TWENTY-FIFTH GÖKOVA
GEOMETRY / TOPOLOGY CONFERENCE
May 28 - June 2 (2018)
Gökova, Turkey
List of invited speakers/participants
| C. Wendl | |
L. Starkston | |
D. Nadler | |
| V. Shende | |
S. Galkin | |
G. Dimitroglou | |
| O. Lazarev | |
J. Hom | |
A. Haydys | |
| W. Chen | |
B. Özbağcı | |
N. Kalinin |
| O. Plamenevskaya | |
Y. Li | |
R. Golovko |
| H. Gao | |
M. Kalafat | |
M. Asadi Golmankhaneh |
| Ö. Ceyhan | |
Y. Huang | |
N. Güğümcü |
| F. Zivanovic | |
A. Moreno | |
S. Habibi Esfahani |
| Ü. Yıldırım | |
E. Yıldız | |
B. Acu |
| B. Kartal | |
S. Sakallı | |
K. Sözer |
| S. Finashin | |
| |
M. Bhupal |
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 Organization: NSF (National Science Foundation)
and
TMD (Turkish Mathematical Society).
The participants of 25th Gökova Geometry - Topology Conference
List of Talks
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| Vivek Shende | |
Liouville sectors & axioms for the wrapped Fukaya category (2 talk minicourse)
Once upon a time, there were many invariants called homology with rather different looking constructions. Some involved triangulations, some maps, some covers, some functions, some forms, etc. Fortunately, the Eilenberg-Steenrod axioms mention no such details, yet characterize how any of these theories behaves on CW complexes. The purpose of these lectures is to give analogous axioms for wrapped Fukaya categories of Liouville manifolds. The basic ingredient is the Liouville sector -- symplectic versions of 'pairs', and I will spend some time introducing these objects. I will explain why the Fukaya category is covariant with respect to sector inclusions, how it satisfies certain axioms (which make no mention of holomorphic disks, or in fact even of Lagrangians), and how these axioms determine the Fukaya categories of all Weinstein manifolds. | | | Chris Wendl | |
Some remarks on transversality and symmetry (2 talk minicourse)
Everyone knows that you can't have transversality and symmetry at the same time. A familiar example arises in the construction of holomorphic curve invariants such as Gromov-Witten theory and SFT, where transversality typically cannot be achieved in the presence of multiply covered curves. In this talk, I will explain why the degree of transversality that is achievable without breaking symmetry is in fact much nicer and more useful than commonly known. Applications include surprising regularity results for certain kinds of multiply covered holomorphic curves, plus a proof that super-rigidty holds generically in Calabi-Yau 3-folds, thus localizing their Gromov-Witten invariants to a finite set of disjoint embedded curves. | | | Oleg Lazarev | |
Simplifying Weinstein Morse functions
By work of Cieliebak and Eliashberg, any Weinstein structure on Euclidean space that is not symplectomorphic to the standard symplectic structure necessarily has at least three critical points; an infinite collection of such exotic examples were constructed by McLean. I will explain how to use handle-slides and loose Legendrians to show that this lower bound on the number of critical points is exact; that is, any Weinstein structure on Euclidean space \(\mathbb R^{2n}\) has a compatible Weinstein Morse function with at most three critical points (of index 0, n-1, n). Similarly, any Weinstein structure on the cotangent bundle of the sphere of dimension has a compatible Weinstein Morse function with two critical points. As applications, I will give new proofs of some existing h-principles and present some new constructions of exotic cotangent bundles. | | | Laura Starkston | |
Arboreal skeleta of Weinstein manifolds
A Weinstein manifold retracts using its Liouville flow onto a half-dimensional complex called the skeleton. For cotangent bundles, a skeleton of \(T^*M\) is given by the zero section \(M\), and this smooth Lagrangian uniquely determines its surrounding Weinstein manifold. In general the skeleton has complicated singularities which depend on the way the skeleton embeds into the ambient manifold. We will discuss a good class of skeleta which generalize the cotangent bundle case, where the singularities are combinatorially enumerated, but still represent skeleta of quite general Weinstein manifolds. | | | Jen Hom | |
Knot concordance in homology cobordisms
Every knot in \(S^3\) bounds a non-locally flat PL-disk in \(B^4\). However, Akbulut showed that there exist knots in the boundary of a contractible 4-manifold X that do not bound PL-disks in X, and Adam Levine showed that there are knots in the boundary of a contractible 4-manifold that do not bound PL-disks in any bounding contractible 4-manifold. We show that the group of knots in null-bordant homology spheres modulo non-locally flat PL-concordance is infinitely generated and contains an infinite cyclic subgroup. The proof relies on Heegaard Floer homology. This is joint work with Adam Levine and Tye Lidman. | | | Youlin Li | |
Contact (+1)-surgeries along Legendrian two-component links
In this talk, we prove that the contact Ozsváth-Szabó invariant of a contact 3-manifold vanishes if it can be obtained from the standard contact 3-sphere by contact \((+1)\)-surgery along a Legendrian two-component link \(L=L_1\cup L_2\) with the linking number of \(L_1\) and \(L_2\) being nonzero and \(L_2\) satisfying \(\nu^{+}(L_2)=\nu^{+}(\overline{L_2})=0\). We also give a sufficient condition for the contact 3-maniold obtained from the standard contact 3-sphere by contact \((+1)\)-surgery along a Legendrian two-component link being overtwisted. This is joint work with Fan Ding and Zhongtao Wu. | | | David Nadler | |
Kostant-Sekiguchi homeomorphisms
I'll explain a new geometric relation between the conjugation and transpose of complex matrices. As an application, I'll construct a homeomorphism between real nilpotent matrices and complex symmetric ones. It can be viewed as a refinement of Kronheimer's instant flow. (Joint work with Tsao-Hsien Chen.) | | | Olga Plamenevskaya | |
Planar open books and links of surface singularities
Due to work of Giroux, contact structures on 3-manifolds can be topologically described by their open books decompositions (which in turn can be encoded via fibered links). A contact structure is called planar if it admits an open book with fibers of genus 0. Symplectic fillings of such contact structures can be understood, by a theorem of Wendl, via Lefschetz fibrations with the same planar fiber. Using this together with topological considerations, we prove new obstructions to planarity (in terms of intersection form of a Stein filling or presence of certain symplectic surfaces in a weak symplectic filling) and obtain a few corollaries. In particular, we show that the canonical contact structure on a link of a normal surface singularity is planar if and only if the singularity is minimal (minimality means a restrictive condition on the resolution graph). For hypersurface singularities, planarity is equivalent to having singularity of type \(A_n\). (Joint work with P. Ghiggini and M. Golla.) | | | Roman Golovko | |
Novikov fundamental group
We construct the Novikov fundamental group associated to a cohomology class of a Morse closed 1-form on a closed manifold. One of the applications of this construction is the new lower bounds for the number of index 1 and 2 critical points of closed 1-forms that are essentially different from the classical Morse-Novikov inequalities.
In addition, we construct the Hurewicz morphism in the Novikov settings. If time permits, we will also mention the way to define Novikov higher homotopy groups.This is joint work with Jean-Francois Barraud, Agnes Gadbled and Hong Van Le. | | | Georgios Dimitroglou | |
Augmentations beyond simple sheaves
In dimension three there is a correspondence between augmentations of the Chekanov-Eliashberg algebra of a Legendrian knot, and simple sheaves with microsupport given by the knot, as shown by Ng-Rutherford-Shende-Sivek-Zaslow. We give examples of Legendrians in high dimension only admitting augmentations of a more general type: in one case coefficients in relative homology groups must be used (i.e. a bulk deformation), while in another case coefficients in chains of the based loop space of the Legendrian must be used. We argue why these examples admit no simple sheaves, and speculate about what the appropriate generalisation are in these cases. This is joint work with H. Gao and Y. Huang. | | | Andriy Haydys | |
Orientation and multiplicities for the blow up set of the Seiberg-Witten monopoles with multiple spinors
A sequence of the Seiberg-Witten monopoles with multiple spinors on a three-manifold can converge after a suitable rescaling only on the complement of a subset Z. In this talk I will discuss the infinitezimal orientations and multiplicities of Z. This yields in particular topological constraints on possible limits of the Seiberg-Witten monopoles with multiple spinors. | | | Agustin Moreno | |
Algebraic torsion in higher-dimensional contact manifolds
Using the notion of algebraic torsion due to Latschev-Wendl, in this talk we sketch the construction of an infinite family of contact manifolds with finite and non-zero order of torsion, in any odd dimension. It follows that they are tight and admit no strong symplectic fillings. Time permitting, we discuss some interesting 5-dimensional cases, which have no Giroux torsion (in the sense of Massot-Niederkrueger-Wendl). | | | Bahar Acu | |
Iterated planar contact manifolds
Planar contact manifolds have been intensively studied to understand several aspects of 3-dimensional contact geometry. In this talk, we define "iterated planar contact manifolds", a higher-dimensional analog of planar contact manifolds, by using topological tools such as "open book decompositions" and "Lefschetz fibrations”. We provide some history on existing low-dimensional results regarding Reeb dynamics, symplectic fillings/caps of contact manifolds and explain some generalization of those results to higher dimensions via iterated planar structure. This is partly based on joint work in progress with J. Etnyre and B. Ozbagci. | | | Alex Degtyarev | |
Slopes of colored links
I will discuss various properties of the newly discovered link invariant, which we called slope. In particular, I will address the skein relations for signature in terms of slopes of tangles and the concordance invariance of slopes (as usual, outside of the Knottennullstellen). This is a joint work in progress with Vincent Florens and Ana G. Lecuona. | | | Yang Huang | |
Convex hypersurface theory in contact topology
In this talk I will sketch how convex hypersurfaces, in the sense of Giroux, can be used to understand contact structures in dimension > 3. In particular, I will introduce the fundamental building block of contact structures: bypass attachment. I will also define an overtwisted object called the overtwisted orange. Joint work with K. Honda. | | | Üstün Yıldırım | |
Complex \(G_2\) manifolds
We introduce the notion of complex \(G_2\) manifolds, and complexification \(M_\mathbb C\) of a \(G_2\) manifold \(M\). As an application we show that isotropic deformations of an associative submanifold of a \(G_2\) manifold M inside of its complexification \(M_\mathbb C\) is given by Seiberg-Witten type equations. This is a joint work with S. Akbulut. | | | Sümeyra Sakallı | |
Deformation of Singular Fibers of Genus 2 Fibrations and Small Exotic Symplectic 4-Manifolds
In 1963, Kodaira classified all singular fibers in pencils of elliptic curves, and showed that in such a pencil, each fiber is either an elliptic curve or a rational curve with a node or a cusp, or a sum of rational curves of self-intersection \(-2\). Later Namikawa and Ueno gave geometric classification of all singular fibers in pencils of genus two curves. In this talk, I will give topological constructions of certain singularity types in the Namikawa-Ueno’s list. I will also discuss 2-nodal spherical deformation of certain singular fibers of genus two fibrations. Then by using them I will provide constructions of exotic, minimal, symplectic 4-manifolds homeomorphic but not diffeomorphic to \(\mathbb {CP}^2\#6(-\mathbb{CP}^2), \mathbb{CP}^2\#7(-\mathbb{CP}^2)\) and \(3\mathbb{CP}^2\#k(-\mathbb{CP}^2)\) for \(k = 16,...,19\). This is a joint work with Anar Akhmedov. | |
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