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1. Hecke categories, idempotents, and commuting stacks

   https://doi.org/10.1007/s00209-024-03674-7  

   Gorsky, Eugene; Neguţ, Andrei (March 2025, Mathematische Zeitschrift)

   Abstract We formulate a connection between a topological and a geometric category. The former is the idempotent completion of the (horizontal) trace of the affine Hecke category, while the latter is the equivariant derived category of the (semi-nilpotent) commuting stack. This provides a more precise and improved version of our proposal in Gorsky and Neguț (Proc Lond Math Soc (3) 126(6): 2013–2056, 2023). 

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2. Higher order Kirillov--Reshetikhin modules for 𝐔 _q ( A _n ⁽¹⁾ ), imaginary modules and monoidal categorification

   https://doi.org/10.1515/crelle-2023-0068  

   Brito, Matheus; Chari, Vyjayanthi (October 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We study the family of irreducible modules for quantum affine $𝔰{𝔩}_{n+1}${\mathfrak{sl}_{n+1}}whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to $A_m${A_{m}}with $m\le n${m\leq n}. These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category ${\mathcal{𝒞}}^{-}${\mathscr{C}^{-}}. This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) Kirillov–Reshetikhin module with its dual always contains an imaginary module in its Jordan–Hölder series and give an explicit formula for its Drinfeld polynomial. Together with the results of [D. Hernandez and B. Leclerc,A cluster algebra approach toq-characters of Kirillov–Reshetikhin modules,J. Eur. Math. Soc. (JEMS) 18 2016, 5, 1113–1159] this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of [E. Lapid and A. Mínguez,Geometric conditions for $\mathrm{\square }$\square-irreducibility of certain representations of the general linear group over a non-archimedean local field,Adv. Math. 339 2018, 113–190]. Finally, we use our methods to give a family of imaginary modules in type $D_4${D_{4}}which do not arise from an embedding of $A_r${A_{r}}with $r\le 3${r\leq 3}in $D_4${D_{4}}. 

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3. Chromoplast differentiation: a central role for plastoglobule lipid droplets comes into focus

   https://doi.org/10.1111/nph.18700  

   Lundquist, Peter K. (January 2023, New Phytologist)

   This article is a Commentary onMorelliet al. (2023),237: 1696–1710. 

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4. Homomorphisms to 3–Manifold Groups

   https://doi.org/10.1007/s00039-025-00713-1  

   Groves, Daniel; Hull, Michael; Liang, Hao (June 2025, Geometric and Functional Analysis)

   Abstract We prove foundational results about the set of homomorphisms from a finitely generated group to the collection of all fundamental groups of compact 3–manifolds and answer questions of Agol–Liu (J. Am. Math. Soc. 25(1):151–187, 2012) and Reid–Wang–Zhou (Acta Math. Sin. Engl. Ser. 18(1):157–172, 2002). 

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5. On the stability of homogeneous equilibria in the Vlasov–Poisson system on R3

   https://doi.org/10.1088/1361-6382/acebb0  

   Ionescu, A D; Pausader, B; Wang, X; Widmayer, K (August 2023, Classical and Quantum Gravity)

   The goal of this article is twofold. First, we investigate the linearized Vlasov–Poisson system around a family of spatially homogeneous equilibria in the unconfined setting. Our analysis follows classical strategies from physics (Binney and Tremaine 2008, Galactic Dynamics,(Princeton University Press); Landau 1946, Acad. Sci. USSR. J. Phys.10,25–34; Penrose 1960,Phys. Fluids,3,258–65) and their subsequent mathematical extensions (Bedrossian et al 2022, SIAM J. Math. Anal.,54,4379–406; Degond 1986,Trans. Am. Math. Soc., 294,435–53; Glassey and Schaeffer 1994,Transp. Theory Stat. Phys.,23, 411–53; Grenier et al 2021, Math. Res. Lett., 28,1679–702; Han-Kwan et al, 2021, Commun. Math. Phys. 387, 1405–40; Mouhot and Villani 2011, Acta Math., 207, 29–201). The main novelties are a unified treatment of a broad class of analytic equilibria and the study of a class of generalized Poisson equilibria. For the former, this provides a detailed description of the associated Green’s functions, including in particular precise dissipation rates (which appear to be new), whereas for the latter we exhibit explicit formulas. Second, we review the main result and ideas in our recent work (Ionescu et al, 2022 on the full global nonlinear asymptotic stability of the Poisson equilibrium in R3 

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