More Like this

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. 

   more » « less
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. 

   more » « less
3. Reducibility Without KAM

   https://doi.org/10.1007/s00220-024-05105-4  

   Argentieri, F; Fayad, B (November 2024, Communications in Mathematical Physics)

   Abstract We prove rotations-reducibility for close to constant quasi-periodic$$SL(2,\mathbb {R})$$ $SL(2,R)$cocycles in one frequency in the finite regularity and smooth cases, and derive some applications to quasi-periodic Schrödinger operators. 

   more » « less
4. 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. 

   more » « less
5. 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. 

   more » « less