Tensor models and embedded Riemann surfaces

Tensor models and, more generally, group field theories are candidates for higher-dimensional quantum gravity, just as matrix models are in the 2d setting. With the recent advent of a 1/N-expansion for coloured tensor models, more focus has been given to the study of the topological aspects of their Feynman graphs. Crucial to the aforementioned analysis were certain subgraphs known as bubbles and jackets. We demonstrate in the 3d case that these graphs are generated by matrix models embedded inside the tensor theory. Moreover, we show that the jacket graphs represent (Heegaard) splitting surfaces for the triangulation dual to the Feynman graph. With this in hand, we are able to re-express the Boulatov model as a quantum field theory on these Riemann surfaces.Comment: 9 pages, 7 fi

Homological Product Codes

Quantum codes with low-weight stabilizers known as LDPC codes have been actively studied recently due to their simple syndrome readout circuits and potential applications in fault-tolerant quantum computing. However, all families of quantum LDPC codes known to this date suffer from a poor distance scaling limited by the square-root of the code length. This is in a sharp contrast with the classical case where good families of LDPC codes are known that combine constant encoding rate and linear distance. Here we propose the first family of good quantum codes with low-weight stabilizers. The new codes have a constant encoding rate, linear distance, and stabilizers acting on at most $\sqrt{n}$ qubits, where $n$ is the code length. For comparison, all previously known families of good quantum codes have stabilizers of linear weight. Our proof combines two techniques: randomized constructions of good quantum codes and the homological product operation from algebraic topology. We conjecture that similar methods can produce good stabilizer codes with stabilizer weight $n^a$ for any $a>0$. Finally, we apply the homological product to construct new small codes with low-weight stabilizers.Comment: 49 page

Characterizing W2,pW^{2,p}~submanifolds by pp-integrability of global curvatures

We give sufficient and necessary geometric conditions, guaranteeing that an immersed compact closed manifold $\Sigma^m\subset \R^n$ of class $C^1$ and of arbitrary dimension and codimension (or, more generally, an Ahlfors-regular compact set $\Sigma$ satisfying a mild general condition relating the size of holes in $\Sigma$ to the flatness of $\Sigma$ measured in terms of beta numbers) is in fact an embedded manifold of class $C^{1,\tau}\cap W^{2,p}$, where $p>m$ and $\tau=1-m/p$. The results are based on a careful analysis of Morrey estimates for integral curvature--like energies, with integrands expressed geometrically, in terms of functions that are designed to measure either (a) the shape of simplices with vertices on $\Sigma$ or (b) the size of spheres tangent to $\Sigma$ at one point and passing through another point of $\Sigma$. Appropriately defined \emph{maximal functions} of such integrands turn out to be of class $L^p(\Sigma)$ for $p>m$ if and only if the local graph representations of $\Sigma$ have second order derivatives in $L^p$ and $\Sigma$ is embedded. There are two ingredients behind this result. One of them is an equivalent definition of Sobolev spaces, widely used nowadays in analysis on metric spaces. The second one is a careful analysis of local Reifenberg flatness (and of the decay of functions measuring that flatness) for sets with finite curvature energies. In addition, for the geometric curvature energy involving tangent spheres we provide a nontrivial lower bound that is attained if and only if the admissible set $\Sigma$ is a round sphere.Comment: 44 pages, 2 figures; several minor correction

More Torsion in the Homology of the Matching Complex

A matching on a set $X$ is a collection of pairwise disjoint subsets of $X$ of size two. Using computers, we analyze the integral homology of the matching complex $M_n$, which is the simplicial complex of matchings on the set $\{1, >..., n\}$. The main result is the detection of elements of order $p$ in the homology for $p \in \{5,7,11,13\}$. Specifically, we show that there are elements of order 5 in the homology of $M_n$ for $n \ge 18$ and for $n \in {14,16}$. The only previously known value was $n = 14$, and in this particular case we have a new computer-free proof. Moreover, we show that there are elements of order 7 in the homology of $M_n$ for all odd $n$ between 23 and 41 and for $n=30$. In addition, there are elements of order 11 in the homology of $M_{47}$ and elements of order 13 in the homology of $M_{62}$. Finally, we compute the ranks of the Sylow 3- and 5-subgroups of the torsion part of $H_d(M_n;Z)$ for $13 \le n \le 16$; a complete description of the homology already exists for $n \le 12$. To prove the results, we use a representation-theoretic approach, examining subcomplexes of the chain complex of $M_n$ obtained by letting certain groups act on the chain complex.Comment: 35 pages, 10 figure

Quantum statistics on graphs

Quantum graphs are commonly used as models of complex quantum systems, for example molecules, networks of wires, and states of condensed matter. We consider quantum statistics for indistinguishable spinless particles on a graph, concentrating on the simplest case of abelian statistics for two particles. In spite of the fact that graphs are locally one-dimensional, anyon statistics emerge in a generalized form. A given graph may support a family of independent anyon phases associated with topologically inequivalent exchange processes. In addition, for sufficiently complex graphs, there appear new discrete-valued phases. Our analysis is simplified by considering combinatorial rather than metric graphs -- equivalently, a many-particle tight-binding model. The results demonstrate that graphs provide an arena in which to study new manifestations of quantum statistics. Possible applications include topological quantum computing, topological insulators, the fractional quantum Hall effect, superconductivity and molecular physics.Comment: 21 pages, 6 figure

On the Expansions in Spin Foam Cosmology

We discuss the expansions used in spin foam cosmology. We point out that already at the one vertex level arbitrarily complicated amplitudes contribute, and discuss the geometric asymptotics of the five simplest ones. We discuss what type of consistency conditions would be required to control the expansion. We show that the factorisation of the amplitude originally considered is best interpreted in topological terms. We then consider the next higher term in the graph expansion. We demonstrate the tension between the truncation to small graphs and going to the homogeneous sector, and conclude that it is necessary to truncate the dynamics as well.Comment: 17 pages, 4 figures, published versio

Permutation combinatorics of worldsheet moduli space

52 pages, 21 figures52 pages, 21 figures; minor corrections, "On the" dropped from title, matches published version52 pages, 21 figures; minor corrections, "On the" dropped from title, matches published versio

The Shape of Pulverized Bituminous Vitrinite Coal Particles

The shape of pulverized bituminous coal particles (vitrinites) was determined by optical and laser light scattering. Vitrain samples were collected from obvious tree remains located in the ceilings of two Appalachian coal mines. Wet sieving produced narrow size cuts. The particles were determined to be oblong or blocky in shape, with average length-to-width ratio of 1.7 and sphericity of 0.78. They were analogous in shape to a square ended, rectangular house brick . The two bituminous coals and different size cuts of each coal had essentially the same shape parameters. Characteristic heating times and terminal velocities were higher by 22 and 20%, respectively compared to spherical particles

"Disputes, Disorders, and Confusion": Authorship, Remediation, and Intellectual Property Regulations in the Digital Age.

In the last two decades, Western media production and consumption have been transformed by the permeation of digital technologies. While not new conventions, remediative techniques such as formal fracturing and intertextuality have become definitive of Digital Age storytelling. The prevalence of these techniques that encourage multimodal reading practices signal an epistemological shift centered on sharing and collaboration. I analyze a subset of contemporary literary works that integrate traditional media forms with Digital Age narrative techniques in ways that challenge individualistic notions of authorship perpetuated by intellectual property regulations. The convergence of feminist methods with legal scholarship, cultural theory, and new media studies guides my inquiries about the evolution of authorship in the Digital Age. This project examines Digital Age storytelling techniques and synergistic audience engagements in a variety of contemporary literary works. Chapter 1 examines the way that Percival Everett’s experimental novel Erasure appeals to digital natives by using remediative techniques to investigate historical controversies about racialized masculinity and African American cultural production. In Chapter 2, I analyze Suzan-Lori Parks’ play with form and challenge to the conventions of drama in her performance text Venus as a means of destabilizing the performance of gendered racism. Chapter 3 explores the citational pleasures of Alan Moore and Melinda Gebbie’s graphic novel Lost Girls that emerge through remediations of public domain children’s stories. My reading of Lost Girls challenges intellectual property regulations that devalue remediative creative methods. Finally, Chapter 4 provides a close reading of the reader appeal created by the complex intertextual blending of artifacts in the DC Comics universe in Grant Morrison’s Batman titles. In addition, I scrutinize work for hire as a complicated model of authorship that sacrifices creators’ rights to generate greater income for publishers. This project explores the growing tension between a public who increasingly validates remediation as a valuable creative method, and private institutions that hold onto outdated definitions of authorship that privilege the monopolization of ideas. Ultimately, this project calls for a more flexible approach to intellectual property that moves away from the privatization of ideas in favor of a more robust public domain.PHDEnglish and Women's StudiesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/98019/1/mollyhat_1.pd

The politics of entrapment : the World Bank, the integrated development model & citizenship at a crossroad

The thesis is about the World Bank and the political dimensions of the new aid architecture. The new aid agenda, which emerged out of the post-Washington Consensus (PWC), has led to debates over its real significance. While Joseph Stiglitz has argued that the new consensus actually marked the end of the era of market fundamentalism, sceptics have rather concluded that it represents nothing more than a discursive shift. The Bank’s own ambiguous narrative on the new aid framework may explain such divergent analysis on the new agenda: On the one hand, the poverty reduction and participatory narrative linked to the model have appeared to be signalling a genuine shift towards a more social policy agenda. On the other hand however, there has been a contradictory move towards a stricter governance model that seems to be specifically designed to insulate policy-makers from vested interests. The overarching objective of this thesis is to shed new light on this apparent paradox inherent in international aid reform. It seeks to untangle the existing confusion between narrative, assumptions and methods entwined with the so-called PWC. Specifically, the crux of this thesis is to address the particular contribution of the World Bank in forging, promoting and implementing the PWC. It focuses both on the theoretical sources that triggered the World Bank, under the leadership of James D. Wolfensohn, to adopt a new model and on the political objectives of this agenda. Within such a research focus, the Bank’s narrative and its new aid allocation mechanisms are seen as one entity under the label ‘Integrated Development Model’ (IDM). The IDM – as a main object of study – allows us to transcend common country-based analyses of the Poverty Reduction Strategies (PRS) experience and to re-position the new aid allocation mechanisms within the larger reordering of power on behalf of market-oriented interests