Gökova Geometry Topology Institute

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===== Research Session on Tropical and Symplectic Geometry =====
=====(Sep 27-Oct 7, 2023) =====

{{:events:2023:23rstropical.jpg|}}


This event will be held in person at Hotel Yucelen.


=== Invited Participants/Speakers ===

|Selman Akbulut (GGTI)|Alexander Degtyarev (Bilkent U)|Georgios Dimitroglu-Rizell (Uppsala U)|
|Tobias Ekholm (Uppsala U)|Sergey Finashin (METU)|Ilia Itenberg (Sorbonne U)|                
|Grigory Mikhalkin (Geneva U)|Mikhail Shkolnikov (ICMS-Sofia)|Umut Varolgunes (Koç U)|

**Organizing Committee**: [[degt@fen.bilkent.edu.tr|Alexander Degtyarev]] [[sfinashin@gmail.com|Sergey Finashin]] [[ilia.itenberg@imj-prg.fr|Ilia Itenberg]] [[g.m@cern.ch|Grigory Mikhalkin]] and [[eylem.yildiz@duke.edu|Eylem Zeliha Yıldız]].

=== List of Talks ===
^Speaker    ^ Title and Abstract ^
| Grigory Mikhalkin | **Tropical trigonometry and wave fronts**\\ The geometric structure in the tropical plane consists of the lattice of integer vectors in the tangent space. It is invariant with respect to translations and lattice automorphisms. In this geometry, any wave front becomes polygonal in an arbitrary small time. Caustics of the tropical wave fronts are closely related to tropical trigonometry, continued fractions, toric singularity resolutions, and even symplectic embedding problems. Based on joint work with Mikhail Shkolnikov.|
| Sergey Finashin | **Topology of real lines on del Pezzo and elliptic surfaces**\\ In our joint work with V.Kharlamov, we describe isotopy classes of real lines on real del Pezzo surfaces of degree 1 and use it to approach a similar task for real rational elliptic surface. In the latter case, studying of the action of real Mordell-Weil group is the key ingredient. In particular, I will discuss the difference between the algebraic and topological sections of real elliptic fibrations.|
| Selman Akbulut | **Using “corks" as tools to prove positive results in 4-manifolds**\\ Traditionally "corks" are used to prove negative results about 4-manifolds; such as building exotic smooth 4-manifold pairs, or constructing exotic concordances between properly-imbedded smooth disk pairs $D^{2} \subset B^{4}$. In this talk, I will show how to use corks in proving positive results about $4$-manifolds. For example, in showing "Shake slice knot is a slice knot” (joint work with Eylem Yildiz), or towards verifying the 4-dimensional smooth Poincare conjecture. Some useful "carving" and "cork" techniques will be discussed in a very informal way.|
| Alex Degtyarev | **Singular real plane sextic curves without real points**\\ It is a common understanding that any reasonable geometric question about $K3$-surfaces can be restated and solved in purely arithmetical terms, by means of an appropriately defined //homological type//. For example, this works well in the study of //singular complex// sextic curves in $\mathbb P^2$ or quartic surfaces in $\mathbb P^3$, as well as in that of //smooth real// ones. However, when the two are combined (both //singular// and //real// curves or surfaces), the approach fails as the "obvious" concept of homological type does not fully reflect the geometry. \\ \\ We show that the situation can be repaired if the curves in question have empty real part or, more generally, have no real singular points; then, one can indeed confine oneself to the homological types consisting of the exceptional divisors, polarization, and real structure.\\ \\ Still, the resulting arithmetical problem is not quite straightforward, but we manage to solve it and obtain a satisfactory classification in the case of //empty// real part; it matches all known results obtained by an alternative purely geometric approach. In the general case of //smooth// real part, we also have a formal classification however, establishing a correspondence between arithmetic and geometric invariants (most notably, the distribution of ovals among the components of a reducible curve) still needs a certain amount of work. This is a joint work with Ilia Itenberg|
| Georgios Dimitroglou-Rizell | **Refined potentials and mutation for conical Lagrangian singularities**\\ We describe computations of refined potential functions associated to disc counts on Lagrangians that exhibit different types of conical singularities. We are particularly interested in the case when the singularities are links given by a Legendrian and its Reeb push-off, which are Lagrangians that admit natural smoothings. The mutations from the SYZ formulation of mirror symmetry is a particular case of such deformations. The talk is partly based upon previous work by the author and Ekholm-Tonkonog and ongoing work with Ghiggini.|   
| Mikhail Shkolnikov | **Tropical wave fronts and caustics of non-convex domain**\\ I will start by giving an overview of the current setup of tropical wave fronts and their caustics developed in joint work with Grigory Mikhalkin, which is currently confined to convex domains on the plane. Extending it to higher dimensions is straightforward, however, going beyond convexity is a challenge that requires a different approach. I will outline three of them, one notably making use of folded symplectic geometry.|  
| Umut Varolgunes | **Zariski descent and deformations**\\ I will prove a Floer theoretic local-to-global theorem for certain covers of Liouville manifolds and discuss potential applications.|       
| Tobias Ekholm | **Skein valued recursion**\\ We show how the Symplectic Field Theory perspective on skein valued curve counts of Lagrangians in open Calabi-Yau $3$-folds naturally give rise to skein valued recursions equations that are lifts of quantum curves which arises as the $U(1)$-skein and augmentation varieties which is semi-classical version of the quantum curve.|
| Ilia Itenberg| **Refined invariants for real elliptic curves and real curves of genus $2$**\\ We suggest new invariants of the projective plane (and, more generally, of certain toric surfaces) that arise from appropriate enumeration of real curves of genus 1 and 2. These invariants admit a refinement (according to the quantum index) similar to the one introduced by Grigory Mikhalkin in the rational case. We discuss the combinatorics of tropical counterparts of the invariants under consideration and establish a tropical algorithm allowing one to compute them.\\ \\ The resulting tropical invariants can be interpreted via a suitable enumeration of separating real tropical curves. This is a joint work with Eugenii Shustin| 


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=== Summary Report ===
 
The research session was focused on several active and promising research directions in Algebraic, Symplectic and Tropical Geometry as well as Geometric Topology. In enumerative geometry the topics included complex and real enumerative problems beyond Del Pezzo surfaces (when the anticanonical class is no longer nef), as well as real enumerative problems in positive genus, with a special emphasis on eliptic curves and their refined enumeration. Refined invariants were also considered from the viewpoint of Symplectic Geometry, in particular, through Skein module approach, as well as through singular Lagrangians. In tropical geometry the principal topic was evolution of tropical wave fronts.

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Gökova Geometry Topology Institute

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