TWENTY-NINTH GÖKOVA
GEOMETRY / TOPOLOGY CONFERENCE
May 27 - June 1 (2024)
Gökova, Türkiye
List of invited speakers/participants (tentative)
| Georgios Dimitroglou Rizell | |
Antoine Toussaint | |
Anton Petrunin |
| Evgeny Shinder | |
Roman Golovko | |
Richard Hind | |
| Mohan Bhupal | |
Sergey Finashin | |
Mikhail Shkolnikov |
| Dimitri Zvonkine | |
Ilia Zharkov | |
Aloïs Pierre Marc Demory |
| Ferihe Atalan | |
İrem Özge Taşpınar | |
Natasha Rozhkovskaya |
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Scientific Committee : D. Auroux, Y. Eliashberg, I. Itenberg, G. Mikhalkin, S. Akbulut, A. Oancea, C. Taubes
Organizing Commitee : T. Önder, S. Koçak, A. Degtyarev, Ö. Kişisel, Y. Ozan, F. Arıkan, E. Z. Yıldız
Supporting Organizations: Santori Family Charitable Foundation 
List of Talks
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| Richard Hind | |
Symplectic packing
A basic question in symplectic geometry is whether one manifold embeds into another, and a natural case to consider is when the domain is a disjoint union of balls.
It is not so hard to show, at least in dimension 4, that any symplectic manifold admits an embedding of a disjoint union of identically sized balls whose images occupy an arbitrarily large proportion of the volume.
This is already in strong contrast to the situation in Riemannian geometry, but a surprising example of symplectic flexibility is that many symplectic manifolds admit volume filling embeddings by finite numbers of sufficiently small balls.
We describe the first examples of symplectic manifolds which do not admit volume filling embeddings by finitely many balls.
The examples can be realized as open subsets of 4-dimensional space, diffeomorphic to a ball, and also provide the first such examples which are not symplectomorphic to the interior of a manifold with smooth boundary.
The proofs rely on Embedded Contact Homology, which gives a relation between symplectic embedding questions and the Minkowski dimension of boundaries. This is joint work with Dan Cristofaro-Gardiner.
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| Georgios Dimitroglou Rizell | |
Mini-course on regular Lagrangians
Lecture 1: Definitions, properties, and examples
Weinstein manifolds and sectors are particularly well-behaved open symplectic manifolds that can be described by a handle-decomposition.
Lagrangian submanifolds of symplectic manifolds play an important role in symplectic topology; we are interested in Lagrangains that are closed or have boundary, i.e. Lagrangian fillings and cobordisms.
Eliashberg-Ganatra-Lazarev introduced the concept of regular Lagrangians in Weinstein manifolds, as those Lagrangians that have a natural description with respect to the Weinstein structure. In the first lecture we define these concepts and show how to construct examples in the ball in the form of a Kirby diagram, starting from a decomposable Lagrangian cobordism.
Lecture 2: The regularity conjecture and non-examples from flexibility
In the second lecture we investigate consequences of a Lagrangian being regular; there are constraints in the cases when the Lagrangian is either closed, a filling, or a cobordism. We also discuss the regularity conjecture and use flexibility results for exhibiting non-regular Lagrangian cobordisms in both low and high dimensions.
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| Evgeny Shinder | |
Open problems on exotic affine spaces
I will give an overview of some long standing conjectures about the affine space \(A^n\), focusing on the Zariski cancellation conjecture, automorphism groups and the characterization of the affine space. I will explain the main approaches to these questions: topological, algebraic and algebro-geometric.
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| İrem Özge Taşpınar | |
Compatible relative open books on relative contact pairs via
generalized square bridge diagrams
Akbulut-Özbağcı and later Arıkan gave algorithms both of which construct an explicit compatible open book decomposition on a closed contact 3-manifold which results from a contact \( (\pm 1) \)-surgery on a Legendrian link in the standard contact 3-sphere by using square
bridge position. In this talk, we introduce the “generalized square bridge position” for a Legendrian link in the standard contact 5-sphere and partially generalize this result to the dimension five via an algorithm which constructs relative open book decompositions on
relative contact pairs.
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| Sergey Finashin | |
Monodromy factorisation of lines on del Pezzo surfaces
(This is a joint work with M.Bhupal, in progress.) An anti-canonical Lefschetz pencil on a del Pezzo surface, X (plane blown up at 9-d points), as is well-known, can be represented by a factorisation of the boundary Dehn twist into a product of twelve internal Dehn twists in the mapping class group of a torus punctured at d points.
The lines (which are by definition rational (-1)-curves that can be blown down) in X can be represented as refinements of the above factorization as we add to a torus one additional (d+1)-st puncture.
We give an explicit description of the 240 such refinements corresponding to the 240 lines for d=1. For d=2 and 3 we describe similarly all 56 and respectively 27 lines in terms of their monodromy factorization.
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| Roman Golovko | |
Instability of Legendrian knotedness and non-regular Lagrangian concordances of knots
We show that family of smoothly non-isotopic Legendrian pretzel knots that all have the same Legendrian invariants as the standard Legendrian unknot have front-spuns that are Legendrian isotopic to the front-spun of the unknot.
Besides that, we construct the first examples of Lagrangian concordances between Legendrian knots that are not regular, and hence not decomposable.
This is joint work with Georgios Dimitroglou Rizell.
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| Eylem Zeliha Yıldız | |
Braids in planar open books and fillable surgeries
We'll give a useful description of braids in \( \# S^1\times S^2 \) using surgery diagrams, which will allow us to address some knots in lens spaces that admit fillable positive contact surgery.
We also demonstrate that smooth 16-surgery to the Pretzel knot \( P(−2,3,7) \) bounds a rational ball, which admits a Stein structure.
This answers a question left open by Thomas Mark and Bülent Tosun.
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| Aloïs Pierre Marc Demory | |
Topology of three- and four-dimensional maximal real algebraic varieties
The study of the topological properties of real algebraic varieties, i.e. complex algebraic varieties endowed with an antiholomorphic involution, is a problem that can be traced back to the work of A. Harnack and D. Hilbert on real algebraic plane curves at the end of the 19th century.
We are particularly interested in the topology of maximal real algebraic varieties, for which the homology of the fixed points set of the involution is as rich as possible. However, there are very few known examples of maximal real algebraic varieties of dimension $3$ and higher.
In this talk, we contribute to filling this gap by studying three- and four-dimensional maximal real algebraic varieties obtained by considering double coverings of projective spaces ramified along maximal real algebraic hypersurfaces, and endowed with involutions obtained by lifting the involution of the basis.
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| Ali Ulaş Özgür Kişisel | |
Amoeba measures of random plane curves
In this talk, I will discuss results about the expected areas of amoebas of random complex plane curves, obtained in our joint work with J-Y. Welschinger.
It will be shown that the expected area of the amoeba of a degree \(d \) plane curve is less than \( 3\ln(d)^2/2+9\ln(d)+9 \), and that the ratio of this expected area to \( \ln(d)^2\) is bounded below by \( 3/4\) as \( d\) goes to infinity.
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| Dimitri Zvonkine | |
Mini-course
Lecture 1: The Quantum Hall effect: a model of electrons on a surface leads to a problem with algebraic geometry
Modelling electrons on a closed conducting surface unexpectedly leads to questions in complex algebraic geometry.
We will explain how the magnetic field becomes a holomorphic line bundles, ground states (that is, lowest energy eigenstates of the Hamiltonian) are holomorphic sections of this line bundle, the electric field changes the holomorphic structure of the line bundle making it move in the Picard group, and, most importantly, the electric current is related to the first Chern class of the resulting bundle over the Picard group.
Thus from a physical problem we get to a mathematical problem of computing characteristic classes of vector bundles called the Laughlin bundles.
Lecture 2: The Laughlin bundes the their characteristic classes
These characteristic classes are computed by applying the Grothendieck-Riemann-Roch formula to a symmetric power of a Riemann surface. We will need to investigate the cohomology of such symmetric powers, compute the Todd class of its tangent vector bundle, and apply some combinatorial ingenuity to express the answer in a closed form.
I will start with a short introduction to the Grothendieck-Riemann-Roch formula.
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| Alex Degtyarev | |
Singular real plane sextic curves without smooth real part
It is well known that the deformation type of (properly understood; say, quasi-polarized) complex \( K3\)-surfaces with singularities is determined by their so-called homological type, and a similar statement holds for smooth real \( K3\)-surfaces.
However, it is equally well known, with plenty of examples, that, if the two are mixed and real surfaces with singularities are considered, the naïve naked homological type is no longer enough: one needs to compute the so-called fundamental polyhedra which are often infinite.
On the example of double planes ramified at sextic curves I will show that the homological type still suffices to determine the equisingular equivariant deformation family provided that the real part is smooth, i.e., all singular points split into pairs of complex conjugate.
This fact has been used to obtain the complete classification of singular real plane sextic curves with smooth real part.
The purpose of this talk is a gentle introduction to the theory of \( K3 \)-surfaces and its real aspects.
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| Anton Petrunin | |
Tubed embeddings
We consider the following question:
When does a Riemannian manifold admit an embedding with a uniformly thick tubular neighborhood in another Riemannian manifold of large dimension?
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| Mikhail Shkolnikov | |
Lines on surfaces and non-commutative phase tropicalization
An amoeba of a subvariety of a complex algebraic torus is an image under the logarithmic projection, forgetting the phase of each coordinate and a tropicalization is a limit of rescaled amoebas.
Non-commutative amoebas generalize the concept of classical amoebas by replacing an algebraic torus with a possibly non-commutative complex group, and the logarithm map by the projection to the quotient by the maximal compact subgroup.
In our paper with Grigory Mikhalkin, we studied the case of the complex three-dimensional group PSL(2,C). For families of curves inside this group, we have described the shape of their non-commutative tropicalizations, which appear to be unions of spheres and geodesic segments organized via a floor diagram.
We have discovered that amoebas of surfaces in PSL(2,C), in contrast with classical amoebas, have a connected complement, which implies that one cannot expect a rich structure of their tropicalization. In my talk, I will discuss a way to get more information from a degeneration procedure of surfaces by preserving the phase, i.e. via a non-commutative version of a phase tropicalization.
To illustrate how it can be applied, I will demonstrate a classical fact that a general surface of a degree greater than three doesn't contain lines. Based on a joint work with Peter Petrov.
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| Antoine Toussaint | |
Type of phase tropical surfaces
The type of a real algebraic variety is determined by the homology class realized by its real points inside the complex points. We study this important topological feature of a real algebraic variety in the case of phase tropical surfaces. These surfaces are 4-manifolds equivariantly homeomorphic to the complex part of a T-hypersurface obtained by primitive patchworking (a special case of Viro's method), which admits a stratified fibration onto the tropical hypersurface dual to the triangulation of the patchworking.
By studying the real structures of these phase tropical surfaces, we explain how to generalize the notion of twisted edges (which is well known in the case of curves since the work of B. Haas) to a phase tropical surface and use it to lift the tropical homology to the homology of the phase tropical hypersurface. As an application, we obtain a combinatorial criterion on the tropical hypersurface which determines the type of a phase tropical surface.
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