TWENTY-EIGHTH GÖKOVA
GEOMETRY / TOPOLOGY CONFERENCE
May 29 - June 3 (2023)
Gökova, Türkiye
List of invited speakers/participants (tentative)
| Georgios Dimitroglou Rizell | |
Felix Schlenk | |
Anton Petrunin |
| Umut Varolgunes | |
Kyler Siegel | |
Tobias Ekholm |
| Richard Hind | |
Roberto Ladu | |
Yusuf Barış Kartal | |
| Roman Golovko | |
Gang Tian | |
Rostislav Matveev | |
| Oleg Viro | |
Vivek Shende | |
Evgeny Shinder | |
| Xiaobo Liu | |
Wenyuan Yang | |
Vladimir Fock | |
| Feng Wang | |
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Huabin Ge | |
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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: This conference is partially supported by 
  .
The participants of 28th Gökova Geometry - Topology Conference
List of Talks
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| Roberto Ladu | |
Corks, protocorks and Seiberg-Witten invariants
Corks are the key to a systematic understanding of exotic phenomena in dimension 4. However, understanding the class of corks in its entirety has proved to be an hard problem due to the lack of classification results. In order to attack this problem we introduce a family of 4-manifolds, called protocorks, a common denominator for all corks. I will discuss corks and protocorks and show how from the study of protocorks we obtain a result about the variation of Seiberg-Witten invariant valid for any cork. This will allow us to define an invariant of corks as the U-torsion order of a special element in the monopole Floer homology of the cork boundary. Time permitting, I will also discuss some open problems connected to symplectic geometry.
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| Evgeny Shinder | |
Gromov's cancellation question in birational algebraic geometry
In 1999 Gromov has asked to which extent an open embedding of an algebraic variety \(U\) into \(X\) determines the (birational) isomorphism class of the complement \(X \setminus U\). It was understood for a long time that this question has a natural interpretation in terms of the Grothendieck ring of varieties but until recently no counterexamples were known.
I will rephrase Gromov's question as nonvanishing of certain subgroups of the Grothendieck groups, and then explain which of these groups are non zero, thus effectively classifying all possible negative answers to Gromov's question. This talk is based on the work of Hassett--Lai 2016 (where the first counterexample can be deduced from), Shinder--Lin--Zimmermann 2020 (the surface case), Shinder--Lin 2022 (the general case).
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| Georgios Dimitroglou Rizell | |
Rational convex surfaces with only hyperbolic complex tangencies
We combine techniques from symplectic topology and complex analysis to construct examples of rationally convex surfaces of genus g that are totally real outside of \(2g-2\) hyperbolic tangencies. We construct fillable examples in the standard contact sphere, as well as non-fillable examples. This is joint work with Mark Lawrence.
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| Felix Schlenk | |
Monotone Lagrangian tori from toric geometry I
For every Markov triple (a,b,c) there is a monotone Lagrangian torus \( T_{a,b,c} \)in the complex projective plane, and any two of these are symplectically inequivalent, as was independently proved by Galkin-Mikhalkin and Vianna.
We show that there are exactly three homology classes in the complement of \( T_{a,b,c} \) that are represented by symplectically embedded spheres, and that the spheres in each class are unique up to symplectic isotopy.
After removing these spheres, we obtain for each Markov triple three symplectically inequivalent tori \( T_{a,b,c}^j \), \( j=0,1,2 \), in the ball.
The same can be done in \( S^2 \times S^2 \) where we now find four homology classes represented by a wedge of symplectic \(2\)-spheres, which give rise to four symplectic equivalence classes of monotone tori in the cube. This time, however, many symplectic equivalence classes of tori in \( S^2 \times S^2 \) fall into two Hamiltonian inequivalent classes, giving rise to eight Hamiltonian equivalence classes in the cube.
This is all joint work with Grisha Mikhalkin.
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| Richard Hind | |
Monotone Lagrangian tori from toric geometry II
In a second part, we make a quantitative study of these tori.
In the case of \( \mathbb CP^2\) and the ball, we ask for the smallest symplectically embedded ball that contains such a torus (its "outer radius"). It turns out that this numerical invariant distinguishes the tori with \(\min\, \{a,b,c\}=1\), and that it rapidly tends to the capacity of \( \mathbb CP^2\) for increasing \( \min \{b,c\}\). We also consider the problem of finding the smallest ball in which these tori ( seen in \( \mathbb R^4 \) ) become Hamiltonian isotopic to the Clifford or the Chekanov torus.
This is all joint work with Grisha Mikhalkin.
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| Anton Petrunin | |
Immersions with small normal curvatures after Gromov
We will discuss immersions small normal curvatures from
closed smooth manifolds to a unit ball in large-dimensional Euclidean space.
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| Xiaobo Liu | |
Intersection numbers on moduli spaces of curves and Schur Q-polynomials.
Generating functions of intersection numbers of certain tautological classes on moduli spaces of stable curves provide geometric solutions to integrable systems. Notable examples are the Kontsevich-Witten tau function and Brezin-Gross-Witten tau function. Both of them provide solutions to the KdV hierarchy.
Using matrix models, Mironov-Morozov gave a formula expressing Kontsevich-Witten tau function as an expansion of Schur's Q-polynomial with simple coefficients. A similar formula was also conjectured by Alexandrov for Brezin-Gross-Witten tau function. In this talk I will describe two proofs of these formulas using Virasoro constraints and cut-and-join operators.
These proofs do not depend on matrix models and have the potential to be generalized to study more general geometric model. The talk is based on joint works with Chenglang Yang.
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| Selman Akbulut | |
Shake Slice conjecture, 4D Smooth Poincare conjecture, Smale conjecture.
We will discuss answers to these conjectures through the use of the old but not well understood "carving" and "cork" techniques. I expect to have extra audience participation, since some of the contributing experts will be in the audience.
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| Umut Varolgunes | |
Involutive covers of symplectic manifolds and closed string mirror symmetry.
Consider a closed graded symplectic manifold M with a finite involutive cover (notion will be reviewed). This gives a canonical spectral sequence that starts from the relative SH of the cover and converges to the quantum cohomology of M.
I will discuss the compatibility of this SS with various algebraic structures, the consequences of degeneration at the earliest reasonable page and what it all means in the mirror symmetry context. If time permits I will outline a local to global computation of the A-side Yukawa coupling that is a reinterpretation of its equivalence to the B-side Yukawa coupling in mirror symmetry to illustrate the technique.
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Vivek Shende,
Tobias Ekholm | |
Mini course
Lecture 1: Skein valued curve counting.
We explain how to invariantly count holomorphic curves of all genera with Lagrangian boundary conditions in Calabi-Yau 3-folds.
In the process we discover a natural geometric occurence of the HOMFLYPT skein relations, and prove the Ooguri-Vafa conjecture relating the HOMFLYPT invariant of a knot to the count of curves ending on a certain associated Lagrangian (the knot conormal transplanted to the resolved conifold).
Lecture 2: Counting bare curves.
We describe ghost bubble censorship and sketch the perturbation theory needed for skein valued curve counting.
Lecture 3: Skein valued recursion relations.
We show how curves at infinity give skein valued recursion relations in several basic geometric situation and apply them to prove the Ooguri-Vafa conjecture for general representation.
We also show how skein valued recursion is related to certain skein modules of cleanly intersecting 3-manifolds.
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| Gang Tian | |
Ricci flow on Fano manifolds.
In this talk, I will discuss a long-standing problem on type II singularity of Ricci flow and Fano manifolds.
I will report on some recent progress and related results.
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| Oleg Viro | |
Real loci of non-real varieties.
A smooth submanifold A of a manifold X is called cooriented,
if its normal bundle is oriented. While oriented submanifolds
realize integer homology classes, cooriented submanifolds realize
integer cohomology classes. Cooriented and oriented submanifolds are
very much like cocycles and cycles with integer coefficients. In
particular, they may have integer intersection or linking numbers even
in a non-oriented ambient manifold.
Coorientations naturally appear in real algebraic geometry.
This happens if X is a real variety with complexification \( X_C\),
and \(A\) is a transverse intersection of \(X\) with a
complex subvariety \(V\) of \(X_C\). Then \(A\) is a real subvariety of \(X\) of a very
special kind. For example, if \(X\) is a real projective space and \(A\) has
codimension \(2\), then \(A\) is the base of a pencil of real hypersurfaces.
The coorientation of \(A\) is related to the geometry of the pencil. There
is an explicit upper bound for the linking number of \(A\) with a real
curve \(B\) bounding in its complexification.
Evening talk: Changing paradigmas in sciences.
A survey of theories (or, rather, conjectures), which have not become mainstream, and, opposing to them, commonly accepted theories, which may leave the mainstream.
I intend to bring up facts and arguments from neurobiology, paleontology, evolution of Earth, cosmology and history.
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| Kyler Siegel | |
On infinitesimal symplectic cobordisms.
Many symplectic cobordisms can be broken up into elementary building blocks which we call "infinitesimal cobordisms".
This suggests an intriguing approach to understanding global pseudoholomorphic curve counts by gluing together curves in each elementary piece.
I will explain a result (joint with G. Mikhalkin) which gives an explicit enumeration of curves in infinitesimal cobordisms between ellipsoids.
This gives a new but rather mysterious perspective on various curve counting problems, which I will illustrate in some simple examples.
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| Feng Wang | |
Kahler-Ricci flow on spherical Fano manifolds.
Spherical varieties is a large class of varieties with symmetry.
The relations between the geometric properties and combinatorial data have been investigated extensively.
In this talk, I will present the recent progress of the Kahler-Ricci flow on spherical Fano manifolds.
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| Yusuf Barış Kartal | |
A Morse-Bott approach to equivariant Floer homotopy type and cyclotomic structures.
SFloer homotopy type refines the Floer homology by associating a space (a spectrum to be precise) to a Hamiltonian, whose homology gives the Hamiltonian Floer homology.
In particular, one expects the existing structures on the latter to lift as well, such as the circle actions.
On the other hand, constructing a genuine circle action even in the Morse theoretic setting is problematic: one usually cannot choose Morse-Smale pairs/Floer data that is invariant under the circle action.
In this talk, we show how to extend the framework of Floer homotopy theory to the Morse-Bott setting, in order to tackle this problem. In the remaining time, we explain how to construct Frobenius maps/power maps on the Floer homotopy type that is compatible with the circle action, and how to relate this to the structures on the free loop spaces of exact Lagrangian submanifolds.
Joint work in progress with Shaoyun Bai and Laurent Cote.
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| Huabin Ge | |
Combinatorial Ricci flows and the hyperbolization of a class of compact 3–manifolds.
It is still not known that whether a hyperbolic 3-manifold admits a geometric ideal triangulation.
In this talk, We will show our program to hyperbolize and further obtain geometric triangulations of 3-manifolds.
To be precise, we will show the connections between 3D-combinatorial Ricci flows and geometric ideal triangulations,
that is, for a particular triangulation, the flow converges if and only if the triangulation is geometric. Further we will show
for a compact 3–manifold M with boundary admitting an ideal triangulation T with valence at least 10 at all edges,
there exists a unique complete hyperbolic metric with totally geodesic boundary, so that T is isotopic to a geometric
decomposition of M. The talk is based on joint works with Ke Feng and Bobo Hua.
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| Wenyuan Yang | |
Generic 3-manifolds are hyperbolic.
In this talk, we first introduce various models to study what a generic 3-manifold looks like.
We then focus on the Heegaard splitting model of 3-manifolds, equipped with geometric complexity using Teichmuller metric.
The main result is that the Hempel distance of a generic Heegaard splitting goes linearly to the infinity. In particular, generic 3-manifolds are hyperbolic in this model.
This represents the joint work with Suzhen Han (AMSS) and Yanqing Zou(ECNU).
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