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Using Kirchberg-Phillips' classification of purely infinite C*-algebras by K-theory, we prove that the isomorphism types of crossed product C*-algebras associated to certain hyperbolic 3-manifold groups acting on their Gromov boundary only depend on the manifold's homology. As a result, we obtain infinitely many pairwise non-isomorphic hyperbolic groups all of whose associated crossed products are isomorphic. These isomomorphisms are not of dynamical nature in the sense that they are not induced by isomorphisms of the underlying groupoids., Comment: 11 pages; revision related to Theorem A: Bram Mesland and Haluk \c{S}eng\"un brought to our attention that a result analogous to Theorem A was obtained by them in the case of non-compact manifolds, using identical techniques (reference within the article)

We explore classifiability of crossed products of actions of countable amenable groups on compact, metrizable spaces. It is completely understood when such crossed products are simple, separable, unital, nuclear and satisfy the UCT: these properties are equivalent to the combination of minimality and topological freeness, and the challenge in this context is establishing $\mathcal{Z}$-stability. While most of the existing results in this direction assume freeness of the action, there exist numerous natural examples of minimal, topologically free (but not free) actions whose crossed products are classifiable. In this work, we take the first steps towards a systematic study of $\mathcal{Z}$-stability for crossed products beyond the free case, extending the available machinery around the small boundary property and almost finiteness to a more general setting. Among others, for actions of groups of polynomial growth with the small boundary property, we show that minimality and topological freeness are not just necessary, but also \emph{sufficient} conditions for classifiability of the crossed product. Our most general results apply to actions that are essentially free, a property weaker than freeness but stronger than topological freeness in the minimal setting. Very recently, M. Joseph produced the first examples of minimal actions of amenable groups which are topologically free and not essentially free. While the current machinery does not give any information for his examples, we develop ad-hoc methods to show that his actions have classifiable crossed products., Comment: 31 pages

The need to understand the structure of hierarchical or high-dimensional data is present in a variety of fields. Hyperbolic spaces have proven to be an important tool for embedding computations and analysis tasks as their non-linear nature lends itself well to tree or graph data. Subsequently, they have also been used in the visualization of high-dimensional data, where they exhibit increased embedding performance. However, none of the existing dimensionality reduction methods for embedding into hyperbolic spaces scale well with the size of the input data. That is because the embeddings are computed via iterative optimization schemes and the computation cost of every iteration is quadratic in the size of the input. Furthermore, due to the non-linear nature of hyperbolic spaces, Euclidean acceleration structures cannot directly be translated to the hyperbolic setting. This paper introduces the first acceleration structure for hyperbolic embeddings, building upon a polar quadtree. We compare our approach with existing methods and demonstrate that it computes embeddings of similar quality in significantly less time. Implementation and scripts for the experiments can be found at https://graphics.tudelft.nl/accelerating-hyperbolic-tsne.

In this paper, we give a complete, two-way characterization, of when a noncommutative crossed product $A \rtimes_\lambda G$ is simple, in the case of $G$ being an FC-hypercentral group. This is a large class of amenable groups that contains all virtually nilpotent groups, and in the finitely-generated setting, coincides with the set of groups which have polynomial growth. We further completely characterize the ideal intersection property under the assumption that the group is FC, meaning that every element has a finite conjugacy class. Finally, for minimal actions of arbitrary discrete groups on unital C*-algebras, we are able to characterize when the crossed product $A \rtimes_\lambda G$ is prime., Comment: 63 pages, 2 figures; Cleaned up statements of main results. Added section 8 on the non-unital case and section 9 on examples

We introduce the notion of tracial amenability for actions of discrete groups on unital, tracial C$^*$-algebras, as a weakening of amenability where all the relevant approximations are done in the uniform trace norm. We characterize tracial amenability with various equivalent conditions, including topological amenability of the induced action on the trace space. Our main result concerns the structure of crossed products: for groups containing the free group $F_2$, we show that outer, tracially amenable actions on simple, unital, $\mathcal{Z}$-stable C$^*$-algebras always have purely infinite crossed products. Finally, we give concrete examples of tracially amenable actions of free groups on simple, unital AF-algebras., Comment: v2: 21 pages. This version has been accepted to Mathematische Annalen

We establish $\mathcal{Z}$-stability for crossed products of outer actions of amenable groups on $\mathcal{Z}$-stable $C^*$-algebras under a mild technical assumption which we call McDuff property with respect to invariant traces. We obtain such result using a weak form of dynamical comparison, which we verify in great generality. We complement our results by proving that McDuffness with respect to invariant traces is automatic in many cases of interest. This is the case, for instance, for every action of an amenable group $G$ on a classifiable $C^*$-algebra $A$ whose trace space $T(A)$ is a Bauer simplex with finite dimensional boundary $\partial_e T(A)$, and such that the induced action $G\curvearrowright \partial_eT(A)$ is free. If $G = \mathbb{Z}^d$ and the action $G\curvearrowright \partial_eT(A)$ is free and minimal, then we obtain McDuffness with respect to invariant traces, and thus $\mathcal{Z}$-stability of the corresponding crossed product, also in case $\partial_e T(A)$ has infinite covering dimension., Comment: 26 pages

We show that all amenable, minimal actions of a large class of nonamenable countable groups on compact metric spaces have dynamical comparison. This class includes all nonamenable hyperbolic groups, many HNN-extensions, nonamenable Baumslag-Solitar groups, a large class of amalgamated free products, lattices in many Lie groups, $\widetilde{A}_2$-groups, as well as direct products of the above with arbitrary countable groups. As a consequence, crossed products by amenable, minimal and topologically free actions of such groups on compact metric spaces are Kirchberg algebras in the UCT class, and are therefore classified by $K$-theory., Comment: v2: 30 pages, 1 figure. To appear in J. Reine Angew. Math. (Crelle)

We develop the notion of Rokhlin dimension for partial actions of finite groups, extending the well-established theory for global systems. The partial setting exhibits phenomena that cannot be expected for global actions, usually stemming from the fact that virtually all averaging arguments for finite group actions completely break down for partial systems. For example, fixed point algebra and crossed product are not in general Morita equivalent, and there is in general no local approximation of the crossed product $A\rtimes G$ by matrices over $A$. By using decomposition arguments for partial actions of finite groups, we show that a number of structural properties are preserved by formation of crossed products, including finite stable rank, finite nuclear dimension, and absorption of a strongly self-absorbing $C^*$-algebra. For partial actions with the Rokhlin property, being an AF-algebra is also preserved. Some of our results are new even in the global case. We also study the Rokhlin dimension of globalizable actions: while in general it differs from the Rokhlin dimension of its globalization, we show that they agree if the coefficient algebra is unital. For topological partial actions on spaces of finite covering dimension, we show that finiteness of the Rokhlin dimension is equivalent to freeness, thus providing a large class of examples to which our theory applies., Comment: V2: minor changes, 29 pages. This version was accepted in Ergodic Theory Dynam. Syst

We propose a general framework to study language emergence through signaling games with neural agents. Using a continuous latent space, we are able to (i) train using backpropagation, (ii) show that discrete messages nonetheless naturally emerge. We explore whether categorical perception effects follow and show that the messages are not compositional., Comment: ACL 2020

We define the decomposition property for partial actions of discrete groups on $C^*$-algebras. Decomposable partial systems appear naturally in practice, and many commonly occurring partial actions can be decomposed into partial actions with the decomposition property. For instance, any partial action of a finite group is an iterated extension of decomposable systems. Partial actions with the decomposition property are always globalizable and amenable, regardless of the acting group, and their globalization can be explicitly described in terms of certain global sub-systems. A direct computation of their crossed products is also carried out. We show that partial actions with the decomposition property behave in many ways like global actions of finite groups (even when the acting group is infinite), which makes their study particularly accessible. For example, there exists a canonical faithful conditional expectation onto the fixed point algebra, which is moreover a corner in the crossed product in a natural way. (Both of these facts are in general false for partial actions of finite groups.) As an application, we show that freeness of a topological partial action with the decomposition property is equivalent to its fixed point algebra being Morita equivalent to its crossed product. We also show by example that this fails for general partial actions of finite groups., Comment: V2: minor changes, 26 pages. This is the accepted version

We bound the nuclear dimension of crossed products associated to some partial actions of finite groups or $\mathbb{Z}$ on finite dimensional locally compact Hausdorff second countable spaces. Our results apply to globalizable partial actions, finite group partial actions, minimal partial automorphisms, and partial automorphisms acting on zero-dimensional spaces, or a class of one dimensional spaces, containing $1$-dimensional CW complexes. This extends work on global systems by Hirshberg and Wu., Comment: 25 pages, 1 figure. To appear in J. Funct. Anal. Minor changes, accepted version

Multilink is a computational model for word retrieval in monolingual and multilingual individuals under different task circumstances (Dijkstra et al., 2018). In the present study, we added lateral inhibition to Multilink's lexical network. Parameters were fit on the basis of reaction times from the English, British, and Dutch Lexicon Projects. We found a maximum correlation of 0.643 (N=1,205) on these data sets as a whole. Furthermore, the simulations themselves became faster as a result of adding lateral inhibition. We tested the fitted model to stimuli from a neighbourhood study (Mulder et al., 2018). Lateral inhibition was found to improve Multilink's correlations for this study, yielding an overall correlation of 0.67. Next, we explored the role of lateral inhibition as part of the model's task/decision system by running simulations on data from two studies concerning interlingual homographs (Vanlangendonck et al., in press; Goertz, 2018). We found that, while lateral inhibition plays a substantial part in the word selection process, this alone is not enough to result in a correct response selection. To solve this problem, we added a new task component to Multilink, especially designed to account for the translation process of interlingual homographs, cognates, and language-specific control words. The subsequent simulation results showed patterns remarkably similar to those in the Goertz study. The isomorphicity of the simulated data to the empirical data was further attested by an overall correlation of 0.538 (N=254) between reaction times and simulated model cycle times, as well as a condition pattern correlation of 0.853 (N=8). We conclude that Multilink yields an excellent fit to empirical data, particularly when a task-specific setting of the inhibition parameters is allowed.

Since their inception in the 30's by von Neumann, operator algebras have been used in shedding light in many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two was sought since their emergence in the late 60's. We connect these seemingly separate type of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and $C^*$-algebras with additional $C^*$-algebraic structure. Our approach naturally applies to algebras arising from $C^*$-correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs., Comment: 1 figure, 28 pages. To appear in Compositio Mathematica

Neural systems can be modeled as networks of functionally connected neural elements. The resulting network can be analyzed using mathematical tools from network science and graph theory to quantify the system's topological organization and to better understand its function. While the network-based approach is common in the analysis of large-scale neural systems probed by non-invasive neuroimaging, few studies have used network science to study the organization of networks reconstructed at the cellular level, and thus many very basic and fundamental questions remain unanswered. Here, we used two-photon calcium imaging to record spontaneous activity from the same set of cells in mouse auditory cortex over the course of several weeks. We reconstruct functional networks in which cells are linked to one another by edges weighted according to the correlation of their fluorescence traces. We show that the networks exhibit modular structure across multiple topological scales and that these multi-scale modules unfold as part of a hierarchy. We also show that, on average, network architecture becomes increasingly dissimilar over time, with similarity decaying monotonically with the distance (in time) between sessions. Finally, we show that a small fraction of cells maintain strongly-correlated activity over multiple days, forming a stable temporal core surrounded by a fluctuating and variable periphery. Our work provides a careful methodological blueprint for future studies of spontaneous activity measured by two-photon calcium imaging using cutting-edge computational methods and machine learning algorithms informed by explicit graphical models from network science. The methods are easily extended to additional datasets, opening the possibility of studying cellular level network organization of neural systems and how that organization is modulated by stimuli or altered in models of disease., Comment: 15 pages, 3 figures

Many combinatorial problems can be solved in time $O^*(c^{tw})$ on graphs of treewidth $tw$, for a problem-specific constant $c$. In several cases, matching upper and lower bounds on $c$ are known based on the Strong Exponential Time Hypothesis (SETH). In this paper we investigate the complexity of solving problems on graphs of bounded cutwidth, a graph parameter that takes larger values than treewidth. We strengthen earlier treewidth-based lower bounds to show that, assuming SETH, Independent Set cannot be solved in $O^*((2-\varepsilon)^{cutw})$ time, and Dominating Set cannot be solved in $O^*((3-\varepsilon)^{cutw})$ time. By designing a new crossover gadget, we extend these lower bounds even to planar graphs of bounded cutwidth or treewidth. Hence planarity does not help when solving Independent Set or Dominating Set on graphs of bounded width. This sharply contrasts the fact that in many settings, planarity allows problems to be solved much more efficiently.

Explicit signaling between threads is a perennial cause of bugs in concurrent programs. While there are several run-time techniques to automatically notify threads upon the availability of some shared resource, such techniques are not widely-adopted due to their run-time overhead. This paper proposes a new solution based on static analysis for automatically generating a performant explicit-signal program from its corresponding implicit-signal implementation. The key idea is to generate verification conditions that allow us to minimize the number of required signals and unnecessary context switches, while guaranteeing semantic equivalence between the source and target programs. We have implemented our method in a tool called Expresso and evaluate it on challenging benchmarks from prior papers and open-source software. Expresso-generated code significantly outperforms past automatic signaling mechanisms (avg. 1.56x speedup) and closely matches the performance of hand-optimized explicit-signal code.