Riemann, Topology, and Physics

Abstract

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Topological Formations in Chiral Nematic Droplets

“…The sum of the winding numbers of defects of a director field with non-zero tangential components on a closed surface is a conserved quantity: the Poincaré-Hopf theorem states that it is equal to the Euler characteristic of the surface, which can be calculated from Eq. (2.15) [26,31]. This means that on a spherical particle with planar anchoring the defects must have a total winding number equal to 2 and on a torus-like particle with planar anchoring, the sum of the winding numbers of defects must be 0.…”

Phase diagram of SALR fluids on spherical surfaces

“…In the case of cluster and bubble phases, topological disclination defects can be understood as lattice points having a defective number of nearest neighbors 41 , whereas stripe patterns also contain two disclination defects located at the poles of the sphere 42 . While the variety of geometries encountered in the phase diagram manifest these topological defects in different ways, all obey the same conservation law which states that 43…”

La dialéctica de la totalidad y la mereo-topología de la teoría de conjuntos. Roy Alfaro Vargas

“…En términos matemáticos, la topología (también llamada analysis situs) tiene como objeto las características que no cambian a través de diversas transformaciones (Monastyrsky, 2008). De este modo, en el mismo sentido que "La topología puede arrojar nuevas perspectivas dentro de la investigación de objetos familiares de las ciencias sociales, por medio de proyectar cómo tales objetos cambian y cómo se relacionan" (Shields, 2012, p. 48), así la topología permite la posibilidad "de concebir la esfera social en términos de más altos órdenes de una estructura abstracta" (Phillips, 2013, p. 135).…”