Asymptotic and Spectral Analysis of Applied Non-self-adjoint Problems

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The objective of this project is the development of mathematical tools for studying several problems emerging from physics and modern mathematical biology. The first two problems concern propagation of waves in non-homogeneous media and optical fibers. The results can be used for detection of hidden objects, nondestructive testing and estimation of the quality of signal transmission in the noisy environment. Further, a mathematical model will be developed for molecular motors used by biophysicists to describe processes in living cells. The model will justify certain hypotheses in the area and will describe quantitatively the parameters for which the hypotheses hold. The last problem concerns the mathematical analysis of modern models of population dynamics aiming to explain the mechanism for non-local and fast propagation of new species. The existence of stable populations with competition between species will be also explained.On a more technical level, the project includes problems in homogenization, spectral theory of non-standard Hamiltonians, and singularly perturbed partial differential equations needed to address the applied problems mentioned above. In particular, the following results will be obtained. The Weyl law for the counting function of eigenvalues of (non-self-adjoint) interior transmission problem will be established. Localization theorem will be proved for waveguides with randomly perturbed surfaces. The homogenization principle will be justified for molecular motors. Front propagation will be studied for general contact processes.