Compact I/O-Efficient Representation of Separable Graphs and Optimal Tree Layouts

Compact and I/O-efficient data representations play an important role in efficient algorithm design, as memory bandwidth and latency can present a significant performance bottleneck, slowing the computation by orders of magnitude. While this problem is very well explored in e.g. uniform numerical data processing, structural data applications (e.g. on huge graphs) require different algorithm-dependent approaches. Separable graph classes (i.e. graph classes with balanced separators of size $\mathcal{O}(n^c)$ with $c < 1$) include planar graphs, bounded genus graphs, and minor-free graphs. In this article we present two generalizations of the separator theorem, to partitions with small regions only on average and to weighted graphs. Then we propose I/O-efficient succinct representation and memory layout for random walks in(weighted) separable graphs in the pointer machine model, including an efficient algorithm to compute them. Finally, we present a worst-case I/O-optimal tree layout algorithm for root-leaf path traversal, show an additive (+1)-approximation of optimal compact layout and contrast this with NP-completeness proof of finding an optimal compact layout

Dynamics on the PSL(2,C)-character variety of a compression body

Let M be a nontrivial compression body without toroidal boundary components. We study the dynamics of the group of outer automorphisms of the fundamental group of M on the PSL(2,C)-character variety of M.Comment: 31 pages, 5 figure

Spectral rigidity of group actions on homogeneous spaces

Actions of a locally compact group G on a measure space X give rise to unitary representations of G on Hilbert spaces. We review results on the rigidity of these actions from the spectral point of view, that is, results about the existence of a spectral gap for associated averaging operators and their consequences. We will deal both with spaces X with an infinite measure as well as with spaces with an invariant probability measure. The spectral gap property has several striking applications to group theory, geometry, ergodic theory, operator algebras, graph theory, theoretical computer science, etc

The Space of Actions, Partition Metric and Combinatorial Rigidity

We introduce a natural pseudometric on the space of actions of $d$-generated groups, such that the zero classes are exactly the weak equivalence classes and the metric identification with respect to this pseudometric is compact. We analyze convergence in this space and prove that every class contains an action that properly satisfies every combinatorial type condition that it satisfies with arbitrarily small error. We also show that the class of every free non-amenable action contains an action that satisfies the measurable von Neumann problem. The results have analogues in the realm of unitary representations as well

Physical Diffeomorphisms in Loop Quantum Gravity

We investigate the action of diffeomorphisms in the context of Hamiltonian Gravity. By considering how the diffeomorphism-invariant Hilbert space of Loop Quantum Gravity should be constructed, we formulate a physical principle by demanding, that the gauge-invariant Hilbert space is a completion of gauge- (i.e. diffeomorphism-)orbits of the classical (configuration) variables, explaining which extensions of the group of diffeomorphisms must be implemented in the quantum theory. It turns out, that these are at least a subgroup of the stratified analytic diffeomorphisms. Factoring these stratified diffeomorphisms out, we obtain that the orbits of graphs under this group are just labelled by their knot classes, which in turn form a countable set. Thus, using a physical argument, we construct a separable Hilbert space for diffeomorphism invariant Loop Quantum Gravity, that has a spin-knot basis, which is labelled by a countable set consisting of the combination of knot-classes and spin quantum numbers. It is important to notice, that this set of diffeomorphism leaves the set of piecewise analytic edges invariant, which ensures, that one can construct flux-operators and the associated Weyl-operators. A note on the implications for the treatment of the Gauss- and the Hamilton-constraint of Loop Quantum Gravity concludes our discussion.Comment: 25 pages, 2 figures, LaTe

Compact Support Biorthogonal Wavelet Filterbanks for Arbitrary Undirected Graphs

In our recent work, we proposed the design of perfect reconstruction orthogonal wavelet filterbanks, called graph- QMF, for arbitrary undirected weighted graphs. In that formulation we first designed "one-dimensional" two-channel filterbanks on bipartite graphs, and then extended them to "multi-dimensional" separable two-channel filterbanks for arbitrary graphs via a bipartite subgraph decomposition. We specifically designed wavelet filters based on the spectral decomposition of the graph, and stated necessary and sufficient conditions for a two-channel graph filter-bank on bipartite graphs to provide aliasing-cancellation, perfect reconstruction and orthogonal set of basis (orthogonality). While, the exact graph-QMF designs satisfy all the above conditions, they are not exactly k-hop localized on the graph. In this paper, we relax the condition of orthogonality to design a biorthogonal pair of graph-wavelets that can have compact spatial spread and still satisfy the perfect reconstruction conditions. The design is analogous to the standard Cohen-Daubechies-Feauveau's (CDF) construction of factorizing a maximally-flat Daubechies half-band filter. Preliminary results demonstrate that the proposed filterbanks can be useful for both standard signal processing applications as well as for signals defined on arbitrary graphs. Note: Code examples from this paper are available at http://biron.usc.edu/wiki/index.php/Graph FilterbanksComment: Submitted for review in IEEE TS

Inducing Primitive Ideals

We study conditions on a $C^*$-dynamical system $(A,G,\alpha)$ under which induction of primitive ideals (resp. irreducible representations) from stabilizers for the action of $G$ on the primitive ideal space \Prim(A) give primitive ideals (resp. irreducible representations) of the crossed product $A\rtimes_\alpha G$. The results build on earlier results of Sauvageot and others, and will correct a (possibly overly optimistic) statement of the first author.Comment: 17 Page

Separable representations of higher-rank graphs

In this monograph we undertake a comprehensive study of separable representations (as well as their unitary equivalence classes) of $C^*$-algebras associated to strongly connected finite $k$-graphs $\Lambda$. We begin with the representations associated to the $\Lambda$-semibranching function systems introduced by Farsi, Gillaspy, Kang, and Packer in \cite{FGKP}, by giving an alternative characterization of these systems which is more easily verified in examples. We present a variety of such examples, one of which we use to construct a new faithful separable representation of any row-finite source-free $k$-graph. Next, we analyze the monic representations of $C^*$-algebras of finite $k$-graphs. We completely characterize these representations, generalizing results of Dutkay and Jorgensen \cite{dutkay-jorgensen-monic} and Bezuglyi and Jorgensen \cite{bezuglyi-jorgensen} for Cuntz and Cuntz-Krieger algebras respectively. We also describe a universal representation for non-negative monic representations of finite, strongly connected $k$-graphs. To conclude, we characterize the purely atomic and permutative representations of $k$-graph $C^*$-algebras, and discuss the relationship between these representations and the classes of representations introduced earlier.Comment: 105 page

Naimark's problem for graph C*-algebras

Naimark's problem asks whether a C*-algebra that has only one irreducible *-representation up to unitary equivalence is isomorphic to the C*-algebra of compact operators on some (not necessarily separable) Hilbert space. This problem has been solved in special cases, including separable C*-algebras and Type I C*-algebras. However, in 2004 Akemann and Weaver used the diamond principle to construct a C*-algebra with $\aleph_1$ generators that is a counterexample to Naimark's Problem. More precisely, they showed that the statement "There exists a counterexample to Naimark's Problem that is generated by $\aleph_1$ elements." is independent of the axioms of ZFC. Whether Naimark's problem itself is independent of ZFC remains unknown. In this paper we examine Naimark's problem in the setting of graph C*-algebras, and show that it has an affirmative answer for (not necessarily separable) AF graph C*-algebras as well as for C*-algebras of graphs in which each vertex emits a countable number of edges.Comment: Version II Comments: Minor changes. Small typos correcte

Representations of higher-rank graph C∗C^*-algebras associated to Λ\Lambda-semibranching function systems

In this paper, we discuss a method of constructing separable representations of the $C^*$-algebras associated to strongly connected row-finite $k$-graphs $\Lambda$. We begin by giving an alternative characterization of the $\Lambda$-semibranching function systems introduced in an earlier paper, with an eye towards constructing such representations that are faithful. Our new characterization allows us to more easily check that examples satisfy certain necessary and sufficient conditions. We present a variety of new examples relying on this characterization. We then use some of these methods and a direct limit procedure to construct a faithful separable representation for any row-finite source-free $k$-graph.Comment: This paper constitutes a partial replacement of arXiv:1709.00592; the latter will not be submitted for publicatio