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Workshop "Real geometry in the footsteps of Gabriel Cramer"
Monday, October 28, 16:30, Battelle, Ilia Itenberg, “Planes in four-dimensional cubics”.
Tuesday, October 29, 11:45, Battelle, Nikita Kalinin, Talk 1
Tuesday, October 29,14:45, Battelle, Kristin Shaw, Minicourse
Wednesday, October 30, 10:30, Battelle, Kristin Shaw, Minicourse
Thursday, October 21, 11:00,Battelle, Kristin Shaw, Minicourse
Thursday, October 21,14:30, Battelle, Nikita Kalinin, Talk 2.
Abstracts:
Kristin Shaw, 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
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 Based on (unpublished) survey of Felix Schlink.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).