From Quantum Gravity to Combinatorial Hives: Addressing Longstanding Puzzles with Novel Approaches

Abstract

A thesis in two parts, this manuscript concerns the applications of contemporary techniques in optimization, network theory, combinatorial topology, and other branches of discrete mathematics to longstanding problems in theoretical physics and algebraic representation theory. We begin with an initial inquiry into the wild world of quantum gravity, reviewing the paradigm of causal dynamical triangulations as a preface to introducing our own novel approach for investigating the emergence of classical geometries from the combinatorial background. In an effort to address the question of whether macroscopic geometric order can be generated from fundamental building blocks of spacetime, we introduce, expand upon, and test the limits of an extremely general combinatorial model for quantum geometry. Along the way, we illustrate interesting model phenomenology such as an intrinsic UV scale cutoff, and, through simulations and numerical analysis, demonstrate the appearance of extended triangulations and probe the far-unconstrained regime of our model for its limiting behavior. As tools used to further characterize our theory, we introduce new algorithms for sampling on spaces of abstract simplicial complexes, and provide theoretical justification for the numerical toolkit that was constructed in the course of this study. Concluding, we state our open questions regarding this model, and propose applications of the technologies that we have developed outside the scope of quantum gravity and into the practical space of network dynamics. We then take an abrupt turn into the realm of combinatorial hives, a mathematical construction introduced by Knutson and Tao. After detailing the fundamentals, we examine two distinct applications of these hives: studying the geometric properties of matrix ensembles, and computing Littlewood-Richardson coefficients. To each end, we propose analytical models that additionally map onto tractable computational algorithms, and demonstrate the experimental confirmation of these algorithms. Our work includes a theoretical counterproof and local numerical counterexamples to the Appleby-Whitehead theorem on hives from Hermitian matrix pairs, as well as the first computational implementation that generates hives with almost certain probability in select matrix ensembles. We also include dual algorithms for estimating deep Littlewood-Richardson coefficients that we provide as tools that the research community may find useful.