Homology for higher-rank graphs and twisted C*-algebras

We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a homological point of view, with their topological counterparts. We show how to twist the C*-algebra of a k-graph by a T-valued 2-cocycle and demonstrate that examples include all noncommutative tori. In the appendices, we construct a cubical set \tilde{Q}(\Lambda) from a k-graph {\Lambda} and demonstrate that the homology and topological realisation of {\Lambda} coincide with those of \tilde{Q}(\Lambda) as defined by Grandis.Comment: 33 pages, 9 pictures and one diagram prepared in TiK

Interconnection networks for parallel and distributed computing

Parallel computers are generally either shared-memory machines or distributed- memory machines. There are currently technological limitations on shared-memory architectures and so parallel computers utilizing a large number of processors tend tube distributed-memory machines. We are concerned solely with distributed-memory multiprocessors. In such machines, the dominant factor inhibiting faster global computations is inter-processor communication. Communication is dependent upon the topology of the interconnection network, the routing mechanism, the flow control policy, and the method of switching. We are concerned with issues relating to the topology of the interconnection network. The choice of how we connect processors in a distributed-memory multiprocessor is a fundamental design decision. There are numerous, often conflicting, considerations to bear in mind. However, there does not exist an interconnection network that is optimal on all counts and trade-offs have to be made. A multitude of interconnection networks have been proposed with each of these networks having some good (topological) properties and some not so good. Existing noteworthy networks include trees, fat-trees, meshes, cube-connected cycles, butterflies, Möbius cubes, hypercubes, augmented cubes, k-ary n-cubes, twisted cubes, n-star graphs, (n, k)-star graphs, alternating group graphs, de Bruijn networks, and bubble-sort graphs, to name but a few. We will mainly focus on k-ary n-cubes and (n, k)-star graphs in this thesis. Meanwhile, we propose a new interconnection network called augmented k-ary n- cubes. The following results are given in the thesis.1. Let k ≥ 4 be even and let n ≥ 2. Consider a faulty k-ary n-cube Q(^k_n) in which the number of node faults f(_n) and the number of link faults f(_e) are such that f(_n) + f(_e) ≤ 2n - 2. We prove that given any two healthy nodes s and e of Q(^k_n), there is a path from s to e of length at least k(^n) - 2f(_n) - 1 (resp. k(^n) - 2f(_n) - 2) if the nodes s and e have different (resp. the same) parities (the parity of a node Q(^k_n) in is the sum modulo 2 of the elements in the n-tuple over 0, 1, ∙∙∙ , k - 1 representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang, Tan and Hsu, and by Fu. Furthermore, we extend known results, obtained by Kim and Park, for the case when n = 2.2. We give precise solutions to problems posed by Wang, An, Pan, Wang and Qu and by Hsieh, Lin and Huang. In particular, we show that Q(^k_n) is bi-panconnected and edge-bipancyclic, when k ≥ 3 and n ≥ 2, and we also show that when k is odd, Q(^k_n) is m-panconnected, for m = (^n(k - 1) + 2k - 6’ / ‘_2), and (k -1) pancyclic (these bounds are optimal). We introduce a path-shortening technique, called progressive shortening, and strengthen existing results, showing that when paths are formed using progressive shortening then these paths can be efficiently constructed and used to solve a problem relating to the distributed simulation of linear arrays and cycles in a parallel machine whose interconnection network is Q(^k_n) even in the presence of a faulty processor.3. We define an interconnection network AQ(^k_n) which we call the augmented k-ary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube Q(^k_n) has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n-cube Q(^k_n) - is a Cayley graph (and so is vertex-symmetric); has connectivity 4n - 2, and is such that we can build a set of 4n - 2 mutually disjoint paths joining any two distinct vertices so that the path of maximal length has length at most max{{n- l)k- (n-2), k + 7}; has diameter [(^k) / (_3)] + [(^k - 1) /( _3)], when n = 2; and has diameter at most (^k) / (_4) (n+ 1), for n ≥ 3 and k even, and at most [(^k)/ (_4) (n + 1) + (^n) / (_4), for n ^, for n ≥ 3 and k odd.4. We present an algorithm which given a source node and a set of n - 1 target nodes in the (n, k)-star graph S(_n,k) where all nodes are distinct, builds a collection of n - 1 node-disjoint paths, one from each target node to the source. The collection of paths output from the algorithm is such that each path has length at most 6k - 7, and the algorithm has time complexity O(k(^3)n(^4))

On several varieties of cacti and their relations

Motivated by string topology and the arc operad, we introduce the notion of quasi-operads and consider four (quasi)-operads which are different varieties of the operad of cacti. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. Using the recognition principle of Fiedorowicz, we prove that spineless cacti are equivalent as operads to the little discs operad. It turns out that in terms of spineless cacti Cohen's Gerstenhaber structure and Fiedorowicz' braided operad structure are given by the same explicit chains. We also prove that spineless cacti and cacti are homotopy equivalent to their normalized versions as quasi-operads by showing that both types of cacti are semi-direct products of the quasi-operad of their normalized versions with a re-scaling operad based on R>0. Furthermore, we introduce the notion of bi-crossed products of quasi-operads and show that the cacti proper are a bi-crossed product of the operad of cacti without spines and the operad based on the monoid given by the circle group S^1. We also prove that this particular bi-crossed operad product is homotopy equivalent to the semi-direct product of the spineless cacti with the group S^1. This implies that cacti are equivalent to the framed little discs operad. These results lead to new CW models for the little discs and the framed little discs operad.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-13.abs.htm

Coverings and Truncations of Graded Selfinjective Algebras

Let $\Lambda$ be a graded self-injective algebra. We describe its smash product \Lambda# k\mathbb Z^* with the group $\mathbb Z$, its Beilinson algebra and their relationship. Starting with $\Lambda$, we construct algebras with finite global dimension, called $\tau$-slice algebras, we show that their trivial extensions are all isomorphic, and their repetitive algebras are the same \Lambda# k\mathbb Z^*. There exist $\tau$-mutations similar to the BGP reflections for the $\tau$-slice algebras. We also recover Iyama's absolute $n$-complete algebra as truncation of the Koszul dual of certain self-injective algebra.Comment: Manuscript revised, introduction and abstract rewritte

The moduli space of cactus flower curves and the virtual cactus group

The space $\mathfrak{ft}_n = \mathbb C^n/\mathbb C$ of $n$ points on the line modulo translation has a natural compactification $\overline{\mathfrak{ft}}_n$ as a matroid Schubert variety. In this space, pairwise distances between points can be infinite; we call such a configuration of points a ``flower curve'', since we imagine multiple components joined into a flower. Within $\mathfrak{ft}_n$, we have the space $F_n = \mathbb C^n \setminus \Delta / \mathbb C$ of $n$ distinct points. We introduce a natural compatification $\overline F_n$ along with a map $\overline F_n \rightarrow \overline{\mathfrak{ft}}_n$, whose fibres are products of genus 0 Deligne-Mumford spaces. We show that both $\overline{\mathfrak{ft}}_n$ and $\overline F_n$ are special fibers of $1$-parameter families whose generic fibers are, respectively, Losev-Manin and Deligne-Mumford moduli spaces of stable genus $0$ curves with $n+2$ marked points. We find combinatorial models for the real loci $\overline{\mathfrak{ft}}_n(\mathbb R)$ and $\overline F_n(\mathbb R)$. Using these models, we prove that these spaces are aspherical and that their equivariant fundamental groups are the virtual symmetric group and the virtual cactus groups, respectively. The deformation of $\overline F_n(\mathbb{R})$ to a real locus of the Deligne-Mumford space gives rise to a natural homomorphism from the affine cactus group to the virtual cactus group.Comment: 69 pages, 3 figure

Applications of Topological Data Analysis to Statistical Physics and Quantum Field Theories

This thesis motivates and examines the use of methods from topological data analysis in detecting and analysing topological features relevant to models from sta-tistical physics and particle physics.In statistical physics, we use persistent homology as an observable of three dif-ferent variants of the two-dimensional XY model in order to identify relevant topo-logical features and study their relation to the phase transitions undergone by each model. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a new way of computing the persistent homology of lattice spin model configurations and demonstrate its use in detecting topological defects called vortices. By considering the fluctuations in the output of logistic regression and k-nearest neighbours mod-els trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finite-size scaling behaviour and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length.In particle physics, we investigate the use of persistent homology as a means to detect and quantitatively describe center vortices in SU(2) lattice gauge theory in a gauge-invariant manner. The sensitivity of our method to vortices in the deconfined phase is confirmed by using twisted boundary conditions which inspires the definition of a new phase indicator for the deconfinement phase transition. We also construct a phase indicator without reference to twisted boundary conditions using a k-nearest neighbours classifier. Finite-size scaling analyses of both persistence-based indicators yield accurate estimates of the critical β and critical exponent of correlation length for the deconfinement phase transition. We also use persistent homology to study the stability of vortices under gradient flow and the classification of different vortex surface geometries

M2-brane Probe Dynamics and Toric Duality

We study the dynamics of a single M2-brane probing toric Calabi-Yau four-fold singularity in the context of the recently proposed M-theory crystal model of AdS4/CFT3 dual pairs. We obtain an effective abelian gauge theory in which the charges of the matter fields are given by the intersection numbers between loops and faces in the crystal. We argue that the probe theory captures certain aspects of the CFT3 even though the true M2-brane CFT is unlikely to be a usual gauge theory. In particular, the moduli space of vacua of the gauge theory coincides precisely with the Calabi-Yau singularity. Toric duality, partial resolution, and a possibility of new RG flows are also discussed.Comment: 50 pages, 24 figures; v2. title changed, abstract and introduction clarified. to appear in Nucl.Phys.