2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/129521We survey the history of totally positive matrices and the generalization to Lie groups. We describe a reduction of a bilinear form to a canonical form (generalizing the case of symplectic nondegenerate forms) using ideas from total positivity; we also place this in a Lie group context. We give a short exposition of results of Fock and Goncharov on the study of homomorphisms of the fundamental group of a closed surface into a Lie group.8 pagesRepresentation Theorytext