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The paper is the survey of the modern results and applications of the theoryof homotopes. The notion of a well-tempered element in an associative algebrais introduced and it is proven that the category of representations of thehomotope constructed by a well-tempered element is the heart of a suitablyglued t-structure. Hochschild and global dimensions of the homotopes arecalculated. The homotopes constructed by generalized Lapalce operators inPoincare groupoids of graphs are studied. It is shown that they are quotientsof Temperley-Lieb algebras of general graphs. The perverse sheaves on apunctured disc and on a 2 dimensional sphere with a double point are identifiedwith representations of suitable homotopes. Relations of the theory toorthogonal decompositions of sl(n, C) into the sum of Cartan subalgebras, toclassifications of configurations of lines, to mutually unbiased bases, toquantum protocols, to generalized Hadamard matrices are discussed.

Theory of homotopes in application to mutually unbiased bases, harmonic analysis on graphs and perverse sheaves