"Multi-Way factor Analysis (MWA) is attracting growing interest in many disciplines of engineering, as described in this proposal. Because the applications are much more numerous than those that serve as focus for this project, the tools developed in the framework of the project will have major impact. MWA is probably the simplest extension of the well-known (linear) Factor Analysis. However, despite its extremely wide panel of applications and its apparently simple expression, it still, surprisingly, lacks theoretical background. The reason is that this identification problem disguises challenges of unexpected magnitude. In fact, several tensor problems still remain open for several decades, and the difficulties should not be overlooked. Yet, the lack of identifiability results (existence, uniqueness) prevents the design of efficient numerical algorithms.The first objective is to address these theoretical problems and develop appropriate identification algorithms. Multilinear models underlying MWA are shown to be closely related to tensor algebra and multivariate polynomials, so that tools can be borrowed from Algebraic Geometry, with the goal of developing theoretical solutions and numerical algorithms. The second objective is to apply these solutions to practical problems in various realms of application. In particular, it centers on creating modified models, better matched to analysis of health (e.g., EEG) or environmental data (e.g., water resources, microbial ecosystems), to analyzing their identifiability, and to developing corresponding identification algorithms. The final goal is to design a device able to detect and track suspect or toxic molecules in river or tap waters.This very challenging research proposal is positioned in close collaboration with specialists in the above-mentioned application fields, and is at the core of European population health."