Tosun  
A. Daemi  
S.C. Lau 
S. Courte  
B. Fang  
W. Chen  
M. Tange  
R. Golovko  
R. Zentner  
F. Arikan  
G. Dimitroglou  
S. Finashin  
M. Bhupal  
S. Galkin  
N. Kurnosov  
C. Ferahlar  
 
H. Arguz  
Scientific Committee : D. Auroux, Y. Eliashberg, I. Itenberg, G. Mikhalkin, S. Akbulut, A. Oancea
Organizing Commitee : T. Önder, S. Koçak, A. Degtyarev, Ö. Kişisel, Y. Ozan, T. Etgü, S. Salur
Supporting Organizations: NSF (National Science Foundation),
and
TMD (Turkish Mathematical Society)
and
ERC (European Research Council).
Application
Application period has ended on February 17, 2017.
Preliminary List of Talks

Tobias Ekholm  
Mini Course (Lectures 1 and 2): Floer disks on simple Lagrangian spheres and applications We study the moduli space of Floer holomorphic disks with boundary on a Lagrangian homtopy sphere immersion with a single double point. This moduli space gives a parallelizable nullcobordism of the homotopy sphere restricting its smooth structure. We discuss applications of the construction to cotangent bundles of homotopy spheres and to homotopy classes of Lagrangian spheres. The talks report on joint work with Smith and Kragh.   
Mini Course (Lectures 3 and 4): Legendrian surgery and partially wrapped Floer cohomology We discuss an extension of the Legendrian surgery results of BourgeoisEkholmEliashberg to the setting of partially wrapped Floer cohomology. The extension involves in particular a Legendrian differential graded algebra with loop space coefficients and under simple connectivity assumptions gives rise to dualities between ordinary and wrapped Floer cohomology. We furthermore discuss various applications of the results to mirror symmetry and to knot contact homology. The talks report on joint work with Lekili and Ng and Shende.   Raphael Zentner  
Holonomy perturbations and irreducible representations of integer homology 3spheres in SL(2,C) We prove that the splicing of any two nontrivial knots in the 3sphere admits an irreducible SU(2)representation of its fundamental group. This uses instanton gauge theory, and in particular a nonvanishing result of KronheimerMrowka and some new results that we establish for holonomy perturbations of the ASD equation. Using a result of Boileau, Rubinstein and Wang (which builds on the geometrization theorem of 3manifolds), it follows that the fundamental group of any integer homology 3sphere different from the 3sphere admits irreducible representations of its fundamental group in SL(2,C).   Bulent Tosun  
Obstructing pseudoconvex embeddings of Brieskorn spheres into complex 2space A Stein manifold is a complex manifold with particularly nice convexity properties. In real dimensions above 4, existence of a Stein structure is essentially a homotopical question, but for 4manifolds the situation is more subtle. An important question that has been circulating among contact and symplectic topologist for some time asks: whether every contractible smooth 4manifold admits a Stein structure? In this talk we will provide examples that answer this question negatively. Moreover, along the way we will provide new evidence to a closely related conjecture of Gompf, which asserts that a nontrivial Brieskorn homology sphere, with either orientation, cannot be embedded in complex 2space as the boundary of a Stein submanifold. This is a joint work with Tom Mark.   Roman Golovko  
Towards the strong Arnold conjecture In the 1960’s, V.I. Arnold announced several fruitful conjectures in symplectic topology concerning the number of fixed point of a Hamiltonian diffeomorphism in both the absolute case (concerning periodic Hamiltonian orbits) and the relative case (concerning Hamiltonian chords on a Lagrangian submanifold).
The strongest form of Arnold conjecture for a closed symplectic manifold (sometimes called the strong Arnold conjecture) says that the number of fixed points of a generic Hamiltonian diffeomorphism of a closed symplectic manifold X is greater or equal than the number of critical points of a Morse function on X.
We will discuss the stable version of Arnold conjecture, which is closely related to the strong Arnold conjecture. This is joint work with Georgios Dimitroglou Rizell.   Renzo Cavalieri  
Graph formulas for tautological cycles The tautological ring of the moduli space of curves is a subring of the Chow ring that, on the one hand, contains many of the classes represented by "geometrically defined" cycles (i.e. loci of curves that satisfy certain geometric properties), on the other has a reasonably manageable structure. By this I mean that we can explicitly describe a set of additive generators, which are indexed by suitably decorated graphs. The study of the tautological ring was initiated by Mumford in the '80s and has been intensely studied by several groups of people. Just a couple years ago, Pandharipande reiterated that we are making progress in a much needed development of a "calculus on the tautological ring", i.e. a way to effectively compute and compare expressions in the tautological ring. An example of such a "calculus" consists in describing formulas for geometrically described classes (e.g. the hyperelliptic locus) via meaningful formulas in terms of the combinatorial generators of the tautological ring. In this talk I will explain in what sense "graph formulas" give a good example of what the adjective "meaningful" meant in the previous sentence, and present a few examples of graph formulas. The original work presented is in collaboration with Nicola Tarasca and Vance Blankers.   Nikon Kurnosov  
KugaSatake construction and cohomology of hyperkahler manifolds Let M be a simple hyperkahler manifold. KugaSatake construction gives an embedding of H^{2}(M,C) into the second cohomology of a torus, compatible with the Hodge structure. We construct a torus T and an embedding of the graded cohomology space H^{*}(M,C) → H^{*+l}(T,C) for some l, which is compatible with the Hodge structures and the Poincare pairing. This work is joint with A.Soldatenkov and M.Verbitsky.   Weimin Chen  
Contact splittings of symplectic rational homology CP^{2}. Let (X,ω) be a symplectic 4manifold with rational homology of CP^{2}. We consider splittings of X by a hypersurface of contact type M, where we require M to be a rational homology 3sphere. We shall go over a few basic properties of such splittings, and then explain how this consideration provides a unified approach to a number of interesting questions in lowdimensional topology, symplectic geometry, as well as in algebraic geometry.   Siu Cheong Lau  
Mirror construction for a pairofpants decomposition Seidel and Sheridan proved homological mirror symmetry for a pairofpants. In my previous joint work with Cho and Hong, we gave a systematic construction of localized mirror functor from a given immersed Lagrangian, which in particular produces the LandauGinzburg mirror for a pairofpants. In this talk we consider a pairofpants decomposition and explain how to glue the local LandauGinzburg mirrors together from Floertheoretical considerations.   Georgios Dimitroglou  
The wrapped Fukaya category of a Weinstein manifold is generated by the Lagrangian cocore discs In a joint work with B. Chantraine, P. Ghiggini, and R. Golovko we decompose any object in the wrapped Fukaya category as a twisted complex built from the cocores of the critical (i.e. halfdimensional) handles in a Weinstein handle decomposition. The main tools used are the Floer homology theories of exact Lagrangian immersions, of exact Lagrangian cobordisms in the SFT sense (i.e. between Legendrians), as well as relations between these theories. Note that exact Lagrangians admit Legendrian lifts, and that appropriate Lagrange surgeries can be seen to give rise to an exact Lagrangian cobordism of the aforementioned type.   Bohan Fang  
BKMP remodeling conjecture and its application The BouchardKlemmMarinoPasquetti remodeling conjecture expresses all genus GromovWitten openclosed invariants in terms of the EynardOrantin topological recursion on its mirror curve. I will explain this conjecture and discuss some of its application, such as the modularity of GromovWitten invariants. This talk is based on the joint work with ChiuChu Melissa Liu and Zhengyu Zong.   Aliakbar Daemi  
Instantons and the homology cobordism group The set of 3manifolds with the same homology as the 3dimensional sphere, modulo an equivalence relation called homology cobordance, forms a group. The additive structure of this group is given by taking connected sum. This group is called the homology cobordism group and plays a special role in low dimensional topology and knot theory. In this talk, I will explain how instantons can be used to construct an infinite family of invariants of the homology cobordism group. The relationship between these invariants and Froyshov’s celebrated invariant will be discussed. I will also talk about some topological applications.   Patrick Massot  
Invariant norms on contact transformation groups Inspired by the Hofer and Viterbo distances on groups of Hamiltonian diffeomorphisms, several recent works define and study invariant distances on groups of contact transformations. In some cases, contact and symplectic rigidity results allow to prove that such a distance is unbounded, hence sees a significant part of the group. But understanding when this happens is not obvious. For instance this happens for the standard contact structures on projective spaces but not for their universal covers, the standard spheres.
In joint work in progress with Sylvain Courte, we shed new lights on this puzzle by uncovering links with Giroux's theory of open book decompositions and Murphy’s theory of loose Legendrian embeddings.   Motoo Tange  
Cork twisting and exotic 4manifolds It is classically wellknown that for any two closed, simplyconnected exotic 4manifolds there exists a contractible (Stein) 4manifold C such that the two exotic 4manifolds are related to each other by removing and regluing C. A pair of such C and the boundary diffeomorphism (gluing map) t is called cork. In this talk we give a review of recent developments of corks by some 4dimensional topologists: finite order corks, Zcorks, Z^{n}corks, and constraints for infinite cork twists.   Sylvain Courte  
Generating functions and sheaves for Legendrian links in R^{3} To a (generic) oneparameter family of functions (f_{x}) on a manifold M we associate the graph of all critical values: this is the front projection of a Legendrian link L in R^{3} and (f_{x}) is called a generating function for L. Which Legendrian links admit a generating function? How many up to equivalence? To deal with such questions it is natural to associate to a generating function a sheaf on R^{2} microsupported on the Legendrian. We will discuss to what extent this is a bijective correspondence. This is joint work (in progress) with Vivek Shende.   Firat Arikan  
Tight contact structures on some closed hyperbolic threemanifolds We show the existence of tight contact structures on infinitely many hyperbolic threemanifolds obtained via Dehn surgeries along sections of hyperbolic surface bundles over circle. This is a joint work with M. Secgin.   Cuneyt Ferahlar  
Vanishing theorems on a compact complex manifold, applications to Hopf conjecture and complex structures on 6sphere Vanishing theorems are important tools in algebraic geometry and differential geometry as they give information about the structure of varieties and manifolds. I generalize a Weitzenbock formula of Wu on Kahler manifolds for complex manifolds and use this to get vanishing theorems on compact complex manifolds under certain conditions, then obtain information on the plurigenera, such as geometric genus, arithmetic genus, irregularity. Hopf conjecture is one of the wellknown questions in differential geometry that relates the local geometry of an even dimensional compact orientable manifold to its global topology. Utilizing the vanishing theorems and the Frolicher spectral sequence, I show that the conjecture holds for compact complex manifolds with positive sectional, holomorphic bisectional and isotropic curvature under certain conditions. Finally, another interesting question is whether there exists a complex structure on S^{6} or not. By using a theorem of Gray that if S^{6} had a complex structure then its (0,1)Hodge number must be strictly positive and by combining this with the vanishing of the irregularity result proved, David L. Johnson and myself prove that under the specified conditions for the vanishing of the irregularity, S^{6} can not have a complex structure by Gray's theorem.   Hulya Arguz  
Log geometric techniques for open invariants in mirror symmetry We will discuss an algebraicgeometric approach to the symplectic Fukaya category via log GromovWitten theory and tropical geometry. Our main object of study will be the degeneration of elliptic curves, namely the Tate curve. We will also discuss a construction of a splitgenerating set of real Lagrangians using log geometric techniques. This is joint work with Bernd Siebert, with general ideas based on discussions of Bernd Siebert and Mohammed Abouzaid. The symplectic aspects we will overview is joint work in progress with Dmitry Tonkonog.  
