Title: Singular traces in noncommutative geometry and analysis

Abstract: In the present talk, I survey the theory of singular traces and its applications to noncommutative geometry and analysis. In the first part of the talk, I outline J. Dixmier's original construction of singular traces, their employment in A. Connes' noncommutative geometry and recent advances in the theory. In the second part, I discuss the applications of singular traces to the perturbation theory.

Brief description of the Speaker: Alex Usachev's research lies at the intersection of functional analysis and noncommutative geometry, with a particular emphasis on spectral theory and that of singular traces. His work focuses on developing various approaches to the construction of noncommutative integrals, allowing their computation for wide classes of operators, e.g., pseudodifferential, commutators with multiplication operators, and Hankel operators.