Geometric and Combinatorial Properties of Lattice Polytopes Defined from Graphs

Polytopes are geometric objects that generalize polygons in the plane and polyhedra in 3-dimensional space. Of particular interest in geometric combinatorics are families of lattice polytopes defined from combinatorial objects, such as graphs. In particular, this dissertation studies symmetric edge polytopes (SEPs), defined from simple undirected graphs. In 2019, Higashitani, Jochemko, and Michalek gave a combinatorial description of the hyperplanes that support facets of a symmetric edge polytope in terms of certain labelings of the underlying graph.Using this framework, we explore the number of facets that can be attained by the symmetric edge polytopes for graphs with certain structure. First, we establishformulas or bounds for the number of facets attained by families of sparse, connected graphs, and give conjectures concerning the maximum and minimum facet counts for more general families. We also consider the number of facets of SEPs arising from graphs generated by several random graph models and investigate a conjectured connection between facet counts for SEPs and clustering metrics on their underlying graphs

Arguesian Identities in Invariant Theory

AbstractHaving been motivated by an example of Doubilet, Rota, and Stein [Stud. Appl. Math.56(1976), 185–216], we present a technique for constructing geometric identities in a Grassmann–Cayley algebra. Each identity represents a projective invariant closely related to the Theorem of Desargues in the plane and its generalizations to higher dimensional projective space. The construction employs certain combinatorial properties of matchings in bipartite graphs. We also prove a dimension independence result for Arguesian identities, thereby connecting the identities with lattice theory

Generic rigidity with forced symmetry and sparse colored graphs

We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of minimally rigid periodic frameworks with fixed-area fundamental domain and fixed-angle fundamental domain.Comment: 21 pages, 2 figure

Quantum Gravity and Matter: Counting Graphs on Causal Dynamical Triangulations

An outstanding challenge for models of non-perturbative quantum gravity is the consistent formulation and quantitative evaluation of physical phenomena in a regime where geometry and matter are strongly coupled. After developing appropriate technical tools, one is interested in measuring and classifying how the quantum fluctuations of geometry alter the behaviour of matter, compared with that on a fixed background geometry. In the simplified context of two dimensions, we show how a method invented to analyze the critical behaviour of spin systems on flat lattices can be adapted to the fluctuating ensemble of curved spacetimes underlying the Causal Dynamical Triangulations (CDT) approach to quantum gravity. We develop a systematic counting of embedded graphs to evaluate the thermodynamic functions of the gravity-matter models in a high- and low-temperature expansion. For the case of the Ising model, we compute the series expansions for the magnetic susceptibility on CDT lattices and their duals up to orders 6 and 12, and analyze them by ratio method, Dlog Pad\'e and differential approximants. Apart from providing evidence for a simplification of the model's analytic structure due to the dynamical nature of the geometry, the technique introduced can shed further light on criteria \`a la Harris and Luck for the influence of random geometry on the critical properties of matter systems.Comment: 40 pages, 15 figures, 13 table