Discrete Mathematics and Combinatorics Commons^™

Generating Polynomials Of Exponential Random Graphs, Mohabat Tarkeshian

Electronic Thesis and Dissertation Repository

The theory of random graphs describes the interplay between probability and graph theory: it is the study of the stochastic process by which graphs form and evolve. In 1959, Erdős and Rényi defined the foundational model of random graphs on n vertices, denoted G(n, p) ([ER84]). Subsequently, Frank and Strauss (1986) added a Markov twist to this story by describing a topological structure on random graphs that encodes dependencies between local pairs of vertices ([FS86]). The general model that describes this framework is called the exponential random graph model (ERGM).

In the past, determining when a probability distribution has strong …

Topological Recursion And Random Finite Noncommutative Geometries, Shahab Azarfar

Electronic Thesis and Dissertation Repository

In this thesis, we investigate a model for quantum gravity on finite noncommutative spaces using the topological recursion method originated from random matrix theory. More precisely, we consider a particular type of finite noncommutative geometries, in the sense of Connes, called spectral triples of type ${(1,0)} \,$, introduced by Barrett. A random spectral triple of type ${(1,0)}$ has a fixed fermion space, and the moduli space of its Dirac operator ${D=\{ H , \cdot \} \, ,}$ ${H \in {\mathcal{H}_N}}$, encoding all the possible geometries over the fermion space, is the space of Hermitian matrices ${\mathcal{H}_N}$. A distribution of the …

Arrangements Of Submanifolds And The Tangent Bundle Complement, Priyavrat Deshpande

Electronic Thesis and Dissertation Repository

Drawing parallels with the theory of hyperplane arrangements, we develop the theory of arrangements of submanifolds. Given a smooth, finite dimensional, real manifold $X$ we consider a finite collection $\A$ of locally flat codimension $1$ submanifolds that intersect like hyperplanes. To such an arrangement we associate two posets: the \emph{poset of faces} (or strata) $\FA$ and the \emph{poset of intersections} $L(\A)$. We also associate two topological spaces to $\A$. First, the complement of the union of submanifolds in $X$ which we call the \emph{set of chambers} and denote by $\Ch$. Second, the complement of union of tangent bundles of these …