Title: Homotopic structures in algebra, geometry and mathematical physics Author: Eva Černohorská Department: Mathematical Institute of Charles University Supervisor: RNDr. Martin Markl, DrSc., Institute of Mathematics of the Academy of Sciences of the Czech Republic, Mathematical Institute of Charles University Abstract: The aim of this thesis was to generalize the result that associative algebras on finite dimensional vector spaces can be described using differentials on free algebras. This result is limited by the duality of vector spaces. If we assume that the underlying space has a linear topology, then we can use the duality between discrete and linearly compact (profinite) vector spaces. To generalize the notion of an algebra, we need to recall the completed tensor product on linear vector spaces. Since this topics does not seem to be sufficiently covered by the literature, this thesis could serve also as a comprehensive text on linear vector spaces and their completed tensor products. We prove that also A∞ structures on linearly compact vector spaces could be represented by differentials on a free algebra. Keywords: Strongly homotopy associative algebra, linear topological vector space, Pontryagin duality, completed tensor product, differential