Several areas of theoretical and mathematical physics are addressed. The physical applications are broadly ranging, although the techniques are mainly those of statistical and quantum mechanics. The quantization of discrete breathers is a priority. What began some years ago as an unexplained anomaly in the decay of luminescence in doped alkali halides, has blossomed into a general study of nonlinear interactions (related to solitons) both at the classical and quantum levels. The particular quantum issue that the PI expects to explore in yet greater depth, is the stability of these breathers against quantum decay. This has been studied both through numerical diagonalization in a phonon basis and through the use of the Feynman path integral. For the latter, the PI has exploited one of the most basic techniques of the path integral, the elimination of quadratic degrees of freedom (as Feynman did for the polaron), although once that was done, particular methods needed to be developed for this application. In work on nonequilibrium statistical mechanics new results on the relation between phase transitions and the eigenvectors of the transition matrix (for a stochastic process) have been found. A particular geometric construction, based on a limited number of eigenvectors, can be used to compute the probability that an arbitrary point in the state space reaches any particular basin of attraction. It can also be used to visualize the structure of metastable phases, particularly when they bear a hierarchical relation to one another (as is believed to obtain for spin glasses). As far as is known, this construction is new. Other work on nonequilibrium systems, for example, investigating the ways that reservoirs can induce complexity in open systems, is also planned. In fundamental areas of physics the PI will continue to explore time-related issues. Some are related to his 'opposite arrows' work of a few years ago, others deal with quantum transitions and experimental tests of theories of quantum measurement. A recent result of the PI on the equilibration of wave-packet spread in an interacting system, leads to general questions of whether the von Neumann entropy maximization that lies behind that result can also be invoked to established unanticipated low levels of entanglement with respect to other degrees of freedom. Besides its scientific content, this work has broad impact in two principal ways: public education on foundational issues (emanating from the PI's recent 'arrow of time' exposure), and on another front, unique cultural experiences for the several Clarkson undergraduates who have done, and will do, research at the optical crystals laboratory of the PI's collaborators in Prague.
