Periodic Reporting for period 5 - FHiCuNCAG (Foundations for Higher and Curved Noncommutative Algebraic Geometry)
Reporting period: 2024-12-01 to 2025-09-30
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 with links to theoretical physics, 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. In one direction, ideas from topology enhance our geometrical understanding, for instance in the study of higher linear categories (see (3) below). In the other direction, geometrical concepts propel the development of new operadic tools in topology, like our box operads inspired by prestacks (see (1) below). The overall objective of the project is to solve open problems in deformation theory in synergy with the development of new machinery.
Next we describe the main outcomes of the project.
In the categorical approach to geometry put forth by Van den Bergh, Kontsevich, Kaledin and others, categories represent spaces in the absence of actual points. The result is sometimes called a noncommutative space, because it typically involves noncommutative algebra. On the one hand, quantum mechanics can famously be described through noncommutative “deformation quantisation” of classical mechanics. On the other hand, the
categorical structures studied in our project occur as models in string theory, and our goal is precisely to study their deformation theory, or, in physical terms, how they can be quantised. Our main results involve several particular categorical models, and can be summarised as follows:
(1) Prestacks are structures in which the local geometry of a space can be organised algebraically. Joint with Hoang Dinh Van and Lander Hermans, we established the complete higher combinatorial structure on their deformation complexes, a question that has been under investigation in particular cases since the 1980s. Our solution makes use of certain patchworks of rectangles (see picture0).
(2) The most widely studied and applicable contemporary model for noncommutative spaces is that of differential graded categories. However, their deformation theory faces the “curvature problem” (Keller - Lowen, 2009; Lurie, 2011) by which not only points, but also sheaves may disappear from the classical picture. Joint with Alessandro Lehmann, we completely solved this problem for first order deformations, hereby introducing a novel conception of “categorical deformation”.
(3) Joint with Arne Mertens, we proposed the candidate new model of “quasi-categories in modules” to overcome certain difficulties in the homotopy theory of differential graded categories. To this end we established the novel framework of templicial modules, which are linear analogues of simplicial sets (see picture1). Joint with Violeta Borges Marques and Arne Mertens, we initiated the deformation theory of these objects.
In the categorical approach to geometry put forth by Van den Bergh, Kontsevich, Kaledin and others, categories represent spaces in the absence of actual points. The result is sometimes called a noncommutative space, because it typically involves noncommutative algebra. On the one hand, quantum mechanics can famously be described through noncommutative “deformation quantisation” of classical mechanics. On the other hand, the
categorical structures studied in our project occur as models in string theory, and our goal is precisely to study their deformation theory, or, in physical terms, how they can be quantised. Our main results involve several particular categorical models, and can be summarised as follows:
(1) Prestacks are structures in which the local geometry of a space can be organised algebraically. Joint with Hoang Dinh Van and Lander Hermans, we established the complete higher combinatorial structure on their deformation complexes, a question that has been under investigation in particular cases since the 1980s. Our solution makes use of certain patchworks of rectangles (see picture0).
(2) The most widely studied and applicable contemporary model for noncommutative spaces is that of differential graded categories. However, their deformation theory faces the “curvature problem” (Keller - Lowen, 2009; Lurie, 2011) by which not only points, but also sheaves may disappear from the classical picture. Joint with Alessandro Lehmann, we completely solved this problem for first order deformations, hereby introducing a novel conception of “categorical deformation”.
(3) Joint with Arne Mertens, we proposed the candidate new model of “quasi-categories in modules” to overcome certain difficulties in the homotopy theory of differential graded categories. To this end we established the novel framework of templicial modules, which are linear analogues of simplicial sets (see picture1). Joint with Violeta Borges Marques and Arne Mertens, we initiated the deformation theory of these objects.
The results obtained in the project constitute major progress beyond the state of the art in noncommutative geometry and deformation theory. In particular, the results listed in (1), (2) and (3) above are breakthroughs in the field, solving open problems and establishing new frameworks with a high potential for further future applications.