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TWENTY-SEVENTH GÖKOVA
GEOMETRY / TOPOLOGY CONFERENCE

May 30 - June 4 (2022)
Gökova, Turkey

List of invited speakers/participants

G. Mikhalkin       I. Itenberg       V. Kharlamov
V. Fock       M. Can       W. Chen
M. Bhupal       I. Zharkov       S. Finashin      
N. Rozhkovskaya            F. C. Köse      
                 

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 TMD (Turkish Math Society), and ANR (French National Research Agency).



The participants of 27th Gökova Geometry - Topology Conference

List of Talks
Grigory Mikhalkin   Exoteric symplectic capacities
The talk will survey a new elementary definition of two series of higher symplectic capacities called "exoteric". One series is based on curves of genus \(0\), and it is very close to some of the rational SFT capacities defined by Kyler Siegel. The other series is based on curves of arbitrary genera, and is most interesting (and most computable) in dimension \(4\), where it turns out to be very close to the ECH capacities defined by Michael Hutchings. By "being very close" I mean that the capacities share the same properties, and produce the same answers in known examples. In the same time, neither the definitions of exoteric capacities, nor their computations in simple examples, use advanced theories like SFT, ECH or Seiberg-Witten.
Weimin Chen   What is the maximal number of \((-2)\) spheres in blown up \(\mathbb CP^2\)
It is known by a theorem of D. Ruberman that for any \(N \geq 2\), \(\mathbb CP^2\) blown up N times contains a disjoint union of N smoothly embedded \((-2)\)-spheres. This number is clearly the maximal number of such \((-2)\)-spheres. We ask: if the \((-2)\)-spheres are required to be symplectic, what is the maximal number of such \((-2)\)-spheres? (It turns out that the maximal number is the same if we instead require the \((-2)\) -spheres are a Lagrangian sphere). In this talk, we shall give some partial answers to this question, explaining it as a special example of a more general, ongoing project (joint with Çağrı Karakurt) which aims to address the question of existence and classification of certain configurations of symplectic surfaces in a rational symplectic \(4\)-manifold.
Mohan Bhupal   Unbraided wiring diagrams for Stein fillings of lens spaces
In previous work, we constructed a planar Lefschetz fibration on each Stein filling of any lens space equipped with its canonical contact structure. In this talk, we describe an algorithm to draw an unbraided wiring diagram whose associated planar Lefschetz fibration obtained by the method of Plamenevskaya and Starkston, where the lens space with its canonical contact structure is viewed as the contact link of a cyclic quotient singularity, is equivalent to the Lefschetz fibration we constructed on each Stein filling of the lens space at hand. Coupled with the work of Plamenevskaya and Starkston, we obtain the following result as a corollary: The wiring diagram we describe can be extended to an arrangement of symplectic graphical disks in \(\mathbb C^2\) with marked points, including all the intersection points of these disks, so that by removing the proper transforms of these disks from the blowup of \(\mathbb C^2\) along those marked points one recovers the Stein filling along with the Lefschetz fibration. Moreover, the arrangement is related to the decorated plane curve germ representing the cyclic quotient singularity by a smooth graphical homotopy.As another application, we set up an explicit bijection between the Stein fillings of any lens space with its canonical contact structure, and the Milnor fibers of the corresponding cyclic quotient singularity, which was first obtained by Nemethi and Popescu-Pampu, using different methods. This is a joint work with Burak Özbağcı.
Ilia Itenberg   Real enumerative invariants and their refinement
The talk is devoted to several real and tropical enumerative problems. We suggest new invariants of the projective plane (and, more generally, of toric surfaces) that arise as results of an appropriate enumeration of real elliptic curves. These invariants admit a refinement (according to the quantum index) similar to the one introduced by Grigory Mikhalkin in the rational case. We discuss tropical counterparts of the elliptic invariants under consideration and establish a tropical algorithm allowing one to compute them. This is a joint work with Eugenii Shustin.
Feride Ceren Köse   Does the Jones polynomial detect the unknot?
It is still an open question whether there is a nontrivial knot with Jones polynomial equal to that of the unknot. I will talk about what has been done so far to answer this question and explain how it is also related to other famous open questions in topology. I will then introduce a construction of knots which was used to produce the first examples of nontrivial knots with trivial Alexander polynomials, and distinct knots with the same Jones polynomials, knot Floer and Khovanov homologies. Then, I will describe a family of knots proposed by Tanaka using this construction such that if any were amphichiral, they would have trivial Jones polynomials and hence could answer the question in the negative. However, I will show that such a knot is always trivial and rule out that possibility to answer the question.
Vladimir Fock   Higher complex structures on Riemann surfaces
Higher complex structure is a differential geometric structure on a two-dimensional surface generalizing complex structure. The moduli of such structures is (conjecturally for \(n>3\) ) isomorphic to the higher Teichmüller space. We will discuss this structures as well as its cotangent bundle which has a combinatorial description related to Hecke algebras, spectral networks and affine Weyl groups (Joint work with A.Thomas and V.Tatitscheff).
Eylem Zeliha Yıldız   On shake slice knots
We will discuss r−shake slice knots. For each 0-shake slice knot we’ll construct an associated \(4\)-manifold which embeds in \(S^4\). This construction allows us to approach the “\(0\)-shake slice implies slice” conjecture, and establishes a connection to corks. This is a joint work in progress with Selman Akbulut.
Viatcheslav Kharlamov   New examples of wall-crossing invariant counting of real curves
One of the basic known examples of wall-crossing invariant counts is the count of real lines on real cubic surfaces. A conceptual treatment of this example has already led to a development of an integer valued real Schubert calculus. In this talk (based on a joint work with S. Finashin) I intend to describe how this example with cubic surfaces can be generalized in a different direction: from straight lines on cubic surfaces to straight lines, and even curves of arbitrary degree, on other del Pezzo surfaces. I intend also to talk about some remarkable properties of the invariants obtained, like a direct connection with the complex Gromov-Witten invariants and computability by surprisingly simple explicit formulas.
Mahir Can   Graded Nash Manifolds
The category of affine Nash manifolds and mappings lies between the category of smooth manifolds and smooth mappings and the category of nonsingular real algebraic varieties with smooth rational mappings. The importance of this category was first noticed by John Nash who used it to show that any compact real analytic manifold can be given a real algebraic manifold structure. Thanks to Tognoli and Akbulut-King, today we know more; every smooth manifold is diffeomorphic to a smooth real algebraic variety. In this talk, we will make an introduction to our general program of extending many related results to the graded setting. In particular we will introduce and discuss the categories of graded (affine) Nash manifolds. If time permits, we will discuss graded Nash groups and their classification in dimension one.
Alex Degtyarev   Conics on polarized \(K3\)-surfaces
Generalizing Barth and Bauer, denote by \(N_{2n}(d)\) the maximal number of smooth degree \(d\) rational curves that can lie on a smooth \(2n\)-polarized \(K3\)-surface \(X\) in \(P^{n+1}\). Originally, the question was raised in conjunction with smooth quartics in \(P^3\), which are \(K3\)-surfaces. The problem is very classical, going back to Schur (1882) and Segre (1943), and a series of recent papers by quite a few contributors gives us the precise values of \(N_{2n}(1)\) for all \(n\). On the other hand, the only known value for \(d=2\) is \(N_6(2)=285\). I will discuss my recent discoveries that support the following conjecture on the conic counts in the remaining interesting degrees. Conjecture. One has \(N_2(2)=8910\), \(N_4(2)=800\), and \(N_8(2)=176\). (Modulo a few final checks, the conjecture \(N_4(2)=800\) has been settled.) The approach used does not distinguish (till the very last moment) between reducible and irreducible conics. However, extensive experimental evidence suggests that all conics are irreducible whenever their number is large enough. Conjecture. There exists a bound \(N_{2n}^*(2) < N_{2n}(2)\) such that, whenever a smooth \(2n\)-polarized \(K3\)-surface \(X \subset P^{n+1}\) has more than \(N_{2n}^*(2)\) conics, it has no lines and, in particular, all conics on \(X\) are irreducible. We know that \(249 \leq N_{6}^*(2)\leq 260 \) is indeed well defined, and it seems feasible that \(N_{2}^*(2) \geq 8100\) and \(N_{4}^*(2) \geq 720\) are also defined; furthermore, conjecturally, the lower bounds above are the exact values (attained at surfaces maximizing the number of lines).
Sergey Finashin   Geometry of the Welschinger weights in the signed count of rational curves
In a series of works with V.Kharlamov we explored a real version of GW-theory, and analyzed the invariants obtained by a signed count of real rational curves. We gave several equivalent descriptions of the signs involved into this count and I will discuss some geometric ones, which look quite elementary, but effective.
Natasha Rozhkovskaya   Polynomial tau-functions of the KP hierarchy
The Kadomtsev-Petviashvili (KP) hierarchy is a family of evolution equations of soliton type. That means the hierarchy is explicitly solvable, and that the set of solutions includes wave-like functions with interesting behavior that gives them the name of `solitons'. The symmetries of soliton hierarchies are studied by methods of different areas of mathematics (such as mathematical physics, algebraic geometry, representation theory, numerical analysis), revealing a number of interesting functions to be connected to solutions of soliton type equations (for example, Weierstrass p-function, characters of representations of general linear group, eigenvalues of certain central elements in representation theory). Using representation theory approach we will describe explicitly all polynomial tau-function of the KP hierarchy as coefficients of generating functions.
Özgür Kişisel   On amoebas of random plane curves
Due to a theorem of Passare and Rullgard, the area of the amoeba of a degree \(d\) algebraic curve in the complex projective plane is bounded above by \(\pi^2 d^2/2\) and the curves attaining the bound - special Harnack curves - have been characterized by Mikhalkin. In this talk, reporting on joint work with Turgay Bayraktar, I will argue that the expected area of a randomly chosen complex algebraic curve, with respect to the Kostlan distribution, is bounded above by a constant times \(d\). This result also generalizes in a natural way to half dimensional complete intersections in toric varieties with an arbitrary Newton polytope.
Ilia Zharkov    Poncelet, plane polygons and Calabi-Yau's
Two involutions on an elliptic curve are behind the famous Poncelet theorem about two conics and lines passing through one and tangent to the other. I'll review several other examples of this phenomenon, quadrics in space, Bolzmann billiard, plane quadrilaterals, PL symplectomorphisms of the plane, and a few more. Some of these examples generalize to a very special higher dimensional Calabi-Yau varieties, I'll give a few examples. The talk is based on some recent conversations with M. Kontsevich.
Selman Akbulut    Disproving Smale conjecture via infinite order cork
Smale asked whether there is a homotopy equivalence \(Diff(S^4)≃SO(5)\). Watanabe disproved this by showing that their higher homotopy groups are different. Here we prove this more directly by showing \(\pi_0(Diff(S^4))≠0\), otherwise a certain loose-cork could not possibly be a loose-cork. To do this I will discuss infinite order corks induced by knot concordances.
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Last updated: May 2022
Web address: GokovaGT.org/2022