Gökova Geometry Topology Institute

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===== Research Session on Real, Tropical, and Symplectic Geometry, and Low-Dimensional Manifolds =====
=====(Oct 1-Oct 10, 2026)=====


The research session will focus on recent progress and new challenges in tropical, symplectic, and real algebraic geometry, as well as the study of 4-dimensional smooth manifolds. Topics will include classical problems concerning (complex or real) plane singular curves, the analysis of separating morphisms of real curves, theta-characteristics of real curves, Lagrangian and Legendrian embeddings, and refined enumerative invariants. Additionally, the session will address recent developments in the Gluck twisting problem on the 4-sphere.

{{:events:2026:26rstropicalandgt.jpeg|}}

=== List of Participants ===

|Selman Akbulut (GGTI)| Sergey Finashin (METU) | Vassily Gorbounov (HSE U)|
|Ilia Itenberg (Sorbonne U) | Ernesto Lupercio (Cinvestav) |Grigory Mikhalkin (Geneva U) |
|Stepan Orevkov (IMT UPS) | Mikhail Shkolnikov (IMI BAS) |Eylem Z Yıldız (GGTI & Muğla U)|
| | Ferihe Atalan (Atılım U) | |


=== List of Talks ===
^Speaker    ^ Title and Abstract ^
|Sergey Finashin | **Wall-crossing invariant counting of real lines and other rational curves on rational surfaces** \\ This is a joint project with V.Kharlamov. Our “integral sign counting” of real rational real algebraic curves (combination of Welschinger invariants) produces a bivariant sequence of wall-crossing invariants, which are defined using real conic bundle structures and Pin structures on surfaces, but eventually are independent not only of their choices, but also of the real structure on a rational surface. I gave a historical motivating introduction tracing the subject from the classical count of lines on real cubic surfaces.|
|Ilia Itenberg | **Two rules of signs in enumeration of real curves**\\ The main goal of the talk is to relate two different Welschinger-type signs that were used for definition of real enumerative invariants of toric surfaces (the invariants considered being relative to the toric boundary). The relation between the two signs intertwines Grigory Mikhalkin's quantum index and topology of real and complex point sets of counted real curves. (Joint work with Eugenii Shustin.)|
|Grigory Mikhalkin | **Real θ-characteristics on algebraic curves **\\ In the talk we looked at an explicit description of the quadratic form corresponding to a real θ-characteristic on an algebraic curve (via the corresponding spin-structure). We also considered some applications of this new description, and reviewed some earlier results on real θ-characteristics of algebraic curves.|
|Stepan Orevkov | **On caustics of an elliptic billiard and geodesic flow on ellipsoid**\\ The $n$-th caustic of the geodesic flow emanated from a point $p$ on a Riemannian manifold is the set of the $n$-th conjugate points of the geodesics emanated from $p$. The "Last geometric statement of Jacobi" (proven by Itoh and Kiyohara in 2004) states that the first caustic of a geodesic flow on an ellipsoid emanated from a non-umbilic point has exactly four cusps. In this talk, I showed that the $n$-th caustic also has exactly four cusps (joint result with Gil Bor). I also showed that the $n$-th caustic of the family of billiard trajectories on an ellipse emanated from a non-focal point has exactly four cusps for any $n$. The fact that they have at least four cusps was previously proved by Bor, Spivakovsky, and Tabachnikov.| 
| Mikhail Shkolnikov | **Monge–Ampère equation as a scaling limit of many-point tropical relaxation** \\ I reviewed the basics of tropical optics, focusing in particular on various ways to relax unstable configurations of tropical series being smooth at perturbation points. We spent some time in dimension one, where apart from the probability one scaling limit of relaxations we saw what are the fluctuations near the limit being described as a Gaussian field defined on the segment. In dimension two, I sketched the singularity theoretic mechanism for the emergence of $SL(2,R)$ symmetry through the macroscopic observable being described as a solution of Monge–Ampère equation with Dirichlet boundary condition and the source being a density of the perturbation profile supported in the interior of the planar convex domain.|  
| | **New results on tropical zeta functions** \\ In this continuation of the previous year's discussion, I gave the updates about new results on residues of tropical zeta functions of convex domains and their implications for various geometric and number theoretic asymptotic problems, following the joint preprint with Kalinin and Lupercio -- in particular, we went again through the definition and the symplectic blow up formula allowing to write planar tropical zetas as Dirichlet generating function of the sizes of exceptional divisors in the infinite chain resulting in a given domain. In addition, I spent some time discussing the newest results which are not publicly available yet, such as the universal 1/2 residue for $C^3$ smooth domains and third term correction in the perimeter of tropical wave fronts of the Bárány-Vershik-Sinai domain at small times.|
|Eylem Zeliha Yıldız | **Gluck twisting 2-sphere in the 4-sphere**\\ We reviewed recent results concerning Gluck twist of the 4-sphere along smoothly embedded 2-spheres. We then presented a proof that if a smoothly embedded 2-sphere in $S^4$ has exactly two local minima, then the 4-manifold obtained by performing a Gluck twist along this sphere admits a handle decomposition with no 3-handles. This is part of an ongoing joint project with Selman Akbulut.|



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**Organizing Committee**: [[sfinashin@gmail.com|Sergey Finashin]] [[ilia.itenberg@imj-prg.fr|Ilia Itenberg]] [[grigory.mikhalkin@unige.ch|Grigory Mikhalkin]] and [[eylemzeliha@gmail.com|Eylem Zeliha Yıldız]].
Gökova Geometry Topology Institute

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