C*-algebras (and sometimes more general topological algebras) are basic pointed compact noncommutative spaces, whose study has led to striking applications to topology, geometry, and mathematical physics. Currently my research focusses on generalized (co)homology theories and other higher invariants of noncommutative spaces that are not necessarily compact (but compactly generated from an ∞-categorical perspective). Until the appearance of noncommutative spectra only a handful of them were known, namely, K-theory, local cyclic homology, and a few more. The stable ∞-category of noncommutative spectra parametrizes in a suitable sense all generalized (co)homology theories of noncommutative spaces. The global structure of this stable ∞-category has yet to be thoroughly explored. This viewpoint provides a conceptual understanding of the existing theories and creates (hopefully) a synergy between operator algebras, topology, and higher category theory.
Understanding the common features as well as the differences between various objects, theories, and even points-of-view has helped in accomplishing major advances in science. Specialising this to geometry or topology one can ask when are two given spaces the same or, if they are different, then is there a simple way to ascertain that. The spaces of interest arise from diverse sources, like (but not limited to) physics and number theory. It is extremely beneficial to have effectively computable invariants that can tell them apart. Typical examples are furnished by Hamiltonian systems in mechanics or spacetimes in string theories. It must be noted that the set of all integers or complex numbers also gives rise to a perfectly legitimate geometric object of study. In the same vein the set of 2x2 matrices over the integers or complex numbers produces a space, which is noncommutative in a certain sense. There is a sophisticated and adequate framework to treat all such spaces on an equal footing from the perspective of geometry and topology, where K-theory and cyclic homology serve as some basic invariants that can be used to differentiate them. Neither K-theory nor cyclic homology can tell the difference between the set of complex numbers and 2x2 matrices over them, but there are more intricate invariants that can be deployed for this purpose. Of course, in practice such spaces appear in a disguised form and often these invariants (coupled with some highly non-trivial mathematical results) are used to recognise them and tell them apart. I am mostly interested in problems of this nature and developing some technical tools to investigate such issues.