The project deals with the fundamental research in differential geometry and its applications in analytical mechanics and mathematical physics. The principal attention will be paid to the Weil algebra technique in differential geometry of fiberedmanifold s. Further, the following topics will be dealt with: the objects of Weilian type on certain functional bundles, relations to the geometry of variational calculus, natural bundles over fibered manifolds, prolongation of connections, fiber productpreservin g bundle functors of the jet type, exchange natural transformations of cotangent and Weil functors. Further types of natural differential operators will be studied. In the area of applications of differential geometry in mathematical physics,the geometri c problems related with covariant classical and quantum mechanics will be studied. In particular, concrete natural operators in Galilei and Einstein quantum mechanics will be investigated and the gauge version of the reduction method for

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