The project lies at the intersection of operator algebras, operator theory and dynamicalsystems, with elements of experimental mathematics and applications to noncommutative structures,completely positive dynamics, spectral analysis of functional operators, and potentially to the theory offunctional-differential equations and quantum physics. It is centered around a development of aninnovative powerful mathematical apparatus based on a construction of objects (of both dynamical andcombinatorial nature) dual to semigroups of C*-correspondences over generically noncommutativealgebras. This will allow a detailed analysis of vast class of objects defined in terms of generators andrelations, inaccessible through the existing methods. This specifically concerns algebras arising recentlyin connection with number and ring theory [Cun08, Li10], thermodynamical phenomena [BC95, HLS12],quantum spaces [RS12], quantum field theory [DR89, Kwa13] and many more.The envisaged applicationsare broadly related to the analysis of quantum structures andevolutions of systems modeling complicated physical processes containing dynamical contributoryfactors as well as interaction with outer media, e.g. the process of motion and transformation of particles.The asymptotic and ergodic properties of such systems are described by spectral properties of theappropriate operators (which can be presented in the form (1) below) [CL99, ABL10]. I expect to achievea substantial contribution to the theory of the latter on the basis of the elaborated general results.
|Effective start/end date||01/10/2014 → 30/03/2016|