Noncommutative Algebraic Topology

Algebraic Topology is the study of spaces using algebraic methods. By a basic duality principle, spaces correspond to commutative algebras. In Noncommutative Geometry, one studies noncommutative spaces that underlie noncommutative algebras. Such algebras show up naturally in a plethora of situations, for instance, when one consider symmetries or deformations of spaces.The study of noncommutative spaces is not only interesting in itself, but also has important applications to various other branches of mathematical and physical sciences. For instance, analysis of the topological K-theory of noncommutative spaces underlying discrete groups allows one to deduce results in pure algebra, topology and geometry. Moreover, in some case, the best results are accessible only through noncommutative geometric methods. In mathematical physics, noncommutative (operator) algebras arise as algebra of observes. In fact, this is how noncommutative geometry was initiated by von Neumann in the first place. Some current research in theoretical physics focus on investigating the space-time as a noncommutative space.My proposal concerns the study of noncommutative algebraic topology, more precisely, the extension of algebraic topological methods to noncommutative geometry. This involves rewriting homotopy theory for noncommutative spaces in terms of modern algebraic topological machinery and making systematic use of various standard techniques such as completion and localization. More concretely, my research proposal consists of four interconnected projects that focus on homotopical algebra, algebraic and connective K-theories for noncommutative spaces and Banach KK-theory, respectively.The research fellowship is to be held at Cardiff University, because it has a strong tradition of Noncommutative Geometry and Mathematical Physics. Also Cardiff University is the largest partner of Wales Institute of Mathematical and Computational Sciences and leader of its Mathematical Physics cluster, whose members include the prestigious string theory group at Swansea and quantum control group at Aberystwyth. In order both to promote successful research and discuss future directions with experts in the relevant fields, I plan to make scientific trips to Muenster University and University of Sheffield.

As a research in pure mathematics, my research will mostly have academic impact. As can be seen from how number theory is used in modern cryptography or how Fourier analysis is used in signal processing, research in pure mathematics could have profound social and economic impact. However, such applications depend on developments in the future and cannot be predicted. Therefore I will concentrate on the academic impact of my research. My research will advance the field of noncommutative geometry by adding new results and, maybe more importantly, new techniques, enhancing the knowledge economy. By making systematic use of algebraic topological methods, I hope to introduce fresh ideas and new perspectives to old problems in noncommutative geometry. At the same time, this will enable me to offer new and interesting problems to algebraic topologists, that are based on examples coming from operator algebras. Furthermore, new developments in noncommutative geometry should result in developments in topology and geometry, extending already known connections. Hence my proposed research should enhance the synergy between algebraic topology and noncommutative geometry, contributing towards the health of the disciplines.