# Workshop "Real geometry in the footsteps of Gabriel Cramer"

** Geneva, 2019 , October 28th - November 1st, Villa Battelle **

Monday, October 28, 16:30, Battelle, Ilia Itenberg, “Planes in four-dimensional cubics”.

Tuesday, October 29, 11:45, Battelle, Nikita Kalinin, “Symplectic packing problem”.

Tuesday, October 29, 14:45, Battelle, Kristin Shaw, “Poincaré duality for tropical manifolds”.

Wednesday, October 30, 10:30, Battelle, Kristin Shaw, “Poincaré duality for tropical manifolds”.

Thursday, October 31, 14:30,Battelle, Kristin Shaw, “Poincaré duality for tropical manifolds”.

Thursday, October 31, 16:00, Battelle, Nikita Kalinin,“Symplectic packing problem and Nagata's conjecture”.

Friday, November 1, 15:00, Battelle, Nikita Kalinin, “Symplectic packing problem and Nagata's conjecture”.

**Abstracts:**

Ilia Itenberg (Sorbonne University)

** Planes in four-dimensional cubics **

We discuss possible numbers of 2-planes in a smooth cubic hypersurface in the 5-dimensional projective space. We show that, in the complex case, the maximal number of planes is 405, the maximum being realized by the Fermat cubic. In the real case, the maximal number of planes is 357.

The proofs deal with the period spaces of cubic hypersurfaces in the 5-dimensional complex projective space and are based on the global Torelli theorem and the surjectivity of the period map for these hypersurfaces, as well as on Nikulin's theory of discriminant forms.

Joint work with Alex Degtyarev and John Christian Ottem.

Kristin Shaw (University of Oslo)

Minicourse: **Poincaré duality for tropical manifolds**

The series of lectures will focus on the different formulations and approaches to Poincaré duality for the tropical homology group of tropical manifolds. Tropical homology is the homology of certain sheaves on polyhedral spaces and was introduced by Itenberg, Katzarkov, Mikhalkin, and Zharkov. The first formulation of tropical Poincaré duality was in terms of a non-degenerate pairing between compactly supported and usual tropical cohomology by Jell, Shaw, and Smacka. This pairing was formulated via the integration of superforms in the sense of Lagerberg. Tropical manifolds also satisfy a version of Poincaré duality for tropical (co)homology with integral coefficients by Jell, Rau, and Shaw. This version is formulated in terms of the cap product with the fundamental class. Another recent approach by Gross and Shokreih is via Verdier duality in the derived category and removes the assumption on the existence of a suitable covering of the tropical manifolds required in the two formulations above.

The proof of these duality statements in all three cases boils down to a local version of Poincaré duality for the tropical (co)homology of matroidal fans which is proved by using the deletion and contraction operations. The matroid property of a fan is sufficient but not necessary for tropical Poincaré duality. I will point out some partial results of Edvard Aksnes on necessary and sufficient conditions for tropical Poincaré duality for fans, and also highlight some consequences of Poincaré duality in the global case.

Nikita Kalinin (HSE University)

Talk 1: **Symplectic packing problem.**
This will be an introductory talk. I will mention several instances of symplectic packing problems and present simple geometric methods for tackling them. Based on (unpublished) survey of Felix Schlink.

Talk 2: **Symplectic packing problem and Nagata’s conjecture.**
Curiously, the question of what is the maximal R such that we can embed k<10 symplectic balls of radius R in CP^2 is related to the question of what is the minimal degree d of an algebraic curve in CP^2 passing through given generic points with given multiplicities. Nagata’s conjecture (still open for all n>10 except squares) states that d>m\sqrt n if we draw a curve through n points of multiplicity m. I will highlight the connections between symplectic packing and Nagata’s conjecture (based on works of McDuff, Polterovich, Biran, among others).