Please use this identifier to cite or link to this item: http://cimat.repositorioinstitucional.mx/jspui/handle/1008/1011
CATEGORICAL INDEPENDENCE AND NONCOMMUTATIVE DISTRIBUTIONS FOR SIMPLICIAL COMPLEXES
CARLOS EDUARDO DIAZ AGUILERA
Acceso Abierto
Atribución-NoComercial
MATEMÁTICAS BÁSICAS
Graphs provide basic and illustrative examples of non-commutative random variables via their adjacency matrices. The main objective of this thesis is to discuss the general question of assigning distributions to graphs and simplicial complexes, for these to be considered as random variables in some non-commutative probability space. We split the problem in three parts. First, we need to decide which matrices associated with simplicial complexes are we considering as random variables. Second, we must specify a expectation or state to endow our matrices with distributions. Third, we may also want to consider collections of compatible conditional expectations, so that we may relate the different distributions for different choices of matrices. Surprisingly, if we restrict the third question, allowing only the simplest types of operator-valued spaces, i.e. projection-valued spaces, then the joint algebraic distribution of incidence and boundary matrices and their duals contains the distributions of most of the matrices that are usually considered while studying simplicial complexes. This includes adjacency matrices and combinatorial Laplace operators, of all dimensions. In addition, the simplicity of the *-algebra generated by the boundary matrix leaves little room for finding self-adjoint operators. Two of the few canonical choices are naturally endowed with analytic distributions that encode the Betti numbers of the considered simplicial complex. It is to remark that a notion of independence is required for the task. In our case, the concept of categorical independence studied was developed in \cite{Ger14}, where it is intended to be the link between all the independence notions in quantum probability spaces. For this work, the categorical indepence of orthogonal spaces is a valuable concept because it enables and encourages the separated study of different orthogonal pieces, which are independent, of an operator. A second objective, is to define a Boolean cumulant type on higher dimensions, i.e. simplicial complexes. We hope that these observations will be helpful for finding simpler and meaningful examples for independent non-commutative random variables, particularly in those general NCP frameworks still lacking of non-artificial realizations.
16-07-2019
Tesis de maestría
OTRAS
Versión aceptada
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