The main objective of the proposal is the creation, and development of a cooperative research network which utilizes the strengths and synergies of the knowledge of the member research groups. This new cooperation symbolizes the coercive power of two branches of mathematics, namely those of ALGEBRA and GEOMETRY, which had been unified first in the creation Decartesian coordinate geometry. 3 of 6 research groups are on the edge of algebraic research, while the others gained essential results and knowledge in geometry. With different backgrounds, new synergies and methodologies will arise and accelerate the research activities.Besides the traditional mobility schemes and distributing ideas on conferences and publications, new methodology of continuous reaction is planned to put in practice by the usage of world wide web, creating the platform of online web workshops at regular times. This will ensure sustainability of the network for long time.We plan to achieve new scientific results on the following topics:• imprimitive transformation groups, affine geometries over paradual near rings;• fundamental theorem of geometric algebra, Novikov’s conjecture and the properties of skew-symmetric and symmetric elements for general involutions in group algebras.• multiplication loops of locally compact topological translation planes; Lie groups which are the groups topologically generated by all left and right translations of topological loops;• the inverse problem of the calculus of variations for second order ordinary differential equations: existence of variational multipliers, in particular, of multipliers satisfying the Finsler homogeneity conditions, and Riemannian and Finsler metrizability;• metric structures associated with Lagrangians and Finsler functions• variational structures in Finsler geometry and applications in physics (general relativity, Feynmam integral);• Hamiltonian structures for homogeneous Lagrangians.