Tensor products in non-commutative geometry and higher deformation theory

Algebraic Geometry is a mathematical discipline based on a symbiotic two-directional dictionary between the fields of Algebra and Geometry. Roughly, it is a dictionary from equations (algebra) to geometrical figures (geometry) and vice versa. For example, given the equation y=x^2, we can draw its corresponding figure, in this case a parabola. We can add a third language to that dictionary, which is the highly abstract field of category theory. To each equation (or figure) we can associate a category, and given the category, we can recover its equations or figure. This dictionary is very useful when we work in commutative algebra, where the multiplication of our equations is commutative. But there exist algebraic structures were the multiplication is no longer commutative with the issue that "drawing" is no longer possible. However the dictionary algebra-category theory is still available. In algebra there is an operation called tensor product, which corresponds to taking the product of geometrical figures in an appropriate sense. In previous research we introduced a tensor product at the level of categories, in order to translate the algebraic operation to the categorical language. In this project we want to analyse further this tensor product of categories and use it to try to understand how the deformation of geometrical figures (both in commutative and "not-drawable" non-commutative directions) behaves.

Keywords:NON-COMMUTATIVE GEOMETRY

Disciplines:Analysis, Applied mathematics in specific fields, General mathematics, History and foundations, Other mathematical sciences and statistics

Project type:Collaboration project