Noncommutative geometry is the geometric approach to the study of noncommutative algebra, which finds its roots in mathematical physics, representation theory of groups, and the study of singular spaces from the world of differential geometry.
Our focus is primarily on noncommutative metric geometry, where we study quantum metric spaces, i.e. noncommutative generalizations of the algebra of Lipschitz functions over metric spaces.
Our purpose is to develop a geometric framework for the study of quantum metric spaces which arise from various fields, such as mathematical physics, dynamical systems, differential geometry and more. A key tool in this framework is a generalization of the Gromov-Hausdorff distance to the noncommutative realm, which enables the exploration of the topology and geometry of classes of quantum metric spaces. We thus become able to construct finite dimensional approximations for C*-algebra, establish the continuity of various families of quantum metric spaces and associated structures, and investigate questions from mathematical physics and C*-algebra theory from a new perspective inspired by metric geometry.
Noncommutative algebra studied by noncommutative geometers typically fit within the realm of functional analysis, i.e., the analysis of infinite dimensional topological vector spaces and related concepts. The techniques used in their study borrow from differential geometry, algebraic and differential topology, topological group theory, abstract harmonic analysis and metric geometry.