Michel Planat, Marcelo M. Amaral, David Chester, Klee Irwin (2022)

Revealing the time structure of physical or biological objects is usually performed thanks to the tools of signal processing like the fast Fourier transform, Ramanujan sum signal processing and many other techniques. For space-time topological objects in physics and biology, we propose a a type of algebraic processing based on schemes in which the discrimination of singularities within objects is based on the space-time-spin group SL(2, C). Such topological objects possess an homotopy structure encoded in their fundamental group and the related SL(2, C) multivariate polynomial character variety contains a plethora of singularities somehow analogous to the frequency spectrum in time structures. Our approach is applied to an Akbulut cork in exotic R4, to an hyperbolic model of topological quantum computing based on algebraic surfaces and to microRNAs in genetics. Such diverse topics reveal the manifold of possibilities of using the concept of a scheme spectrum. Keywords: Finitely generated group; SL(2, C) character variety; algebraic surfaces; schemes; exotic R4; topological quantum computing; microRNAs.