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1. On uniqueness for half-wave maps in dimension 𝑑≥3

   https://doi.org/10.1090/btran/171  

   Eyeson, Eugene; Reyes Farina, Silvino; Schikorra, Armin (February 2024, Transactions of the American Mathematical Society, Series B)

   Extending an argument by Shatah and Struwe [Int. Math. Res. Not. 11 (2002), pp. 555–571] we obtain uniqueness for solutions of the half-wave map equation in dimension $d\ge <\#comment/>3$in the natural energy class. 

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2. Generalized cluster structures related to the Drinfeld double of 𝑮𝑳_𝒏

   Gekhtman, M.; Shapiro, M.; Vainshtein, A. (October 2021, Journal of the London Mathematical Society)

   We prove that the regular generalized cluster structure on the Drinfeld double of 𝐺𝐿𝑛 constructed in Gekhtman, Shapiro, and Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) is complete and compatible with the standard Poisson–Lie structure on the double. Moreover, we show that for 𝑛 = 4 this structure is distinct from a previously known regular generalized cluster structure on the Drinfeld double, even though they have the same compatible Poisson structure and the same collection of frozen variables. Further, we prove that the regular generalized cluster structure on band periodic matrices constructed in Gekhtman, Shapiro, and Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) possesses similar compatibility and completeness properties. 

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3. Invertible topological field theories

   https://doi.org/10.1112/topo.12335  

   Schommer‐Pries, Christopher (June 2024, Journal of Topology)

   A ‐dimensional invertible topological field theory (TFT) is a functor from the symmetric monoidal ‐category of ‐bordisms (embedded into and equipped with a tangential ‐structure) that lands in the Picard subcategory of the target symmetric monoidal ‐category. We classify these field theories in terms of the cohomology of the ‐connective cover of the Madsen–Tillmann spectrum. This is accomplished by identifying the classifying space of the ‐category of bordisms with as an ‐space. This generalizes the celebrated result of Galatius–Madsen–Tillmann–Weiss (Acta Math.202(2009), no. 2, 195–239) in the case , and of Bökstedt–Madsen (An alpine expedition through algebraic topology, vol. 617, Contemp. Math., Amer. Math. Soc., Providence, RI, 2014, pp. 39–80) in the ‐uple case. We also obtain results for the ‐category of ‐bordisms embedding into a fixed ambient manifold , generalizing results of Randal–Williams (Int. Math. Res. Not. IMRN2011(2011), no. 3, 572–608) in the case . We give two applications: (1) we completely compute all extended and partially extended invertible TFTs with target a certain category of ‐vector spaces (for ), and (2) we use this to give a negative answer to a question raised by Gilmer and Masbaum (Forum Math.25(2013), no. 5, 1067–1106. arXiv:0912.4706). 

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4. The isomorphism problem for tensor algebras of multivariable dynamical systems

   https://doi.org/10.1017/fms.2022.73  

   Katsoulis, Elias G.; Ramsey, Christopher (January 2022, Forum of Mathematics, Sigma)

   Abstract We resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically, we show that unitary equivalence after a conjugation for multivariable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis, Inter. Math. Res. Not. 2014 (2014), 1289–1311 relating to work of Arveson, Acta Math. 118 (1967), 95–109 from the 1960s, and extends related work of Kakariadis and Katsoulis, J. Noncommut. Geom. 8 (2014), 771–787. 

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5. The trace of the affine Hecke category

   https://doi.org/10.1112/plms.12523  

   Gorsky, Eugene; Neguț, Andrei (April 2023, Proceedings of the London Mathematical Society)

   Abstract We compare the (horizontal) trace of the affine Hecke category with the elliptic Hall algebra, thus obtaining an “affine” version of the construction of Gorsky et al. (Int. Math. Res. Not. IMRN2022(2022) 11304–11400). Explicitly, we show that the aforementioned trace is generated by the objects as , where denote the Wakimoto objects of Elias and denote Rouquier complexes. We compute certain categorical commutators between the 's and show that they match the categorical commutators between the sheaves on the flag commuting stack that were considered in Neguț (Publ. Math. Inst. Hautes Études Sci. 135 (2022) 337–418). At the level of ‐theory, these commutators yield a certain integral form of the elliptic Hall algebra, which we can thus map to the ‐theory of the trace of the affine Hecke category. 

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