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1. Higher Orbital Integrals, Cyclic Cocycles and Noncommutative Geometry

   https://doi.org/10.1017/fms.2024.115  

   Song, Yanli; Tang, Xiang (January 2025, Forum of Mathematics, Sigma)

   Abstract LetGbe a linear real reductive Lie group. Orbital integrals define traces on the group algebra ofG. We introduce a construction of higher orbital integrals in the direction of higher cyclic cocycles on the Harish-Chandra Schwartz algebra ofG. We analyze these higher orbital integrals via Fourier transform by expressing them as integrals on the tempered dual ofG. We obtain explicit formulas for the pairing between the higher orbital integrals and theK-theory of the reduced group$$C^{*}$$-algebra, and we discuss their application toK-theory. 

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2. Realizing finite groups as automizers

   https://doi.org/10.1515/jgth-2022-0145  

   Bayard, Sylvia; Lynd, Justin (February 2024, Journal of Group Theory)

   Abstract It is shown that any finite group 𝐴 is realizable as the automizer in a finite perfect group 𝐺 of an abelian subgroup whose conjugates generate 𝐺.The construction uses techniques from fusion systems on arbitrary finite groups, most notably certain realization results for fusion systems of the type studied originally by Park. 

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3. Cyclic cocycles and quantized pairings in materials science

   Emil Prodan (February 2023, Proceedings of symposia in pure mathematics)

   The pairings between the cyclic cohomologies and the K-theories of separable C∗-algebras supply topological invariants that often relate to physical response coefficients of materials. Using three numerical simulations, we exemplify how some of these invariants survive throughout the full Sobolev domains of the cocycles. These interesting phenomena, which can be explained by index theorems derived from Alain Connes’ quantized calculus, are now well understood in the independent electron picture. Here, we review recent developments addressing the dynamics of correlated many-fermions systems, obtained in collaboration with Bram Mesland. They supply a complete characterization of an algebra of relevant derivations over the C∗-algebra of canonical anti-commutation relations indexed by a generic discrete Delone lattice. It is argued here that these results already supply the means to generate interesting and relevant states over this algebra of derivations and to identify the cyclic cocycles corresponding to the transport coefficients of the many-fermion systems. The existing index theorems for the pairings of these cocycles, in the restrictive single fermion setting, are reviewed and updated with an emphasis on pushing the analysis on Sobolev domains. An assessment of possible generalizations to the many-body setting is given. 

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4. An intrinsic approach to relative braid group symmetries on ı$$\imath$$quantum groups

   https://doi.org/10.1112/plms.12562  

   Wang, Weiqiang; Zhang, Weinan (November 2023, Proceedings of the London Mathematical Society)

   Abstract We initiate a general approach to the relative braid group symmetries on (universal) quantum groups, arising from quantum symmetric pairs of arbitrary finite types, and their modules. Our approach is built on new intertwining properties of quasi ‐matrices which we develop and braid group symmetries on (Drinfeld double) quantum groups. Explicit formulas for these new symmetries on quantum groups are obtained. We establish a number of fundamental properties for these symmetries on quantum groups, strikingly parallel to their well‐known quantum group counterparts. We apply these symmetries to fully establish rank 1 factorizations of quasi ‐matrices, and this factorization property, in turn, helps to show that the new symmetries satisfy relative braid relations. As a consequence, conjectures of Kolb–Pellegrini and Dobson–Kolb are settled affirmatively. Finally, the above approach allows us to construct compatible relative braid group actions on modules over quantum groups for the first time. 

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5. Batch Multivalid Conformal Prediction

   Jung, Christopher; Noarov, Georgy; Ramalingam, Ramya; Roth, Aaron (January 2023, International Conference on Learning Representations (ICLR))

   We develop fast distribution-free conformal prediction algorithms for obtaining multivalid coverage on exchangeable data in the batch setting. Multivalid coverage guarantees are stronger than marginal coverage guarantees in two ways: (1) They hold even conditional on group membership---that is, the target coverage level holds conditionally on membership in each of an arbitrary (potentially intersecting) group in a finite collection of regions in the feature space. (2) They hold even conditional on the value of the threshold used to produce the prediction set on a given example. In fact multivalid coverage guarantees hold even when conditioning on group membership and threshold value simultaneously. We give two algorithms: both take as input an arbitrary non-conformity score and an arbitrary collection of possibly intersecting groups , and then can equip arbitrary black-box predictors with prediction sets. Our first algorithm is a direct extension of quantile regression, needs to solve only a single convex minimization problem, and produces an estimator which has group-conditional guarantees for each group in . Our second algorithm is iterative, and gives the full guarantees of multivalid conformal prediction: prediction sets that are valid conditionally both on group membership and non-conformity threshold. We evaluate the performance of both of our algorithms in an extensive set of experiments. 

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