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1. Toeplitz quotient C*-algebras and ratio limits for random walks

   Dor-On, Adam (January 2021, Documenta mathematica)

   We study quotients of the Toeplitz C*-algebra of a random walk, similar to those studied by the author and Markiewicz for finite stochastic matrices. We introduce a new Cuntz-type quotient C*-algebra for random walks that have convergent ratios of transition probabilities. These C*-algebras give rise to new notions of ratio limit space and boundary for such random walks, which are computed by appealing to a companion paper by Woess. Our combined results are leveraged to identify a unique symmetry-equivariant quotient C*-algebra for any symmetric random walk on a hyperbolic group, shedding light on a question of Viselter on C*-algebras of subproduct systems. 

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2. Additivity of Higher Rho Invariants and Nonrigidity of Topological Manifolds

   https://doi.org/https://doi.org/10.1002/cpa.21962  

   Weinberger, S.; Xie, Z.; Yu, G. (January 2021, Communications on pure and applied mathematics)

   null (Ed.)

   We prove that the higher rho invariant is a homomorphism from the structure group of a compact manifold to the K-group of certain geometric C*-algebra. In particular, we apply this result to show that the structure group is infinitely generated for a class of manifolds. 

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3. Classifying the Dynamics of Architected Materials by Groupoid Methods

   Mesland, Bram; Prodan, Emil (February 2024, Journal of Geometry and Physics)

   We consider synthetic materials consisting of self-coupled identical resonators carrying classical internal degrees of freedom. The architecture of such material is specified by the positions and orientations of the resonators. Our goal is to calculate the smallest C*-algebra that covers the dynamical matrices associated to a fixed architecture and adjustable internal structures. We give the answer in terms of a groupoid C*-algebra that can be canonically associated to a uniformly discrete subset of the group of isometries of the Euclidean space. Our result implies that the isomorphism classes of these C*-algebras split these architected materials into classes containing materials that are identical from the dynamical point of view. 

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4. Structure-preserving GANs

   Birrell, J.; Katsoulakis, M.; Rey-Bellet, L.; Zhu, W. (January 2022, Proceedings of Machine Learning Research)

   Generative adversarial networks (GANs), a class of distribution-learning methods based on a two-player game between a generator and a discriminator, can generally be formulated as a minmax problem based on the variational representation of a divergence between the unknown and the generated distributions. We introduce structure-preserving GANs as a data-efficient framework for learning distributions with additional structure such as group symmetry, by developing new variational representations for divergences. Our theory shows that we can reduce the discriminator space to its projection on the invariant discriminator space, using the conditional expectation with respect to the sigma-algebra associated to the underlying structure. In addition, we prove that the discriminator space reduction must be accompanied by a careful design of structured generators, as flawed designs may easily lead to a catastrophic “mode collapse” of the learned distribution. We contextualize our framework by building symmetry-preserving GANs for distributions with intrinsic group symmetry, and demonstrate that both players, namely the equivariant generator and invariant discriminator, play important but distinct roles in the learning process. Empirical experiments and ablation studies across a broad range of data sets, including real-world medical imaging, validate our theory, and show our proposed methods achieve significantly improved sample fidelity and diversity—almost an order of magnitude measured in Frechet Inception Distance—especially in the small data regime 

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5. Emergent generalized symmetry and maximal symmetry topological order

   https://doi.org/10.1103/jzfv-ygmr  

   Chatterjee, Arkya; Ji, Wenjie; Wen, Xiao-Gang (September 2025, Physical Review B)

   A characteristic property of a gapless liquid state is its emergent symmetry and dual symmetry, associated with the conservation laws of symmetry charges and symmetry defects respectively. These conservation laws, considered on an equal footing, can't be described simply by the representation theory of a group (or a higher group). They are best described in terms of {\it a topological order (TO) with gappable boundary in one higher dimension}; we call this the {\it symTO} of the gapless state. The symTO can thus be considered a fingerprint of the gapless state. We propose that a largely complete characterization of a gapless state, up to local-low-energy equivalence, can be obtained in terms of its {\it maximal} emergent symTO. In this paper, we review the symmetry/topological-order (Symm/TO) correspondence and propose a definition of {\it maximal symTO}. We discuss various examples to illustrate these ideas. We find that the 1+1D Ising critical point has a maximal symTO described by the 2+1D double-Ising topological order. We provide a derivation of this result using symmetry twists in an exactly solvable model of the Ising critical point. The critical point in the 3-state Potts model has a maximal symTO of double (6,5)-minimal-model topological order. As an example of a noninvertible symmetry in 1+1D, we study the possible gapless states of a Fibonacci anyon chain with emergent double-Fibonacci symTO. We find the Fibonacci-anyon chain without translation symmetry has a critical point with unbroken double-Fibonacci symTO. In fact, such a critical theory has a maximal symTO of double (5,4)-minimal-model topological order. We argue that, in the presence of translation symmetry, the above critical point becomes a stable gapless phase with no symmetric relevant operator. 

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