Periodic Reporting for period 1 - nalimdif (Non-Archimedean limits of differential forms, Gromov-Hausdorff limits and essential skeleta)
Reporting period: 2019-10-01 to 2021-09-30
In the mid-2000s two approaches to the SYZ conjecture --- metric and non-archimedean --- were put forward by Kontsevich and Soibelman. Each approach proposed certain mathematical construction that would take a family of Calabi-Yau manifolds of dimension 2n and produce an n-dimensional sphere ("base of the SYZ fibration") endowed with additional mathematical structures. The structures produced by the two approaches are different, but are both underlied by a common geometric strucuters called singular integral affine structure. Both approaches have been extensively studied each on its own, but the relationship between the two remains mysterious. Konstevich and Soibelman conjectured that the singular affine structures arising from both approaches should be related in a certain way. The aim of this project is to develop mathematical tools that would allow to describe this relationship and study the interaction of mathematical structures in both approaches via the singular integral affine structure that underlies them. Development of these tools will have impact on several fields of pure mathematics that intersect in the mathematical treatment of mirror symmetry.
These theorems were applied to a particular class of families of Calabi-Yau manifolds called Kulikov degenerations, some new results about their properties were obtained. These results were reported to experts at scientific seminars, and published on an open publication pre-print server.
The technical methods underlying these results are closely related to recently developed theory of tropical homology. As a part of the project an international summer school for Masters and early PhD students on the subject of tropical homology was held (online due to COVID pandemic).