Homological Algebra, Its Non Abelian and Categorical Topics: Applications to Homotopy Theory, KTheory, Algebraic Geometry and Galois Theory Project #GM1115 The project was devoted to the study of homological and homotopical properties of algebraic structures using methods and techniques of simplicial , combinatorial and categorical algebra with applications to important fields of mathematics. The following results were obtained. Quillen's algebraic Ktheory was extended to the category of normed algebras over commutative Banach rings and its relationship with topological Ktheory was established. Sufficient conditions were given for the isomorphism of algebraic and topological Kgroups. New descriptions of non abelian homology groups in terms of derived functors were given. Sufficient conditions were established for non abelian homology of groups to be finite, finitely generated, pgroups and torsion groups. Some calculation formulas for lower homology groups were obtained. Graded automorphisms of polytopal semigroup rings were described in terms of toric and elementary automorphisms and symmetries.  
