Causal Space-Times on a Null Lattice

I investigate a discrete model of quantum gravity on a causal null-lattice with \SLC structure group. The description is geometric and foliates in a causal and physically transparent manner. The general observables of this model are constructed from local Lorentz symmetry considerations only. For smooth configurations, the local lattice actions reduce to the Hilbert-Palatini action, a cosmological term and the three topological terms of dimension four of Pontyagin, Euler and Nieh-Yan. Consistency conditions for a topologically hypercubic complex with null 4-simplexes are derived and a topological lattice theory that enforces these non-local constraints is constructed. The lattice integration measure is derived from an \SLC-invariant integration measure by localization of the non-local structure group. This measure is unique up to a density that depends on the local 4-volume. It can be expressed in terms of manifestly coordinate invariant geometrical quantities. The density provides an invariant regularization of the lattice integration measure that suppresses configurations with small local 4-volumes. Amplitudes conditioned on geodesic distances between local observables have a physical interpretation and may have a smooth ultraviolet limit. Numerical studies on small lattices in the unphysical strong coupling regime of large imaginary cosmological constant suggest that this model of triangulated causal manifolds is finite. Two topologically different triangulations of space-time are discussed: a single, causally connected universe and a duoverse with two causally disjoint connected components. In the duoverse, two hypercubic sublattices are causally disjoint but the local curvature depends on fields of both sublattices. This may simulate effects of dark matter in the continuum limit.Comment: Greatly improved version, new numerics, appendices, etc.. 42 pages, 14 figure

Group field theories for all loop quantum gravity

Group field theories represent a 2nd quantized reformulation of the loop quantum gravity state space and a completion of the spin foam formalism. States of the canonical theory, in the traditional continuum setting, have support on graphs of arbitrary valence. On the other hand, group field theories have usually been defined in a simplicial context, thus dealing with a restricted set of graphs. In this paper, we generalize the combinatorics of group field theories to cover all the loop quantum gravity state space. As an explicit example, we describe the GFT formulation of the KKL spin foam model, as well as a particular modified version. We show that the use of tensor model tools allows for the most effective construction. In order to clarify the mathematical basis of our construction and of the formalisms with which we deal, we also give an exhaustive description of the combinatorial structures entering spin foam models and group field theories, both at the level of the boundary states and of the quantum amplitudes.Comment: version published in New Journal of Physic

Propagation and interaction of chiral states in quantum gravity

We study the stability, propagation and interactions of braid states in models of quantum gravity in which the states are four-valent spin networks embedded in a topological three manifold and the evolution moves are given by the dual Pachner moves. There are results for both the framed and unframed case. We study simple braids made up of two nodes which share three edges, which are possibly braided and twisted. We find three classes of such braids, those which both interact and propagate, those that only propagate, and the majority that do neither.Comment: 34 pages, 30 figures, typos corrected, 2 references added, to match the version accepted for publication in Nucl. Phys.

A unified approach to reverse engineering and data selection for unique network identification

Due to cost concerns, it is optimal to gain insight into the connectivity of biological and other networks using as few experiments as possible. Data selection for unique network connectivity identification has been an open problem since the introduction of algebraic methods for reverse engineering for almost two decades. In this manuscript we determine what data sets uniquely identify the unsigned wiring diagram corresponding to a system that is discrete in time and space. Furthermore, we answer the question of uniqueness for signed wiring diagrams for Boolean networks. Computationally, unsigned and signed wiring diagrams have been studied separately, and in this manuscript we also show that there exists an ideal capable of encoding both unsigned and signed information. This provides a unified approach to studying reverse engineering that also gives significant computational benefits.Comment: 21 page

Proceedings of the Workshop on Algorithmic Aspects of Advanced Programming Languages: WAAAPL'99: Paris, France, September 30, 1999

The first Workshop on Algorithmic Aspects of Advanced Programming Languages was held on September 30, 1999, in Paris, France, in conjunction with the PLI'99 conferences and workshops. The choice of programming languages has a huge effect on the algorithms and data structures that are to be implemented in that language. Traditionally, algorithms and data structures have been studied in the context of imperative languages. This workshop considers the algorithmic implications of choosing an advanced functional or logic programming language instead. A total of eight papers were selected for presentation at the workshop, together with an invited lecture by Robert Harper. We would like to thank Dider Remv, general chair of PLI'99, for his assistance in organizing this workshop

Refinement of Interval Approximations for Fully Commutative Quivers

A fundamental challenge in multiparameter persistent homology is the absence of a complete and discrete invariant. To address this issue, we propose an enhanced framework that realizes a holistic understanding of a fully commutative quiver's representation via synthesizing interpretations obtained from intervals. Additionally, it provides a mechanism to tune the balance between approximation resolution and computational complexity. This framework is evaluated on commutative ladders of both finite-type and infinite-type. For the former, we discover an efficient method for the indecomposable decomposition leveraging solely one-parameter persistent homology. For the latter, we introduce a new invariant that reveals persistence in the second parameter by connecting two standard persistence diagrams using interval approximations. We subsequently present several models for constructing commutative ladder filtrations, offering fresh insights into random filtrations and demonstrating our toolkit's effectiveness in analyzing the topology of materials

IST Austria Thesis

This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph. For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton. In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars