arXiv subjects

Jianchao Wu

Publications and source records attributed to Jianchao Wu.

At least 19 recordsLinked to original sources

arXiv2026

Almost elementary étale groupoids

Motivated by Matui and Kerr's work on almost finiteness, we introduce a new finite approximation property for (possibly non-ample) étale groupoids called {almost elementariness}, which unifies and generalizes both almost finiteness and pure infiniteness. This property serves as a dynamical analogue of regularity properties of $C^*$-algebras. In support of this view, we prove that minimal almost elementary groupoids yield tracially $\mathcal{Z}$-stable reduced groupoid $C^*$-algebras. Consequently, we obtain as a corollary that the reduced $C^*$-algebras of all minimal amenable second countable almost finite groupoids in Matui's sense are $\mathcal{Z}$-stable and thus classifiable by the Elliott invariants. Two basic ingredients underlying the definition of almost elementariness are castles in groupoids and groupoid subequivalence, both of which are developed extensively in this work. Notably, building on our flexible framework of castles, we introduce the technique of nesting of castles, which play a key role in the proof of our main theorem. Besides our main theorem on tracial $\mathcal{Z}$-stability, we also discuss in depth the relations between almost elementariness and other properties for groupoids such as effectiveness, the groupoid small boundary property, groupoid strict comparison and Matui's and Kerr's notions of almost finiteness.

math.OA↗

arXiv2026

Almost elementary groupoid models for $C^*$-algebras

The notion of almost elementariness for a locally compact Hausdorff étale groupoid $\mathcal{G}$ with a compact unit space was introduced by the authors as a sufficient condition ensuring the reduced groupoid $C^*$-algebra $C^*_r(\mathcal{G})$ is (tracially) $\mathcal{Z}$-stable and thus classifiable under additional natural assumptions. In this paper, we explore the converse direction and show that many groupoids in the literature serving as models for classifiable $C^*$-algebras are almost elementary. In particular, for a large class $\mathcal{C}$ of Elliott invariants and a $C^*$-algebra $A$ with $\operatorname{Ell}(A)\in \mathcal{C}$, we show that $A$ is classifiable if and only if $A$ possesses a minimal, effective, amenable, second countable, almost elementary groupoid model, which leads to a groupoid-theoretic characterization of classifiability of $C^*$-algebras with certain Elliott invariants. In addition, we demonstrate obstructions to obtaining a transformation groupoid model for the Jiang-Su algebra $\mathcal{Z}$.

math.OA↗

arXiv2026

Fiberwise amenability of étale groupoids

We introduce a new amenability property for étale groupoids, termed \textit{fiberwise amenability}, along with a stronger variant termed \emph{ubiquitous fiberwise amenability}. (Ubiquitous) fiberwise amenability emerges naturally from a coarse-geometric perspective on étale groupoids and, in the special case of transformation groupoids, it coincides precisely with the amenability of the acting group (rather than topological amenability of the action). It is also tightly linked to the existence of invariant measures on the unit space of the groupoid. The coarse-geometric framework for étale groupoids that we develop systematically in this work allows us to establish several foundational properties of (ubiquitous) fiberwise amenability. As an application, we prove a Følner--paradoxical dichotomy for minimal étale groupoids, which will serve as a key tool in a sequel on almost elementariness of étale groupoids.

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arXiv2026

Rokhlin dimension for actions of residually compact groups

We introduce the concept of Rokhlin dimension for actions of residually compact groups on C*-algebras, which extends and unifies previous notions for actions of compact groups, residually finite groups and the reals. We then demonstrate that finite nuclear dimension (respectively, absorption of a strongly self-absorbing C*-algebra) is preserved under the formation of crossed products by residually compact group actions with finite Rokhlin dimension (respectively, finite Rokhlin dimension with commuting towers). Furthermore, if second countable residually compact group contains a non-open cocompact closed subgroup, then crossed products arising from actions with finite Rokhlin dimension are stable. Finally, we study the relationship between the tube dimension of a topological dynamical system and the Rokhlin dimension of the induced C*-dynamical system.

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arXiv2026

Nuclear dimension and virtually polycyclic groups

We show that the nuclear dimension of a (twisted) group C*-algebra of a virtually polycyclic group is finite. This prompts us to make a conjecture relating finite nuclear dimension of group C*-algebras and finite Hirsch length, which we then verify for a class of elementary amenable groups beyond the virtually polycyclic case. In particular, we give the first examples of finitely generated, non-residually finite groups with finite nuclear dimension. A parallel conjecture on finite decomposition rank is also formulated and an analogous result is obtained. Our method relies heavily on recent work of Hirshberg and the second named author on actions of virtually nilpotent groups on $C_0(X)$-algebras.

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arXiv2025

StoryTeller: Improving Long Video Description through Global Audio-Visual Character Identification

Existing large vision-language models (LVLMs) are largely limited to processing short, seconds-long videos and struggle with generating coherent descriptions for extended video spanning minutes or more. Long video description introduces new challenges, such as consistent character identification and plot-level descriptions incorporating both visual and audio information. To address these, we figure out audio-visual character identification, matching character names to each dialogue, as a key factor. We propose StoryTeller, a system for generating dense descriptions of long videos, incorporating both low-level visual concepts and high-level plot information. StoryTeller uses a multimodal large language model that integrates visual, audio, and text modalities to perform audio-visual character identification on minute-long video clips. The results are then fed into a LVLM to enhance consistency of video description. We validate our approach on movie description tasks and introduce MovieStory101, a dataset with dense descriptions for three-minute movie clips. To evaluate long video descriptions, we create StoryQA, a large set of multiple-choice questions for MovieStory101 test set. We assess descriptions by inputting them into GPT-4 to answer these questions, using accuracy as an automatic evaluation metric. Experiments show that StoryTeller outperforms all open and closed-source baselines on StoryQA, achieving 9.5% higher accuracy than the strongest baseline, Gemini-1.5-pro, and demonstrating a +15.56% advantage in human side-by-side evaluations. Additionally, incorporating audio-visual character identification from StoryTeller improves the performance of all video description models, with Gemini-1.5-pro and GPT-4o showing relative improvement of 5.5% and 13.0%, respectively, in accuracy on StoryQA.

cs.CV↗

arXiv2025

Hilbert-Hadamard spaces and the equivariant coarse Novikov conjecture

The equivariant coarse Novikov conjectures stand among a handful profound $K$-theoretic conjectures in noncommutative geometry. Motivated by the quest to verify Novikov-type conjectures for groups of diffeomorphisms, we study in this paper the equivariant coarse Novikov conjectures for spaces that equivariantly and coarsely embed into admissible Hilbert-Hadamard spaces, which are a type of infinite-dimensional nonpositively curved spaces. The paper is split into two parts. We prove in the first part that for any metric space $X$ with bounded geometry and with a proper isometric action $α$ by a countable discrete group $Γ$, if $X$ admits an equivariant coarse embedding into an admissible Hilbert-Hadamard space and $Γ$ is torsion-free, then the equivariant coarse strong Novikov conjecture holds rationally for $(X, Γ, α)$. In the second part, we extend the result in the first part by dropping the torsion-free assumption on $Γ$. To this end, we introduce, for a proper $Γ$-space $X$ with equivariant bounded geometry, a new Novikov-type conjecture that we call the rational analytic equivariant coarse Novikov conjecture, which generalizes the rational analytic Novikov conjecture and asserts the rational injectivity of a certain assembly map associated with a coarse analog of the classifying space $EΓ$. We show that for a proper $Γ$-space $X$ with equivariant bounded geometry, if $X$ admits an equivariant coarse embedding into an admissible Hilbert-Hadamard space, then the rational analytic equivariant coarse Novikov conjecture holds for $(X,Γ,α)$, i.e., the assembly map is a rational injection.

math.KT↗

arXiv2025

The Novikov conjecture, the group of diffeomorphisms and continuous fields of Hilbert-Hadamard spaces

In this paper, we prove the Novikov conjecture for a class of highly non-linear groups, namely discrete subgroups of the diffeomorphism group of a compact smooth manifold. This removes the volume-preserving condition in a previous work. This result is proved by studying operator $K$-theory and group actions on continuous fields of infinite dimensional non-positively curved spaces.

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arXiv2024

The dynamical Cuntz semigroup and ideal-free quotients of Cuntz semigroups

We develop a theory of general quotients for W- and Cu-semigroups beyond the case of quotients by ideals. To this end, we introduce the notion of a normal pair, which allows us to take quotients of W-semigroups in a similar way as normal subgroups arise as kernels of group homomorphisms. We use this to define the dynamical Cuntz semigroup as the universal object induced from an action of a group G on a W-semigroup. In the C*-algebraic framework, under mild assumptions, the universality of this dynamical invariant helps us tap into the structure of the Cuntz semigroup of crossed product C*-algebras.

math.OA↗

arXiv2024

Long thin covers and nuclear dimension

We establish finite nuclear dimension for crossed product C*-algebras arising from various classes of possibly non-free topological actions, including arbitrary actions of finitely generated virtually nilpotent groups on finite dimensional spaces, certain amenable actions of hyperbolic groups, and certain allosteric actions of wreath products. We obtain these results by introducing a new notion of dimension for topological dynamical systems, called the long thin covering dimension, which involves a suitable version of Rokhlin-type towers with controlled overlaps for possibly non-free actions.

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arXiv2021

The UCT problem for nuclear $C^\ast$-algebras

In recent years, a large class of nuclear $C^\ast$-algebras have been classified, modulo an assumption on the Universal Coefficient Theorem (UCT). We think this assumption is redundant and propose a strategy for proving it. Indeed, following the original proof of the classification theorem, we propose bridging the gap between reduction theorems and examples. While many such bridges are possible, various approximate ideal structures appear quite promising.

math.OA↗

arXiv2021

The nuclear dimension of $C^*$-algebras associated to topological flows and orientable line foliations

We show that for any locally compact Hausdorff space $Y$ with finite covering dimension and for any continuous flow $\mathbb{R} \curvearrowright Y$, the resulting crossed product $C^*$-algebra $C_0(Y) \rtimes \mathbb{R}$ has finite nuclear dimension. This generalizes previous results for free flows, where this was proved using Rokhlin dimension techniques. As an application, we obtain bounds for the nuclear dimension of $C^*$-algebras associated to one-dimensional orientable foliations. This result is analogous to the one we obtained earlier for non-free actions of $\mathbb{Z}$. Some novel techniques in our proof include the use of a conditional expectation constructed from the inclusion of a clopen subgroupoid, as well as the introduction of what we call fiberwise groupoid coverings that help us build a link between foliation $C^*$-algebras and crossed products.

math.OA↗

arXiv2021

The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces

We prove that the Novikov conjecture holds for any discrete group admitting an isometric and metrically proper action on an admissible Hilbert-Hadamard space. Admissible Hilbert-Hadamard spaces are a class of (possibly infinite-dimensional) non-positively curved metric spaces that contain dense sequences of closed convex subsets isometric to Riemannian manifolds. Examples of admissible Hilbert-Hadamard spaces include Hilbert spaces, certain simply connected and non-positively curved Riemannian-Hilbertian manifolds and infinite\-/dimensional symmetric spaces. Thus our main theorem can be considered as an infinite-dimensional analogue of Kasparov's theorem on the Novikov conjecture for groups acting properly and isometrically on complete, simply connected and non-positively curved manifolds. As a consequence, we show that the Novikov conjecture holds for geometrically discrete subgroups of the group of volume preserving diffeomorphisms of a closed smooth manifold. This result is inspired by Connes' theorem that the Novikov conjecture holds for higher signatures associated to the Gelfand-Fuchs classes of groups of diffeormorphisms.

math.KT↗

arXiv2020

Context-Aware RCNN: A Baseline for Action Detection in Videos

Video action detection approaches usually conduct actor-centric action recognition over RoI-pooled features following the standard pipeline of Faster-RCNN. In this work, we first empirically find the recognition accuracy is highly correlated with the bounding box size of an actor, and thus higher resolution of actors contributes to better performance. However, video models require dense sampling in time to achieve accurate recognition. To fit in GPU memory, the frames to backbone network must be kept low-resolution, resulting in a coarse feature map in RoI-Pooling layer. Thus, we revisit RCNN for actor-centric action recognition via cropping and resizing image patches around actors before feature extraction with I3D deep network. Moreover, we found that expanding actor bounding boxes slightly and fusing the context features can further boost the performance. Consequently, we develop a surpringly effective baseline (Context-Aware RCNN) and it achieves new state-of-the-art results on two challenging action detection benchmarks of AVA and JHMDB. Our observations challenge the conventional wisdom of RoI-Pooling based pipeline and encourage researchers rethink the importance of resolution in actor-centric action recognition. Our approach can serve as a strong baseline for video action detection and is expected to inspire new ideas for this filed. The code is available at \url{https://github.com/MCG-NJU/CRCNN-Action}.

cs.CV↗

arXiv2020

Straightening warped cones

We provide the converses to two results of J. Roe (Geom. Topol. 2005): first, the warped cone associated to a free action of an a-T-menable group admits a fibred coarse embedding into a Hilbert space, and second, a free action yielding a warped cone with property A must be amenable. We construct examples showing that in both cases the freeness assumption is necessary. The first equivalence is obtained also for other classes of Banach spaces, in particular for $L^p$-spaces.

math.MG↗

arXiv2019

The local-triviality dimension of actions of compact quantum groups

We define the local-triviality dimension for actions of compact quantum groups on unital C*-algebras. The resulting compact quantum principal bundle is said to be locally trivial when this dimension is finite. For commutative C*-algebras, this notion recovers the standard definition of local triviality of compact principal bundles. We prove that actions with finite local-triviality dimension are automatically free. Then we apply this new notion to prove the noncommutative Borsuk-Ulam-type conjecture under the assumption that a compact quantum group admits a non-trivial classical subgroup whose induced action has finite local-triviality dimension. This is a noncommutative extension of the Borsuk-Ulam-type theorem for locally trivial principal bundles.

math.OA↗

arXiv2019

Learning Actor Relation Graphs for Group Activity Recognition

Modeling relation between actors is important for recognizing group activity in a multi-person scene. This paper aims at learning discriminative relation between actors efficiently using deep models. To this end, we propose to build a flexible and efficient Actor Relation Graph (ARG) to simultaneously capture the appearance and position relation between actors. Thanks to the Graph Convolutional Network, the connections in ARG could be automatically learned from group activity videos in an end-to-end manner, and the inference on ARG could be efficiently performed with standard matrix operations. Furthermore, in practice, we come up with two variants to sparsify ARG for more effective modeling in videos: spatially localized ARG and temporal randomized ARG. We perform extensive experiments on two standard group activity recognition datasets: the Volleyball dataset and the Collective Activity dataset, where state-of-the-art performance is achieved on both datasets. We also visualize the learned actor graphs and relation features, which demonstrate that the proposed ARG is able to capture the discriminative relation information for group activity recognition.

cs.CV↗

arXiv2019

Isomorphism of the cubical and categorical cohomology groups of a higher-rank graph

We use category-theoretic techniques to provide two proofs showing that for a higher-rank graph $Λ$, its cubical (co-)homology and categorical (co-)homology groups are isomorphic in all degrees, thus answering a question of Kumjian, Pask and Sims in the positive. Our first proof uses the topological realization of a higher-rank graph, which was introduced by Kaliszewski, Kumjian, Quigg, and Sims. In our more combinatorial second proof, we construct, explicitly and in both directions, maps on the level of (co-)chain complexes that implement said isomorphism. Along the way, we extend the definition of cubical (co-)homology to allow arbitrary coefficient modules.

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