Non-Perturbative Methods in Quantum Field Theory and Quantum Gravity

This thesis considers non-perturbative methods in quantum field theory with applications to gravity and cosmology. In particular, there are chapters on black hole holography, inflationary model building, and the conformal bootstrap.

Expedition in Data and Harmonic Analysis on Graphs

The graph Laplacian operator is widely studied in spectral graph theory largely due to its importance in modern data analysis. Recently, the Fourier transform and other time-frequency operators have been defined on graphs using Laplacian eigenvalues and eigenvectors. We extend these results and prove that the translation operator to the i’th node is invertible if and only if all eigenvectors are nonzero on the i’th node. Because of this dependency on the support of eigenvectors we study the characteristic set of Laplacian eigenvectors. We prove that the Fiedler vector of a planar graph cannot vanish on large neighborhoods and then explicitly construct a family of non-planar graphs that do exhibit this property. We then prove original results in modern analysis on graphs. We extend results on spectral graph wavelets to create vertex-dyanamic spectral graph wavelets whose support depends on both scale and translation parameters. We prove that Spielman’s Twice-Ramanujan graph sparsifying algorithm cannot outperform his conjectured optimal sparsification constant. Finally, we present numerical results on graph conditioning, in which edges of a graph are rescaled to best approximate the complete graph and reduce average commute time.

A Better Place to Be: Republicanism as an Alternative to the Authoritarianism-Democracy Dichotomy

ABSTRACT Title of Dissertation: A BETTER PLACE TO BE: REPUBLICANISM AS AN ALTENATIVE TO THE AUTHORITARIANISM-DEMOCRACY DICHOTOMY Christopher Ronald Binetti, Doctor of Philosophy, and 2016 Dissertation directed by: Dr. Charled Frederick Alford, Department of Government and Politics In this dissertation, I argue that in modern or ancient regimes, the simple dichotomy between democracies and autocracies/dictatorships is both factually wrong and problematic for policy purposes. It is factually wrong because regimes between the two opposite regime types exist and it is problematic because the either/or dichotomy leads to extreme thinking in terms of nation-building in places like Afghanistan. In planning for Afghanistan, the argument is that either we can quickly nation-build it into a liberal democracy or else we must leave it in the hands of a despotic dictator. This is a false choice created by both a faulty categorization of regime types and most importantly, a failure to understand history. History shows us that the republic is a regime type that defies the authoritarian-democracy dichotomy. A republic by my definition is a non-dominating regime, characterized by a (relative) lack of domination by any one interest group or actor, mostly non-violent competition for power among various interest groups/factions, the ability of factions/interest groups/individual actors to continue to legitimately play the political game even after electoral or issue-area defeat and some measure of effectiveness. Thus, a republic is a system of government that has institutions, laws, norms, attitudes, and beliefs that minimize the violation of the rule of law and monopolization of power by one individual or group as much as possible. These norms, laws, attitudes, and beliefs ae essential to the republican system in that they make those institutions that check and balance power work. My four cases are Assyria, Persia, Venice and Florence. Assyria and Persia are ancient regimes, the first was a republic and then became the frightening opposite of a republic, while the latter was a good republic for a long time, but had effectiveness issues towards the end. Venice is a classical example of a medieval or early modern republic, which was very inspirational to Madison and others in building republican America. Florence is the example of a medieval republic that fell to despotism, as immortalized by Machiavelli’s writings. In all of these examples, I test certain alternative hypotheses as well as my own.

Essays on Matching Markets and Their Equilibria

Matching theory and matching markets are a core component of modern economic theory and market design. This dissertation presents three original contributions to this area. The first essay constructs a matching mechanism in an incomplete information matching market in which the positive assortative match is the unique efficient and unique stable match. The mechanism asks each agent in the matching market to reveal her privately known type. Through its novel payment rule, truthful revelation forms an ex post Nash equilibrium in this setting. This mechanism works in one-, two- and many-sided matching markets, thus offering the first mechanism to unify these matching markets under a single mechanism design framework. The second essay confronts a problem of matching in an environment in which no efficient and incentive compatible matching mechanism exists due to matching externalities. I develop a two-stage matching game in which a contracting stage facilitates subsequent conditionally efficient and incentive compatible Vickrey auction stage. Infinite repetition of this two-stage matching game enforces the contract in every period. This mechanism produces inequitably distributed social improvement: parties to the contract receive all of the gains and then some. The final essay demonstrates the existence of prices which stably and efficiently partition a single set of agents into firms and workers, and match those two sets to each other. This pricing system extends Kelso and Crawford's general equilibrium results in a labor market matching model and links one- and two-sided matching markets as well.

Multiple sulfur isotope fractionations in inorganic aqueous systems

New constraints on isotope fractionation factors in inorganic aqueous sulfur systems based on theoretical and experimental techniques relevant to studies of the sulfur cycle in modern environments and the geologic rock record are presented in this dissertation. These include theoretical estimations of equilibrium isotope fractionation factors utilizing quantum mechanical software and a water cluster model approach for aqueous sulfur compounds that span the entire range of oxidation state for sulfur. These theoretical calculations generally reproduce the available experimental determinations from the literature and provide new constraints where no others are available. These theoretical calculations illustrate in detail the relationship between sulfur bonding environment and the mass dependence associated with equilibrium isotope exchange reactions involving all four isotopes of sulfur. I additionally highlight the effect of isomers of protonated compounds (compounds with the same chemical formula but different structure, where protons are bound to either sulfur or oxygen atoms) on isotope partitioning in the sulfite (S4+) and sulfoxylate (S2+) systems, both of which are key intermediates in oxidation-reduction processes in the sulfur cycle. I demonstrate that isomers containing the highest degree of coordination around sulfur (where protonation occurs on the sulfur atom) have a strong influence on isotopic fractionation factors, and argue that isomerization phenomenon should be considered in models of the sulfur cycle. Additionally, experimental results of the reaction rates and isotope fractionations associated with the chemical oxidation of aqueous sulfide are presented. Sulfide oxidation is a major process in the global sulfur cycle due largely to the sulfide-producing activity of anaerobic microorganisms in organic-rich marine sediments. These experiments reveal relationships between isotope fractionations and reaction rate as a function of both temperature and trace metal (ferrous iron) catalysis that I interpret in the context of the complex mechanism of sulfide oxidation. I also demonstrate that sulfide oxidation is a process associated with a mass dependence that can be described as not conforming to the mass dependence typically associated with equilibrium isotope exchange. This observation has implications for the inclusion of oxidative processes in environmental- and global-scale models of the sulfur cycle based on the mass balance of all four isotopes of sulfur. The contents of this dissertation provide key reference information on isotopic fractionation factors in aqueous sulfur systems that will have far-reaching applicability to studies of the sulfur cycle in a wide variety of natural settings.