Fundamental domains for proper affine actions of Coxeter groups in three dimensions

We study proper actions of groups $G \cong \Z/2\Z \ast \Z/2\Z \ast \Z/2\Z$ on affine space of three real dimensions. Since $G$ is nonsolvable, work of Fried and Goldman implies that it preserves a Lorentzian metric. A subgroup $\Gamma < G$ of index two acts freely, and $\R^3/\Gamma$ is a Margulis spacetime associated to a hyperbolic surface $\Sigma$. When $\Sigma$ is convex cocompact, work of Danciger, Gu{\'e}ritaud, and Kassel shows that the action of $\Gamma$ admits a polyhedral fundamental domain bounded by crooked planes. We consider under what circumstances the action of $G$ also admits a crooked fundamental domain. We show that it is possible to construct actions of $G$ that fail to admit crooked fundamental domains exactly when the extended mapping class group of $\Sigma$ fails to act transitively on the top-dimensional simplices of the arc complex of $\Sigma$. We also provide explicit descriptions of the moduli space of $G$ actions that admit crooked fundamental domains.

MULTI-BAND BOSE HUBBARD MODELS AND EFFECTIVE THREE-BODY INTERACTIONS

Experiments with ultracold atoms in optical lattice have become a versatile testing ground to study diverse quantum many-body Hamiltonians. A single-band Bose-Hubbard (BH) Hamiltonian was first proposed to describe these systems in 1998 and its associated quantum phase-transition was subsequently observed in 2002. Over the years, there has been a rapid progress in experimental realizations of more complex lattice geometries, leading to more exotic BH Hamiltonians with contributions from excited bands, and modified tunneling and interaction energies. There has also been interesting theoretical insights and experimental studies on “un- conventional” Bose-Einstein condensates in optical lattices and predictions of rich orbital physics in higher bands. In this thesis, I present our results on several multi- band BH models and emergent quantum phenomena. In particular, I study optical lattices with two local minima per unit cell and show that the low energy states of a multi-band BH Hamiltonian with only pairwise interactions is equivalent to an effec- tive single-band Hamiltonian with strong three-body interactions. I also propose a second method to create three-body interactions in ultracold gases of bosonic atoms in a optical lattice. In this case, this is achieved by a careful cancellation of two contributions in the pair-wise interaction between the atoms, one proportional to the zero-energy scattering length and a second proportional to the effective range. I subsequently study the physics of Bose-Einstein condensation in the second band of a double-well 2D lattice and show that the collision aided decay rate of the con- densate to the ground band is smaller than the tunneling rate between neighboring unit cells. Finally, I propose a numerical method using the discrete variable repre- sentation for constructing real-valued Wannier functions localized in a unit cell for optical lattices. The developed numerical method is general and can be applied to a wide array of optical lattice geometries in one, two or three dimensions.

VOX POPULI: THREE ESSAYS ON THE USE OF SOCIAL MEDIA FOR VALUE CREATION IN SERVICES

Prior research shows that electronic word of mouth (eWOM) wields considerable influence over consumer behavior. However, as the volume and variety of eWOM grows, firms are faced with challenges in analyzing and responding to this information. In this dissertation, I argue that to meet the new challenges and opportunities posed by the expansion of eWOM and to more accurately measure its impacts on firms and consumers, we need to revisit our methodologies for extracting insights from eWOM. This dissertation consists of three essays that further our understanding of the value of social media analytics, especially with respect to eWOM. In the first essay, I use machine learning techniques to extract semantic structure from online reviews. These semantic dimensions describe the experiences of consumers in the service industry more accurately than traditional numerical variables. To demonstrate the value of these dimensions, I show that they can be used to substantially improve the accuracy of econometric models of firm survival. In the second essay, I explore the effects on eWOM of online deals, such as those offered by Groupon, the value of which to both consumers and merchants is controversial. Through a combination of Bayesian econometric models and controlled lab experiments, I examine the conditions under which online deals affect online reviews and provide strategies to mitigate the potential negative eWOM effects resulting from online deals. In the third essay, I focus on how eWOM can be incorporated into efforts to reduce foodborne illness, a major public health concern. I demonstrate how machine learning techniques can be used to monitor hygiene in restaurants through crowd-sourced online reviews. I am able to identify instances of moral hazard within the hygiene inspection scheme used in New York City by leveraging a dictionary specifically crafted for this purpose. To the extent that online reviews provide some visibility into the hygiene practices of restaurants, I show how losses from information asymmetry may be partially mitigated in this context. Taken together, this dissertation contributes by revisiting and refining the use of eWOM in the service sector through a combination of machine learning and econometric methodologies.