Minimum cycle and homology bases of surface embedded graphs

We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the $1$-dimensional ($\mathbb{Z}_2$)-homology classes) of an undirected graph embedded on a surface. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the $1$-skeleton of any graph is exactly its minimum cycle basis. For the minimum cycle basis problem, we give a deterministic $O(n^\omega+2^{2g}n^2+m)$-time algorithm for graphs embedded on an orientable surface of genus $g$. The best known existing algorithms for surface embedded graphs are those for general graphs: an $O(m^\omega)$ time Monte Carlo algorithm and a deterministic $O(nm^2/\log n + n^2 m)$ time algorithm. For the minimum homology basis problem, we give a deterministic $O((g+b)^3 n \log n + m)$-time algorithm for graphs embedded on an orientable or non-orientable surface of genus $g$ with $b$ boundary components, assuming shortest paths are unique, improving on existing algorithms for many values of $g$ and $n$. The assumption of unique shortest paths can be avoided with high probability using randomization or deterministically by increasing the running time of the homology basis algorithm by a factor of $O(\log n)$.Comment: A preliminary version of this work was presented at the 32nd Annual International Symposium on Computational Geometr

Incremental cycle bases for cycle-based pose graph optimization

Pose graph optimization is a special case of the simultaneous localization and mapping problem where the only variables to be estimated are pose variables and the only measurements are inter-pose constraints. The vast majority of PGO techniques are vertex based (variables are robot poses), but recent work has parameterized the pose graph optimization problem in a relative fashion (variables are the transformations between poses) that utilizes a minimum cycle basis to maximize the sparsity of the problem. We explore the construction of a cycle basis in an incremental manner while maximizing the sparsity. We validate an algorithm that constructs a sparse cycle basis incrementally and compare its performance with a minimum cycle basis. Additionally, we present an algorithm to approximate the minimum cycle basis of two graphs that are sparsely connected as is common in multi-agent scenarios. Lastly, the relative parameterization of pose graph optimization has been limited to using rigid body transforms on SE(2) or SE(3) as the constraints between poses. We introduce a methodology to allow for the use of lower-degree-of-freedom measurements in the relative pose graph optimization problem. We provide extensive validation of our algorithms on standard benchmarks, simulated datasets, and custom hardware

Dark energy: EFTs and supergravity

The subject of this thesis is cosmological implications of string compactifications understood in a broad sense. In the first half of the thesis, we will begin by reviewing the four-dimensional description of the tree-level perturbative type IIB action. We will then introduce a number of open questions in cosmology and their relevance with regards to the remainder of the thesis. We will first explore some of these questions from the perspective of effective field theories motivated by supergravity. In particular, we provide a description of a naturally light dark energy field in terms of the clockwork mechanism and the Dvali-Kaloper-Sorbo four-form mixing. We study its possible UV completion and show a no-go for its embedding within perturbative type IIA supergravity. We also discuss the coincidence problem for dynamical models of dark energy consistent with a quintessence field slowly rolling down a potential slope, of the type one would expect from the asymptotics of moduli space. As it rolls, a tower of heavy states will generically descend, triggering a phase transition in the low energy cosmological dynamics after at most a few hundred Hubble times. As a result, dark energy domination cannot continue indefinitely and there is at least a percentage chance that we find ourselves in the first Hubble epoch. In the second half of the thesis, we introduce the effects of perturbative and nonperturbative corrections to the tree-level type IIB action. We then focus on obtaining a viable model of quintessence from the type IIB effective field theory. However, we are able to show that such a model must have a non-supersymmetric Minkowski vacuum at leading order. Furthermore, it must necessarily take the form of axion hilltop quintessence. When we consider the effects of quantum fluctuations during the early Universe, we see that such models must have extremely fine-tuned initial conditions to describe a slow-rolling scalar field at present times. We conclude that quintessence faces more challenges than a true cosmological constant, to the point that quintessence is very unattractive for model building modulo a ruling out of the cosmological constant by observations. Following this line of reasoning, we consider whether other perturbative corrections can generate de Sitter solutions in an appropriate setting. In particular, we consider the effects of higher curvature corrections in the Gauss-Bonnet term. Remarkably, we are able to show that, for the particular setting of a fluxed runaway potential motivated by heterotic supergravity, the curvature corrections reduce the space of solutions

Matrix models and phase transitions in gauge theories

Aspects of gauge theories in two, three and five dimensions are investigated using matrix models. Specifically, we consider pure Yang–Mills theory and its deformations in two dimensions, and supersymmetric Yang–Mills and Chern–Simons-matter theories in three and five dimensions. The random matrix approach allows us to explore a vast range of features of the gauge theories, including phase transitions, one-form symmetries and integrability. Partition functions and Wilson loops are studied in these setups by exploiting their matrix model presentation derived by localization. Two main lines of research are pursued: the computation of exact results at fixed and the quest for quantum phase transitions at large . The partition functions of several three-dimensional quiver Chern–Simons-matter theories are computed exactly using Mordell integrals, and we put forward a character expansion in terms of Schur polynomials, with coefficients given by topological invariants. A correspondence between two matrix models is provided as well, one computing topological invariants in pure Chern–Simons theory and the other arising from a two-dimensional, noncommutative scalar field theory. The correspondence is extended to supermatrix models, with ABJ(M) theory replacing topological Chern–Simons theory in this case. Partition functions and Wilson loop expectation values in three-dimensional = 4 gauge theories are also computed, uncovering a relation with Calogero–Moser integrable systems. Furthermore, we apply localization to five-dimensional supersymmetric Yang–Mills theory on compact product manifolds 3 × Σ, where Σ is a closed oriented Riemann surface, and introduce in this way a novel, “squashed” deformation of q-deformed Yang–Mills theory on Σ. Proceeding in the study of deformations of two-dimensional Yang–Mills theory, we analyze their perturbation by the operator ⊤₸ and prove that Abelianization still holds, although other characteristic properties such as factorization of the partition function break down. The analysis of large quantum phase transitions in matrix models and gauge theories constitutes the core of the thesis. We present a systematic study and classification of phase transitions for supersymmetric gauge theories on three- and five-dimensional spheres of large radius. The transitions are always third order for gauge theories connected to a known superconformal point, but are second order for generic five-dimensional () theories. Several multi-parameter families of unitary matrix models are also considered and their phase diagrams are established. Finally, we show how the Douglas–Kazakov transition of two-dimensional Yang–Mills on the sphere extends to its newly derived deformations. When both ⊤₸ and q-deformations are turned on, the two effects compete, and the system has two phases in the most part of the parameter space, but the weak coupling phase is removed in the regime of strong ⊤₸ -deformation, whereas the strong coupling phase is removed in the strong q-deformation regime

Toward better computation models for modern machines

Modern computers are not random access machines (RAMs). They have a memory hierarchy, multiple cores, and a virtual memory. We address the computational cost of the address translation in the virtual memory and difficulties in design of parallel algorithms on modern many-core machines. Starting point for our work on virtual memory is the observation that the analysis of some simple algorithms (random scan of an array, binary search, heapsort) in either the RAM model or the EM model (external memory model) does not correctly predict growth rates of actual running times. We propose the VAT model (virtual address translation) to account for the cost of address translations and analyze the algorithms mentioned above and others in the model. The predictions agree with the measurements. We also analyze the VAT-cost of cache-oblivious algorithms. In the second part of the paper we present a case study of the design of an efficient 2D convex hull algorithm for GPUs. The algorithm is based on the ultimate planar convex hull algorithm of Kirkpatrick and Seidel, and it has been referred to as the first successful implementation of the QuickHull algorithm on the GPU by Gao et al. in their 2012 paper on the 3D convex hull. Our motivation for work on modern many-core machines is the general belief of the engineering community that the theory does not produce applicable results, and that the theoretical researchers are not aware of the difficulties that arise while adapting algorithms for practical use. We concentrate on showing how the high degree of parallelism available on GPUs can be applied to problems that do not readily decompose into many independent tasks.Moderne Computer sind keine Random Access Machines (RAMs), da ihr Speicher hierarchisch ist und sie sowohl mehrere Rechenkerne als auch virtuellen Speicher benutzen. Wir betrachten die Kosten von Adressübersetzungen in virtuellem Speicher und die Schwierigkeiten beim Entwurf nebenläufiger Algorithmen für moderne Mehrkernprozessoren. Am Anfang unserer Arbeit über virtuellen Speicher steht die Beobachtung, dass die Analyse einiger einfacher Algorithmen (zufällige Zugriffe in einem Array, Binärsuche, Heapsort) sowohl im RAM Modell als auch im EM (Modell für externen Speicher) die tatsächlichen asymptotischen Laufzeiten nicht korrekt wiedergibt. Um auch die Kosten der Adressübersetzung mit in die Analyse aufzunehmen, definieren wir das sogenannte VAT Modell (virtual address translation) und benutzen es, um die oben genannten Algorithmen zu analysieren. In diesem Modell stimmen die theoretischen Laufzeiten mit den Messungen aus der Praxis überein. Zudem werden die Kosten von Cache-oblivious Algorithmen im VAT Modell untersucht. Der zweite Teil der Arbeit behandelt eine Fallstudie zur Implementierung eines effizienten Algorithmus zur Berechnung von konvexen Hüllen in 2D auf GPUs (Graphics Processing Units). Der Algorithmus basiert auf dem ultimate planar convex hull algorithm von Kirkpatrick und Seidel und wurde 2012 von Gao et al. in ihrer Veröffentlichung über konvexe Hüllen in 3D als die erste erfolgreiche Implementierung des QuickHull-Algorithmus auf GPUs bezeichnet. Motiviert wird diese Arbeit durch den generellen Glauben der IT-Welt, dass Resultate aus der theoretischen Informatik nicht immer auf Probleme in der Praxis anwendbar sind und dass oft nicht auf die speziellen Anforderungen und Probleme eingegangen wird, die mit einer Implementierung eines Algorithmus einhergehen. Wir zeigen, wie der hohe Grad an Parallelität, der auf GPUs verfügbar ist, für Probleme nutzbar gemacht werden kann, für die eine Zerlegung in unabhängige Teilprobleme nicht offensichtlich ist

D-brane inflation with perturbative moduli stabilisation

Cosmological inflation is a period of accelerated expansion of the early universe. Nowadays there are plenty of inflationary models which agree with experimental constraints but a complete microscopic understanding of this quasi de Sitter phase is still missing. In this thesis, after reviewing the present situation in Cosmology and Supersymmetry, we present the main possibilities to embed inflation in a UV complete framework coming from String Theory. In particular, we present the main virtues and limitations of RG-induced modulus stabilisation, focusing on D3-antiD3 inflation and discussing whether slow-roll can be obtained in the regime of validity of the effective field theory

The decay width of stringy hadrons

In this paper we further develop a string model of hadrons by computing their strong decay widths and comparing them to experiment. The main decay mechanism is that of a string splitting into two strings. The corresponding total decay width behaves as $\Gamma=\frac\pi2 ATL$ where $T$ and $L$ are the tension and length of the string and $A$ is a dimensionless universal constant. We show that this result holds for a bosonic string not only in the critical dimension. The partial width of a given decay mode is given by $\Gamma_i/\Gamma=\Phi_i\exp(-2\pi Cm_{sep}^2/T)$ where $\Phi_i$ is a phase space factor, $m_{sep}$ is the mass of the "quark" and "antiquark" created at the splitting point, and $C$ is a dimensionless coefficient close to unity. Based on the spectra of hadrons we observe that their (modified) Regge trajectories are characterized by a negative intercept. This implies a repulsive Casimir force that gives the string a "zero point length". We fit the theoretical decay width to experimental data for mesons on the trajectories of $\rho$, $\omega$, $\pi$, $\eta$, $K^*$, $\phi$, $D$, and $D^*_s$, and of the baryons $N$, $\Delta$, $\Lambda$, and $\Sigma$. We examine both the linearity in $L$ and the exponential suppression factor. The linearity was found to agree with the data well for mesons but less for baryons. The extracted coefficient for mesons $A=0.095\pm0.015$ is indeed quite universal. The exponential suppression was applied to both strong and radiative decays. We discuss the relation with string fragmentation and jet formation. We extract the quark-diquark structure of baryons from their decays. A stringy mechanism for Zweig suppressed decays of quarkonia is proposed and is shown to reproduce the decay width of $\Upsilon$ states. The dependence of the width on spin and flavor symmetry is discussed. We further apply this model to the decays of glueballs and exotic hadrons.Comment: v1: 98 pages / v2: minor revisions, references added, 100 pages (41 figures) / v3: final published version, minor corrections, 100 page

Integrability, Recursion Operators and Soliton Interactions

This volume contains selected papers based on the talks,presented at the Conference "Integrability, Recursion Operators and Soliton Interactions", held in Sofia, Bulgaria (29 - 31 August 2012) at the Institute for Nuclear Research and Nuclear Energy of the Bulgarian Academy of Sciences. Included are also invited papers presenting new research developments in the thematic area. The Conference was dedicated to the 65-th birthday of our esteemed colleague and friend Vladimir Gerdjikov. The event brought together more than 30 scientists, from 6 European countries to celebrate Vladimir's scientific achievements. All participants enjoyed a variety of excellent talks in a friendly and stimulating atmosphere. The main topics of the conference were those where Vladimir has contributed enormously during his career: integrable nonlinear partial differential equations, underlying algebraic and geometric structures of the integrable systems, soliton solutions, soliton interactions, quantum integrable systems, discrete integrable systems and applications of the nonlinear models. The papers, included in this volume will be useful to researchers with interests in these areas