Derived geometry and arithmetic
Final Report Abstract
My DFG postdoctoral project is a project of pure Mathematics. Therefore, the outcome of my research has been new theorems of a quite abstract nature. These results do not have immediate economical exploitation, but they increase our understanding of some mathematical objects that are important in some branches of mathematics that lie at the intersection between arithmetic and geometry. My project took place at the Mathematical Institute of the University of Oxford and it has been performed mainly in collaboration with Professor Kremnizer. The main achievement I obtained during my fellowship, in collaboration with Professor Kremnizer, has been the solution of a 30-years old problem called the “sheafyness problem” for Banach rings. This is a technical issue of some geometric spaces used to study problems in arithmetic geometry, that is the field of Mathematics that investigates problems in number theory and arithmetic using methods and ideas from geometry. This problem prevents the application of geometric methods on objects of interest in this kind of research. We have shown that using the ideas of derived analytic geometry I described in my project proposal such a problem can be solved. Therefore, our new methods permit to study the geometry of objects that up to now were impossible to interpret geometrically. This opens up many new possibilities for a deeper understanding and study of such objects with many potential applications to arithmetic.
Publications
- “On the Sheafyness Property of Spectra of Banach Rings”, September 2020, preprint
F. Bambozzi, K. Kremnizer
(See online at https://doi.org/10.48550/arXiv.2009.13926) - “Derived Analytic Geometry for Z-Valued Functions. Part I – Topological Properties”, July 2021
F. Bambozzi, T. Mihara
(See online at https://doi.org/10.48550/arXiv.2107.09004) - “Homotopy Epimorphisms and Derived Tate’s Acyclicity for Commutative C ∗ -algebras”, March 2021
F. Bambozzi, T. Mihara
(See online at https://doi.org/10.48550/arXiv.2103.11722) - “On the uniqueness of invariant states”, Advances in Mathematics, Volume 376, 6 January 2021
F. Bambozzi, S. Murro
(See online at https://doi.org/10.1016/j.aim.2020.107445)
