PhD in Mathematics- Topology and Geometry in Denver United States | University of Denver

Topology examines the concept of continuity, from its fundamental definition to its use in analyzing spatial structures. Geometry extends topology by incorporating analysis and algebra to investigate the characteristics of forms and spatial dimensions.

Noncommutative geometry applies geometric methods to analyze noncommutative algebras, drawing inspiration from mathematical physics, group representation theory, and the examination of singular spaces in differential geometry. Our work centers on noncommutative metric geometry, where we explore quantum metric spaces—noncommutative extensions of Lipschitz function algebras over metric spaces. The goal is to create a geometric framework for analyzing quantum metric spaces emerging from diverse areas like mathematical physics, dynamical systems, and differential geometry. A crucial element of this approach is extending the Gromov-Hausdorff distance to noncommutative settings, allowing us to probe the topological and geometric properties of quantum metric space classes. This enables us to develop finite-dimensional approximations for C*-algebras, demonstrate the continuity of various quantum metric space families and related structures, and address problems in mathematical physics and C*-algebra theory through a metric geometry lens.