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1. The Sphere of Semiadditive Height 1

   https://doi.org/10.1093/imrn/rnad044  

   Yuan, Allen (May 2023, International Mathematics Research Notices)

   Abstract We construct a lift of the $$p$$-complete sphere to the universal height $$1$$ higher semiadditive stable $$\infty $$-category of Carmeli–Schlank–Yanovski, providing a counterexample, at height $$1$$, to their conjecture that the natural functor $$ _n \to \operatorname{\textrm{Sp}}_{T(n)}$$ is an equivalence. We then record some consequences of the construction, including an observation of Schlank that this gives a conceptual proof of a classical theorem of Lee on the stable cohomotopy of Eilenberg–MacLane spaces. 

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2. Topological gauging and double current deformations

   https://doi.org/10.1007/JHEP05(2023)240  

   Dubovsky, Sergei; Negro, Stefano; Porrati, Massimo (June 2023, Journal of High Energy Physics)

   A bstract We study solvable deformations of two-dimensional quantum field theories driven by a bilinear operator constructed from a pair of conserved U(1) currents J a . We propose a quantum formulation of these deformations, based on the gauging of the corresponding symmetries in a path integral. This formalism leads to an exact dressing of the S -matrix of the system, similarly as what happens in the case of a $$ \textrm{T}\overline{\textrm{T}} $$ T T ¯ deformation. For conformal theories the deformations under study are expected to be exactly marginal. Still, a peculiar situation might arise when the conserved currents J a are not well-defined local operators in the original theory. A simple example of this kind of system is provided by rotation currents in a theory of multiple free, massless, non-compact bosons. We verify that, somewhat unexpectedly, such a theory is indeed still conformal after deformation and that it coincides with a TsT transformation of the original system. We then extend our formalism to the case in which the conserved currents are non-Abelian and point out its connection with Deformed T-dual Models and homogeneous Yang-Baxter deformations. In this case as well the deformation is based on a gauging of the symmetries involved and it turns out to be non-trivial only if the symmetry group admits a non-trivial central extension. Finally we apply what we learned by relating the $$ \textrm{T}\overline{\textrm{T}} $$ T T ¯ deformation to the central extension of the two-dimensional Poincaré algebra. 

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3. Polynomial Parametrization for SL2 over Quadratic Number Rings

   https://doi.org/10.1093/imrn/rnz046  

   Larsen, Michael; Nguyen, Dong Quan (March 2019, International Mathematics Research Notices)

   null (Ed.)

   Abstract If $$R$$ is the ring of integers of a number field, then there exists a polynomial parametrization of the set $$\operatorname{SL}_2(R)$$, that is, an element $$A\in{\textrm{SL}}_2(\mathbb{Z}[x_1,\ldots ,x_n])$$ such that every element of $$\operatorname{SL}_2(R)$$ is obtained by specializing $$A$$ via some homomorphism $$\mathbb{Z}[x_1,\ldots ,x_n]\to R$$. 

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4. Morphisms of Character Varieties

   https://doi.org/10.1093/imrn/rnae124  

   Cotner, Sean (June 2024, International Mathematics Research Notices)

   Abstract Let $$k$$ be a field, let $$H \subset G$$ be (possibly disconnected) reductive groups over $$k$$, and let $$\Gamma $$ be a finitely generated group. Vinberg and Martin have shown that the induced morphism $$\underline{\operatorname{Hom}}_{k\textrm{-gp}}(\Gamma , H)//H \to \underline{\operatorname{Hom}}_{k\textrm{-gp}}(\Gamma , G)//G$$ is finite. In this note, we generalize this result (with a significantly different proof) by replacing $$k$$ with an arbitrary locally Noetherian scheme, answering a question of Dat. Along the way, we use Bruhat–Tits theory to establish a few apparently new results about integral models of reductive groups over discrete valuation rings. 

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5. A modulation invariant Carleson embedding theorem outside local L^2

   Di Plinio, Francesco; Ou, Yumeng (April 2016, Journal d’analyse mathématique)

   The article arXiv:1309.0945 by Do and Thiele develops a theory of Carleson embeddings in outer Lp spaces for the wave packet transform of functions in L^p, in the 2≤p≤∞ range referred to as local L^2. In this article, we formulate a suitable extension of this theory to exponents 1<2, answering a question posed in arXiv:1309.0945. The proof of our main embedding theorem involves a refined multi-frequency Calder\'on-Zygmund decomposition. We apply our embedding theorem to recover the full known range of Lp estimates for the bilinear Hilbert transforms without reducing to discrete model sums or appealing to generalized restricted weak-type interpolation. 

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