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We study Beurling-Fourier algebras of $ q $-deformations of compact semisimple Lie groups. In particular, we show that the space of irreducible representations of the function algebras of their Drinfeld doubles is exhausted by the irreducible representations of weighted Fourier algebras associated to a certain family of central weights., Comment: 18 pages

We study an averaging procedure for completely bounded multipliers on a locally compact quantum group with respect to a compact quantum subgroup. As a consequence we show that central approximation properties of discrete quantum groups are equivalent to the corresponding approximation properties of their Drinfeld doubles. This is complemented by a discussion of the averaging of Fourier algebra elements. We compare the biinvariant Fourier algebra of the Drinfeld double of a discrete quantum group with the central Fourier algebra. In the unimodular case these are naturally identified, but we show by exhibiting a family of counter-examples that they differ in general., Comment: 39 pages. Revision, correcting an error in Proposition 8.5 of the first version which affects some statements in section 8, and adding some further material

We study the Haagerup--Kraus approximation property for locally compact quantum groups, generalising and unifying previous work by Kraus--Ruan and Crann. Along the way we discuss how multipliers of quantum groups interact with the $\mathrm{C}^*$-algebraic theory of locally compact quantum groups. Several inheritance properties of the approximation property are established in this setting, including passage to quantum subgroups, free products of discrete quantum groups, and duals of double crossed products. We also discuss a relation to the weak$^*$ operator approximation property. For discrete quantum groups, we introduce a central variant of the approximation property, and relate this to a version of the approximation property for rigid $\mathrm{C}^*$-tensor categories, building on work of Arano--De Laat--Wahl., Comment: 54 pages; accepted version

We define and study quantum permutations of infinite sets. This leads to discrete quantum groups which can be viewed as infinite variants of the quantum permutation groups introduced by Wang. More precisely, the resulting quantum groups encode universal quantum symmetries of the underlying sets among all discrete quantum groups. We also discuss quantum automorphisms of infinite graphs, including some examples and open problems regarding both the existence and non-existence of quantum symmetries in this setting., Comment: 29 pages. Final version

Language models demonstrate both quantitative improvement and new qualitative capabilities with increasing scale. Despite their potentially transformative impact, these new capabilities are as yet poorly characterized. In order to inform future research, prepare for disruptive new model capabilities, and ameliorate socially harmful effects, it is vital that we understand the present and near-future capabilities and limitations of language models. To address this challenge, we introduce the Beyond the Imitation Game benchmark (BIG-bench). BIG-bench currently consists of 204 tasks, contributed by 450 authors across 132 institutions. Task topics are diverse, drawing problems from linguistics, childhood development, math, common-sense reasoning, biology, physics, social bias, software development, and beyond. BIG-bench focuses on tasks that are believed to be beyond the capabilities of current language models. We evaluate the behavior of OpenAI's GPT models, Google-internal dense transformer architectures, and Switch-style sparse transformers on BIG-bench, across model sizes spanning millions to hundreds of billions of parameters. In addition, a team of human expert raters performed all tasks in order to provide a strong baseline. Findings include: model performance and calibration both improve with scale, but are poor in absolute terms (and when compared with rater performance); performance is remarkably similar across model classes, though with benefits from sparsity; tasks that improve gradually and predictably commonly involve a large knowledge or memorization component, whereas tasks that exhibit "breakthrough" behavior at a critical scale often involve multiple steps or components, or brittle metrics; social bias typically increases with scale in settings with ambiguous context, but this can be improved with prompting., Comment: 27 pages, 17 figures + references and appendices, repo: https://github.com/google/BIG-bench

We introduce Hecke algebras associated to discrete quantum groups with commensurated quantum subgroups. We study their modular properties and the associated Hecke operators. In order to investigate their analytic properties we adapt the construction of the Schlichting completion to the quantum setting, thus obtaining locally compact quantum groups with compact open quantum subgroups. We study in detail a class of examples arising from quantum HNN extensions., Comment: 44 pages

Dark patterns utilize interface elements to trick users into performing unwanted actions. Online shopping websites often employ these manipulative mechanisms so as to increase their potential customer base, to boost their sales, or to optimize their advertising efforts. Although dark patterns are often successful, they clearly inhibit positive user experiences. Particularly, with respect to customers' perceived annoyance and trust put into a given brand, they may have negative effects. To investigate respective connections between the use of dark patterns, users' perceived level of annoyance and their expressed brand trust, we conducted an experiment-based survey. We implemented two versions of a fictitious online shop; i.e. one which used five different types of dark patterns and a similar one without such manipulative user interface elements. A total of $n=204$ participants were then forwarded to one of the two shops (approx. $2/3$ to the shop which used the dark patterns) and asked to buy a specific product. Subsequently, we measured participants' perceived annoyance level, their expressed brand trust and their affinity for technology. Results show a higher level of perceived annoyance with those who used the dark pattern version of the online shop. Also, we found a significant connection between perceived annoyance and participants' expressed brand trust. A connection between participants' affinity for technology and their ability to recognize and consequently counter dark patterns, however, is not supported by our data., Comment: 13 pages

Thinking aloud is an effective meta-cognitive strategy human reasoners apply to solve difficult problems. We suggest to improve the reasoning ability of pre-trained neural language models in a similar way, namely by expanding a task's context with problem elaborations that are dynamically generated by the language model itself. Our main result is that dynamic problem elaboration significantly improves the zero-shot performance of GPT-2 in a deductive reasoning and natural language inference task: While the model uses a syntactic heuristic for predicting an answer, it is capable (to some degree) of generating reasoned additional context which facilitates the successful application of its heuristic. We explore different ways of generating elaborations, including fewshot learning, and find that their relative performance varies with the specific problem characteristics (such as problem difficulty). Moreover, the effectiveness of an elaboration can be explained in terms of the degree to which the elaboration semantically coheres with the corresponding problem. In particular, elaborations that are most faithful to the original problem description may boost accuracy by up to 24%.

Motivated by the theory of Cuntz-Krieger algebras we define and study $ C^\ast $-algebras associated to directed quantum graphs. For classical graphs the $ C^\ast $-algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear. We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix block up to $ KK $-equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular that these $ C^\ast $-algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of $ KK $-theory. We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation. We also discuss quantum symmetries of quantum Cuntz-Krieger algebras in general., Comment: 40 pages

This paper takes a first step towards a critical thinking curriculum for neural auto-regressive language models. We introduce a synthetic corpus of deductively valid arguments, and generate artificial argumentative texts to train and evaluate GPT-2. Significant transfer learning effects can be observed: Training a model on three simple core schemes allows it to accurately complete conclusions of different, and more complex types of arguments, too. The language models generalize the core argument schemes in a correct way. Moreover, we obtain consistent and promising results for NLU benchmarks. In particular, pre-training on the argument schemes raises zero-shot accuracy on the GLUE diagnostics by up to 15 percentage points. The findings suggest that intermediary pre-training on texts that exemplify basic reasoning abilities (such as typically covered in critical thinking textbooks) might help language models to acquire a broad range of reasoning skills. The synthetic argumentative texts presented in this paper are a promising starting point for building such a "critical thinking curriculum for language models."

We discuss a number of general constructions concerning additive $ C^* $-categories, focussing in particular on establishing the existence of bicolimits. As an illustration of our results we show that balanced tensor products of module categories over $ C^* $-tensor categories exist without any finiteness assumptions., Comment: 39 pages

We show that complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups, satisfy a categorical version of the Baum-Connes conjecture with trivial coefficients. This approach, based on homological algebra in triangulated categories, is compatible with the previously studied deformation picture of the assembly map, and allows us to define an assembly map with arbitrary coefficients for these quantum groups., Comment: 20 pages. Revised version, to appear in IMRN

Hidden matter that interacts only gravitationally would oscillate at characteristic frequencies when trapped inside of Earth. For small oscillations near the center of the Earth, these frequencies are around 300 $\mu$Hz. Additionally, signatures at higher harmonics would appear because of the non-uniformity of Earth's density. In this work, we use data from a global network of gravimeters of the International Geodynamics and Earth Tide Service (IGETS) to look for these hypothetical trapped objects. We find no evidence for such objects with masses on the order of 10$^{14}$ kg or greater with an oscillation amplitude of 0.1 $r_e$. It may be possible to improve the sensitivity of the search by several orders of magnitude via better understanding of the terrestrial noise sources and more advanced data analysis.

We calculate the Plancherel formula for complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups. As a consequence we obtain a concrete description of their associated reduced group $ C^* $-algebras. The main ingredients in our proof are the Bernstein-Gelfand-Gelfand complex and the Hopf trace formula., Comment: 21 pages. Minor revision. Accepted for publication in Ann. Sci. Ec. Norm. Sup

We define and study an analogue of the Baum-Connes assembly map for complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups. Our starting point is the deformation picture of the Baum-Connes assembly map for a complex semisimple Lie group $ G $, which allows one to express the $ K $-theory of the reduced group $ C^* $-algebra of $ G $ in terms of the $ K $-theory of its associated Cartan motion group. The latter can be identified with the semidirect product of the maximal compact subgroup $ K $ acting on $ \mathfrak{k}^* $ via the coadjoint action. In the quantum case the role of the Cartan motion group is played by the Drinfeld double of the classical group $ K $, whose associated group $ C^* $-algebra is the crossed product of $ C(K) $ with respect to the adjoint action of $ K $. Our quantum assembly map is obtained by varying the deformation parameter in the Drinfeld double construction applied to the standard deformation $ K_q $ of $ K $. We prove that the quantum assembly map is an isomorphism, thus providing a description of the $ K $-theory of complex quantum groups in terms of classical topology. Moreover, we show that there is a continuous field of $ C^* $-algebras which encodes both the quantum and classical assembly maps as well as a natural deformation between them. It follows in particular that the quantum assembly map contains the classical Baum-Connes assembly map as a direct summand., Comment: 26 pages

These notes contain an introduction to the theory of complex semisimple quantum groups. Our main aim is to discuss the classification of irreducible Harish-Chandra modules for these quantum groups, following Joseph and Letzter. Along the way we cover extensive background material on quantized universal enveloping algebras and explain connections to the analytical theory in the setting of locally compact quantum groups., Comment: Minor revisions. To be published in Lecture Notes in Mathematics

The advent of novel measurement instrumentation can lead to paradigm shifts in scientific research. Optical atomic clocks, due to their unprecedented stability and uncertainty, are already being used to test physical theories and herald a revision of the International System of units (SI). However, to unlock their potential for cross-disciplinary applications such as relativistic geodesy, a major challenge remains. This is their transformation from highly specialized instruments restricted to national metrology laboratories into flexible devices deployable in different locations. Here we report the first field measurement campaign performed with a ubiquitously applicable $^{87}$Sr optical lattice clock. We use it to determine the gravity potential difference between the middle of a mountain and a location 90 km apart, exploiting both local and remote clock comparisons to eliminate potential clock errors. A local comparison with a $^{171}$Yb lattice clock also serves as an important check on the international consistency of independently developed optical clocks. This campaign demonstrates the exciting prospects for transportable optical clocks.

We introduce the spatial Rokhlin property for actions of coexact compact quantum groups on $\mathrm{C}^*$-algebras, generalizing the Rokhlin property for both actions of classical compact groups and finite quantum groups. Two key ingredients in our approach are the concept of sequentially split $*$-homomorphisms, and the use of braided tensor products instead of ordinary tensor products. We show that various structure results carry over from the classical theory to this more general setting. In particular, we show that a number of $\mathrm{C}^*$-algebraic properties relevant to the classification program pass from the underlying $\mathrm{C}^*$-algebra of a Rokhlin action to both the crossed product and the fixed point algebra. Towards establishing a classification theory, we show that Rokhlin actions exhibit a rigidity property with respect to approximate unitary equivalence. Regarding duality theory, we introduce the notion of spatial approximate representability for actions of discrete quantum groups. The spatial Rokhlin property for actions of a coexact compact quantum group is shown to be dual to spatial approximate representability for actions of its dual discrete quantum group, and vice versa., Comment: 47 pages; v2 minor corrections. This version is going to appear in J. Funct. Anal

We introduce $C^*$-algebras associated to the foliation structure of a quantum flag manifold. We use these to construct $SL_q(3,\mathbb{C})$-equivariant Fredholm modules for the full quantum flag manifold $X_q = SU_q(3)/T$ of $SU_q(3)$, based on an analytical version of the Bernstein-Gelfand-Gelfand complex. As a consequence we deduce that the flag manifold $ X_q $ satisfies Poincar\'e duality in equivariant $ KK $-theory. Moreover, we show that the Baum-Connes conjecture with trivial coefficients holds for the discrete quantum group dual to $SU_q(3)$., Comment: 43 pages

We compute the $ K $-theory of quantum automorphism groups of finite dimensional $ C^* $-algebras in the sense of Wang. The results show in particular that the $ C^* $-algebras of functions on the quantum permutation groups $ S_n^+ $ are pairwise non-isomorphic for different values of $ n $. Along the way we discuss some general facts regarding torsion in discrete quantum groups. In fact, the duals of quantum automorphism groups are the most basic examples of discrete quantum groups exhibiting genuine quantum torsion phenomena., Comment: 16 pages

We present a modified version of the definition of property RD for discrete quantum groups given by Vergnioux in order to accommodate examples of non-unimodular quantum groups. Moreover we extend the construction of spectral triples associated to discrete groups with length functions, originally due to Connes, to the setting of quantum groups. For quantum groups of rapid decay we study the resulting spectral triples from the point of view of compact quantum metric spaces in the sense of Rieffel., Comment: 19 pages