THIRTIETH GÖKOVA
GEOMETRY / TOPOLOGY CONFERENCE
May 26 - May 31 (2025)
Gökova, Türkiye
Pearl Anniversary
List of invited speakers/participants (tentative)
| Paolo Cascini | |
Eaman Eftekhary | |
Ivan Cheltsov |
| Arthur Renaudineau | |
Frédéric Mangolte | |
Diego Matessi |
| Surena Hozoori | |
Vivek Shende | |
Agustin Moreno |
| Turgay Bayraktar | |
Tatsuki Kuwagaki | |
Sergey Finashin |
| Ke Feng | |
Mikhail Shkolnikov | |
Mohan Bhupal |
| Honghuai Fang | |
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Mark Lawrence |
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Scientific Committee : D. Auroux, Y. Eliashberg, G. Dimitroglou Rizell, I. Itenberg, G. Mikhalkin, S. Akbulut, A. Oancea, C. Taubes, U. Varolgüneş
Organizing Commitee : T. Önder, S. Koçak, A. Degtyarev, Ö. Kişisel, Y. Ozan, F. Arıkan, E. Z. Yıldız
Supporting Organizations: 
Santori Family Charitable Foundation 
The participants of 30th Gökova Geometry - Topology Conference
List of Talks
(Schedule)
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| Turgay Bayraktar | |
Random real algebraic geometry and random ameobas
Classical problems in algebraic geometry concern invariant or extremal properties
of algebraic varieties, whereas in the probabilistic version, we focus on statistical
properties of the fundamental invariants. For example, a real algebraic projective
plane curve of degree \( d\) has at most \( g+1 = (d−1)(d−2)/2+1\) connected components
where \(g\) denotes the genus, which is an extremal property; whereas a random real
algebraic projective degree d plane curve in a suitable precise sense (to be explained
in the talk) has an expected number of connected components of order \(d\). In this
talk, I will discuss some recent results on the statistical properties of connected
components and amoebas of random algebraic varieties. The talk is based on a joint work with Emel Karaca and another joint work with Özgür Kişisel.
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| Paolo Cascini | |
Mini-course on some old and recent results on the MMP
Lecture 1: Classical Results of the MMP
The Minimal Model Program (MMP) is a framework in modern algebraic geometry,
aiming to extend the classification of complex projective surfaces to higher dimensions. In this lecture, I will introduce the classical structure of the MMP,
including the concepts of cone theorem, contractions, flips, and the existence and termination of minimal models.
Lecture 2: Some recent progress on the MMP
In recent years, the techniques of the Minimal Model Program (MMP) have been successfully extended to several other setting, such as in the case of foliations. In this lecture,
I will discuss the adaptation of MMP tools to the study of foliated complex projective varieties. Topics will include foliated versions of flips and divisorial contractions, and the canonical models for foliations. I will also highlight some of the open problems that arise in this context.
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| Ivan Cheltsov | |
Which cubics 3-folds admit rational parametrization?
Every day we deal with geometric objects defined by algebraic equations (circles, parabolas, hyperbolas, splines, spheres, hyperboloids, etc). In many applications, we have to parametrize them using the simplest possible functions - rational functions in several variables.
To find such parametrization maybe tricky. This is a very classical problem - rational parametrization of a circle has been found by Pythagoras, and the same approach gives explicit rational parametrization of a sphere or any geometric object given by one quadratic equation. In more complicated cases, the problem can be very difficult.
Moreover, quite often rational parametrization does not exist - there are many algebraic objects that do not admit rational parametrization. In my talk, I will focus on the existence of a rational parametrization of cubics - geometric objects defined by one equation of degree 3 (cubic curves, cubic surfaces, and cubic 3-folds).
The talk is based on a joint work with Yuri Tschinkel and Zhijia Zhang from New York.
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| Alex Degtyarev | |
Split hyperplane sections on polarized \( K3\)-surfaces
I will discuss a new result which is an unexpected outcome, following a
question by Igor Dolgachev, of a long saga about smooth rational curves on
(quasi-)polarized \( K3\)-surfaces. The best known example of a \( K3\)-surface is
a quartic surface in space. A simple dimension count shows that a typical quartic
contains no lines. Obviously, some of them do and, according to B. Segre, the
maximal number is \(64\) (an example is to be worked out). The key role in
Segre's proof (as well as those by other authors) is played by plane sections
that split completely into four lines, constituting the dual adjacency graph \(K(4)\). A
similar, though less used, phenomenon happens for sextic
\( K3\)-surfaces in \( \mathbb{P}^4\) (complete intersections of a quadric and a cubic): a
split hyperplane section consists of six lines, three from each
of the two rulings, on a hyperboloid (the section of the quadric), thus constituting a \( K(3,3) \).
Going further, in degrees \(8\) and \(10\) one's sense of beauty suggests that
the graphs should be the \(1\)-skeleton of a \(3\)-cube and
Petersen graph, respectfully. However, further advances to higher degrees
required a systematic study of such \(3\)-regular graphs and, to my great
surprise, I discovered that typically there is more than one!
Even for sextics one can also imagine the \(3\)-prism (occurring when the
hyperboloid itself splits into two planes).
The ultimate outcome of this work is the complete classification of the
graphs that occur as split hyperplane sections (a few dozens) and that of the
configurations of split sections within a single surface (manageable starting
from degree \(10\)). In particular, answering Igor's original question, the
maximal number of split sections on a quartic is \(72\), whereas on a sextic
in \( \mathbb{P}^4 \) it
is \(40\) or \(76\), depending on the question asked. The ultimate champion is
the Kummer surface of degree \(12\): it has \(90\) split hyperplane sections.
The tools used (probably, not to be mentioned)
are a fusion of graph theory and number theory, sewn together
by the geometric insight.
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| Eaman Eftekhary | |
Two-tangles and exact triangles in knot Floer homology
Over the past two decades, knot Floer homology has emerged as a powerful tool in knot theory, offering a rich array of applications.
As a categorification of the Alexander polynomial, it not only detects the unknot, the Seifert genus, and the fiberedness of knots but also provides strong lower bounds for invariants such as the 4-ball genus, the unknotting number, and the bridge number.
The skein relation for the Alexander polynomial finds its deeper counterpart in the skein exact triangle for knot Floer homology, a versatile tool for both computation and theoretical exploration.
Additionally, Manolescu's construction of an unoriented skein exact triangle establishes a fascinating connection to Khovanov homology. This raises a natural question: Are there other exact triangles in knot Floer homology?
In this talk, we begin with a brief overview of knot Floer homology and its key applications. We then delve into the construction of skein exact triangles and present several new examples, expanding the scope of this powerful framework.
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| Ke Feng | |
Some progress on geometric ideal triangulation
Glueing ideal tetrahedra plays a crucial role in the construction of hyperbolic 3-manifolds.
While it is still not known that whether a hyperbolic 3-manifold admits a geometric ideal triangulation. In this talk, we will show some of our progress to hyperbolize,
and further obtain geometric triangulations of 3-manifolds. To be precise, we will show the rigidity of hyperbolic polyhedral metrics on 3-manifolds, which is a joint work with Huabin Ge.
And then, we will show the connections between 3D-combinatorial Ricci flows and Thurston's geometric ideal triangulations. At the same time, we will also give some topological conditions
to guarantee the convergence of the combinatorial Ricci flows, and furthermore, the existence of geometric ideal triangulations.
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| Sergey Finashin | |
On wall-crossing invariance of certain sums of Welschinger numbers
We study real enumerative invariants not sensitive to changing of the real structure on del Pezzo surfaces.
These “wall-crossing invariants" are combinations of the Welschinger invariants taken with signs determined by certain Pin-structures on the surfaces.
The recurrence relation for such invariants admit a solution giving explicit formulas. The relation to Gromov-Witten invariants, namely, their second moments, is also revealed.
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| Surena Hozoori | |
Regularity and persistence in non-Weinstein Liouville geometry via hyperbolic dynamics
We explore Mitsumatsu's construction of non-Weinstein Liouville geometric objects based on Anosov 3-flows.
In the generalized framework of Liouville Interpolation Systems and non-singular partially hyperbolic flows. We discuss the subtle phenomena inherited from the regularity and persistence theory of hyperbolic dynamics in the resulting Liouville structures,
and prove dynamical and geometric rigidity results in this context. Among other things, we show that Mitsumatsu's examples characterize 4-dimensional non-Weinstein Liouville geometry with 3-dimensional -persistent transverse skeleton.
Time permitting, we also draw applications to the regularity theory of the weak dominated bundles for non-singular partially hyperbolic 3-flows.
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| Tatsuki Kuwagaki | |
An Introduction to sheaf quantization over the Novikov Ring
Let X be a Weinstein manifold. The theorem of Ganatra–Pardon–Shende establishes an equivalence between the (partially) wrapped Fukaya category of X and the category of microlocal sheaves on X.
To generalize this result to the non-exact setting, we introduce the category SQ(X) of sheaf quantizations over the Novikov ring associated to X, which is expected to be equivalent to the unwrapped Fukaya category of non-exact Lagrangians in X.
In the case where X is moreover holomorphic, SQ(X) is also expected to be closely related to deformation quantization modules, spectral networks, and exact WKB analysis. In this talk, I will provide an introduction to these ideas and outline recent developments.
The talk is partly based on various joint works with Ike, Petr, Shende, and Zhang.
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| Mark Lawrence | |
Polynomial hulls of some knotted tori in \( S^1 \times \mathbb C \)
For a compact set \( 𝐾\subseteq \mathbb C^n \), it is interesting to know when there is an analytic variety in the polynomial hull \( \hat K \). In full generality, there is no hope of a positive
answer, but there are some positive results for manifolds. For some simple classes
of totally real tori in \( S^1 \times \mathbb C \), I can prove some positive results. The method
involves continuity of some holomorphic disc families, Gromov compactness, and
a geometric argument involving linking at a critical level. This is a first step in a
practically unexplored area, which is the intersection of knot theory and
polynomial hull theory.
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| Frédéric Mangolte | |
Real loci of rational Fano threefolds
From the classification of real rational surfaces worked out by Comessatti at the beginning of the 20th century, we get the following striking characterization of real rational surfaces: a geometrically rational real surface is rational if and only if its real locus is non-empty and connected.
The analogous assertion fails in higher dimension.
In a work in progress with Andrea Fanelli, we explore real loci of geometrically rational Fano threefolds in relation to their rationality.
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| Diego Matessi | |
Patchworking and mirror symmetry
Consider a real hypersurface of a toric variety constructed via primitive patchworking.
Then Renaudineau and Shaw construct a spectral sequence which converges to the mod 2 cohomology of the real locus.
The elements of the first page are the tropical homology groups. In this talk, I will discuss of the special case of patchworking on a primitive central triangulation of a reflexive polytope.
This gives examples of real Calabi-Yau manifolds. I will show how to use tropical mirror symmetry to compute some of the maps of the spectral sequence.
For instance, the map \( (1,n-1) \dashrightarrow (2, n-2) \), on the mirror becomes of type \( (1,1) \dashrightarrow (2,2) \). One can show it is a "twisted squaring of divisors".
This is based on joint work with Arthur Renaudineau.
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| Agustin Moreno | |
Exotic contact structures on the sphere and beyond
By work of Eliashberg in dimension 3 and Borman—Eliashberg—Murphy in higher dimensions, there are two flavours of contact structures: tight (geometric/rigid), and overtwisted (topological/flexible).
Tight contact structures are much harder to classify than the overtwisted ones (which satisfy an h-principle, and so are abundant). One way to understand tight structures is via their symplectic fillings.
In this talk, I will explain how in dimension at least 5, the odd-dimensional sphere admits contact structures which are exotic from a symplectic point of view, namely they are tight but admit no (strong) symplectic filling.
This is in stark contrast to the three-dimensional case, as Eliashberg showed there is a unique tight contact structure on the 3-sphere (which is fillable). I will also discuss applications to arbitrary topology.
This is based on joint work with Jonathan Bowden, Fabio Gironella and Zhengyi Zhou.
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| Arthur Renaudineau | |
Real phase structures on tropical varieties
In this talk, we will propose a generalization of combinatorial patchworking in any codimension beyond the case of complete intersections.
It works by reinterpreting the combinatorial patchworking in terms of tropical geometry (due to Grisha Mikhalkin).
We will remind this interpretation carefully and indicate that, in general, a result of a non-singular patchwork gives a topological manifold and that if the tropical variety is approximated by a family of real varieties, this topological manifold is homeomorphic to any member of the family near the tropical limit.
This is a joint work with Johannes Rau and Kris Shaw.
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| Vivek Shende | |
Categorification of quantum groups from Floer homology
Fix a semisimple Lie algebra. We study Fukaya categories of the moduli of monopoles on the product of a surface and the real line, and establish the following results.
(1) When the surface is a strip, the category naturally carries an algebra structure, and recovers the
Khovanov-Lauda-Rouquier categorification of half of the corresponding quantum group.
(2) When the surface is a hexagon, the category provides a coproduct, providing the (long
awaited) monoidal structure on the representation category of the categorified quantum
group.
(3) The fundamental representations and generic Verma modules also have geometric
categorifications; the corresponding surfaces are half disks respectively with a labelled
marked point, or a hole in the interior.
(4) Our (categorified tensor product) of (categorified fundamental representations) thus corresponds
to a half disk with several marked points in the interior; we show it recovers Webster’s
(categorified tensor product of fundamental representations).
(5) Categorification for Lie superalgebras fits in the same framework, although the moduli
spaces involved are no longer identified with monopoles. For g = sl(1|1), the relevant space is
the moduli of vortices on the surface, i.e. its symmetric powers. Our setup correspondingly
recovers and extends the results of Rouquier and Manion on Heegard-Floer thoery.
This is work with Mina Aganagic, Elise LePage, and Peng Zhou.
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| Mikhail Shkolnikov | |
Equiaffine distance functions
We will discuss a natural class of functions associating a positive real number to a point in the interior of a convex domain on the equiaffine plane
(with its symmetry group being affine transformations preserving area), which may be interpreted as a distance from the point to the boundary of the domain.
Its definition stems from a recent suggestion of Conan Leung to average the tropical distance function, previously studied in joint works with Grigory Mikhalkin and Nikita Kalinin, over the space of all tropical structures compatible with the fixed equiaffine structure of the ambient plane.
We conjecture that the equiaffine-equidistant loci of a given compact convex domain converge to an ellipse as the distance approaches its maximal value (a stronger form of this conjecture stated in terms of Mahler area was verified numerically by Ernesto Lupercio).
Remarkably, a proof of an analogous statement for non-compact convex domains with two non-parallel asymptotes, i.e. convergence of level sets to a branch of a hyperbola, is very simple. This proof, other foundational properties, and necessary tropical preliminaries will be explained during the talk.
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Conference main page |
Last updated: December 2025
Web address: GokovaGT.org/2025
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