2. Maestría
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Tesis de la Escuela de Posgrado
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Item Isomorfismo de curvas elípticas mediante el invariante j(Pontificia Universidad Católica del Perú, 2022-04-06) Villajuan Guzman, Richard Andres; Poirier Schmitz, Alfredo BernardoComenzamos con un breve recordatorio sobre algunas nociones de conjuntos algebraicos, morfismos racionales y regulares. Por otro lado, veremos que la forma de Weierstrass de una cúbica tiene asociado dos elementos importantes. El primero es el discriminante τ que nos permite decidir si una cúbica es singular o no. El segundo elemento, muy importante en este trabajo, es el invariante j, cuyo nombre se debe a que éste no varía a pesar de los cambios de coordenadas que se realicen en la curva. Este elemento cobra gran importancia pues nos ayuda a reconocer cuando dos curvas elípticas son isomorfas. Y además, también nos permite contar el número de automorfismos sobre una curva elíptica dada.Item Symmetry breaking in grand unified theories(Pontificia Universidad Católica del Perú, 2016-04-14) Torrejón Maguiña, Miguel Ángel; Jones Pérez, JoelIn this work we review the symmetry breaking mechanism of gauge theories. On the first chapters of this thesis, we review the concept of symmetry as the action of a group that leaves an object invariant, in particular Lagrangians and actions, and then develop the corresponding globally and gauge symmetric theories and the relationship between them. It also reviewed the concept and general framework of the spontaneous breaking of a symmetry for renormalizable potentials. Correspondingly, two main results for global symmetries, Noether’s theorem and Goldstone’s Theorem, are reviewed in a general setting. Chapter 3 is the most important part of this work. The Brout-Englert-Higgs mechanism is explained and used to retrieve the symmetry breaking patterns for the vector and all the second rank tensor irreducible representations of the O(n) and SU(n) groups. In general we will retrieve the vacuum expectation value (vev) for the particular representation and value of the parameters of the potential. Then, for this vev, we calculate the number of massive vector bosons of the theory. Following BEH mechanism and Goldstone’s theorem, this number is equal to the number of broken generators delining thus the particular symmetry breaking pattern. Chapter 4 is a review of the Standard Model with an aim towards Grand Unified Theories (GUTs). Lastly in Chapter 5 we review the group theory of the minimal model SU(5) in a very exhaustive way and use the results of Chapter 3 to see the breaking patterns for this particular GUT.