Algunos fenómenos naturales, desde un enfoque teleológico, escogen trayectorias expeditas, e.g. la refracción de la luz, el plegamiento de biopol´ımeros, otros por el contrario seleccionan caminos ineficientes y extravagantes, e.g. la reflexión de la luz en superficies espejadas cóncavas. Otros, en cambio, eligen caminos que evolucionan sin proseguir estrategias extremas. Los problemas aqu´ı tratados plantean determinar el conjunto de trayectorias admisibles, para lo cual se apela a métodos y modelos sustentados en argumentos lógicos y proposiciones matemáticas. La metodología variacional permite un nexo entre el pensamiento de Hamilton en “Geometría ´Optica” y su diseño del “Icosian Game”. Vínculo que consiste en la identificación de las trayectorias hamiltonianas y cuasi-hamiltonianas reflexivas en las arquitecturas de las redes con nodos en los v´ertices de los n-gonos regulares. Mientras que mediante la aplicaci´on del algoritmo aritmético propuesto se caracterizan las soluciones extremales de diferentes problemas de hamiltonianos cíclicos y no cíclicos óptimos y subóptimos. / In the teleology of natural phenomena it is well known that some processes expedite progress, e.g. the law of refraction, the folding of biopolymers, while, on the contrary other processes perform the pathways of the inefficiency or extravagance, e.g. the law of reflection at the hollow mirrors, and there are processes that involve non-extreme strategies. The studied problems impose to determine the set of the admissible trajectories that require methods and models supported by logical arguments and mathematical statements. The variational procedure allows a link between Hamilton’s thoughts in “Geometric Optics”and his design of the “Icosian Game”. This connection identifies the reflective hamiltonian and quasi-hamiltonian paths in the architecture of the networks built on the vertices of the regular n-gons. The applications of the proposed algorithm deal with the characterization of the pathways that solve different hamiltonian cyclic and non-cyclic extremal path problems.

Understanding the action of an autoequivalence on a triangulated category is generally a very difficult problem. If one can find a stability condition for which the autoequivalence is "compatible", one can explicitly write down the action of this autoequivalence. In turn, the now understood autoequivalence can provide ways of extracting geometric information from the stability condition. In this thesis, we elaborate on what it means for an autoequivalence and stability condition to be "compatibile" and derive a sufficiency criterion. / text