Periodic Reporting for period 3 - FHiCuNCAG (Foundations for Higher and Curved Noncommutative Algebraic Geometry)
Reporting period: 2022-06-01 to 2023-11-30
The central goal of this project is to bridge between noncommutative algebraic geometry on the one hand and algebraic topology on the other hand, by laying new foundations for higher and curved noncommutative geometry. As a project in fundamental mathematics, societal impact should be seen in the long term, in the advancement of science as a whole. Impact within mathematics is situated primarily in the fields of algebra, geometry and topology. The project addresses central problems in deformation theory like Deligne’s Conjecture, and has important links with theoretical physics, in particular quantum and string theory. In one direction, ideas from topology enhance our geometrical understanding, like in the study of higher linear topoi. In the other direction, geometrical concepts propel the development of new operadic tools in topology, like in the cohomological investigation of prestacks as generalised structure sheaves of spaces.
Next we describe the main progress by the team members and their collaborators.
In Van den Bergh’s categorical approach to geometry, categories represent spaces in the absence of actual points. The result is sometimes called a noncommutative space, because it typically involves noncommutative algebra. In the first period of the project, an important focus has been on the development of geometrical tools for large categories. We built a suitable analogue for the product of spaces, called the tensor product. We also set up the right derived framework in order to do deformation theory, which deals with nearby structures, and allows ordinary spaces to get deformed into noncommutative ones. We came up with a new proof of Deligne’s Conjecture for suitable monoidal categories, opening up new possibilities for application. As an instance of direction two above, we established the higher structure governing deformations of prestacks, making use of certain patchworks of rectangles (see picture0). A starting point for several objectives is the famous Gabriel-Popescu theorem, which identifies linear topoi as ideal large categories to do geometry. Our team has taken this result to the derived level, appropriate for deformation theory. As an instance of direction one above, we established the new framework of templicial modules, which are linear analogues of simplicial sets (see picture1).
In Van den Bergh’s categorical approach to geometry, categories represent spaces in the absence of actual points. The result is sometimes called a noncommutative space, because it typically involves noncommutative algebra. In the first period of the project, an important focus has been on the development of geometrical tools for large categories. We built a suitable analogue for the product of spaces, called the tensor product. We also set up the right derived framework in order to do deformation theory, which deals with nearby structures, and allows ordinary spaces to get deformed into noncommutative ones. We came up with a new proof of Deligne’s Conjecture for suitable monoidal categories, opening up new possibilities for application. As an instance of direction two above, we established the higher structure governing deformations of prestacks, making use of certain patchworks of rectangles (see picture0). A starting point for several objectives is the famous Gabriel-Popescu theorem, which identifies linear topoi as ideal large categories to do geometry. Our team has taken this result to the derived level, appropriate for deformation theory. As an instance of direction one above, we established the new framework of templicial modules, which are linear analogues of simplicial sets (see picture1).
In the second period of the project, the newly introduced templicial modules will be used in order to develop higher linear topos theory with similar desirable features as warranted by the Gabriel-Popescu theorem above. We will employ some of our newly developed computational tools to bring Mac Lane cohomology to algebraic geometry, and establish a non-linear deformation theory. Finally, we will establish new foundations in order to deal with the mysterious phenomenon of curvature in algebraic deformation theory.