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1. On the Restriction Maps of the Fourier and Fourier–Stieltjes Algebras Over Locally Compact Groupoids

   https://doi.org/10.1093/imrn/rnaf057  

   DeGaetani, Joseph; Ghandehari, Mahya (March 2025, International Mathematics Research Notices)

   Abstract The Fourier and Fourier–Stieltjes algebras over locally compact groupoids have been defined in a way that parallels their construction for groups. In this article, we extend the results on surjectivity or lack of surjectivity of the restriction map on the Fourier and Fourier–Stieltjes algebras of groups to the groupoid setting. In particular, we consider the maps that restrict the domain of these functions in the Fourier or Fourier–Stieltjes algebra of a groupoid to an isotropy subgroup. These maps are continuous contractive algebra homomorphisms. When the groupoid is étale, we show that the restriction map on the Fourier algebra is surjective. The restriction map on the Fourier–Stieltjes algebra is not surjective in general. We prove that for a transitive groupoid with a continuous section or a group bundle with discrete unit space, the restriction map on the Fourier–Stieltjes algebra is surjective. We further discuss the example of an HLS groupoid, and obtain a necessary condition for surjectivity of the restriction map in terms of property FD for groups, introduced by Lubotzky and Shalom. As a result, we present examples where the restriction map for the Fourier–Stieltjes algebra is not surjective. Finally, we use the surjectivity results to provide conditions for the lack of certain Banach algebraic properties, including the (weak) amenability and existence of a bounded approximate identity, in the Fourier algebra of étale groupoids. 

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2. Cosets of Free Field Algebras via Arc Spaces

   https://doi.org/10.1093/imrn/rnac367  

   Linshaw, Andrew_R; Song, Bailin (January 2023, International Mathematics Research Notices)

   Abstract Using the invariant theory of arc spaces, we find minimal strong generating sets for certain cosets of affine vertex algebras inside free field algebras that are related to classical Howe duality. These results have several applications. First, for any vertex algebra $${{\mathcal {V}}}$$, we have a surjective homomorphism of differential algebras $$\mathbb {C}[J_{\infty }(X_{{{\mathcal {V}}}})] \rightarrow \text {gr}^{F}({{\mathcal {V}}})$$; equivalently, the singular support of $${{\mathcal {V}}}$$ is a closed subscheme of the arc space of the associated scheme $$X_{{{\mathcal {V}}}}$$. We give many new examples of classically free vertex algebras (i.e., this map is an isomorphism), including $$L_{k}({{\mathfrak {s}}}{{\mathfrak {p}}}_{2n})$$ for all positive integers $$n$$ and $$k$$. We also give new examples where the kernel of this map is nontrivial but is finitely generated as a differential ideal. Next, we prove a coset realization of the subregular $${{\mathcal {W}}}$$-algebra of $${{\mathfrak {s}}}{{\mathfrak {l}}}_{n}$$ at a critical level that was previously conjectured by Creutzig, Gao, and the 1st author. Finally, we give some new level-rank dualities involving affine vertex superalgebras. 

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3. Ricci Curvature and Fundamental Groups of Effective Regular Sets

   https://doi.org/10.4208/jms.v58n1.25.01  

   Pan, Jiayin (January 2025, Journal of Mathematical Study)

   For a Gromov-Hausdorff convergent sequence of closed manifolds $$M_i^n\ghto X$$ with $$\Ric\ge-(n-1)$$, $$ \mathrm{diam}(M_i)\le D$$, and $$\mathrm{vol}(M_i)\ge v>0,$$ we study the relation between $$\pi_1(M_i)$$ and $$X$$. It was known before that there is a surjective homomorphism $$\phi_i:\pi_1(M_i)\to \pi_1(X)$$ by the work of Pan--Wei. In this paper, we construct a surjective homomorphism $$\psi_i$$ from the interior of the effective regular set in $$X$$ back to $$M_i$$. These surjective homomorphisms $$\phi_i$$ and $$\psi_i$$ are natural in the sense that their composition $$\phi_i \circ \psi_i$$ is exactly the homomorphism induced by the inclusion map from the effective regular set to $$X$$. 

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4. Cohomological Blowups of Graded Artinian Gorenstein Algebras along Surjective Maps

   https://doi.org/10.1093/imrn/rnac002  

   Iarrobino, Anthony; Macias Marques, Pedro; McDaniel, Chris; Seceleanu, Alexandra; Watanabe, Junzo (February 2022, International Mathematics Research Notices)

   Abstract We introduce the cohomological blowup of a graded Artinian Gorenstein algebra along a surjective map, which we term BUG (blowup Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blowup of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersections. We have included many examples throughout this manuscript. 

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5. A non-Borel special alpha-limit set in the square

   https://doi.org/10.1017/etds.2021.68  

   JACKSON, STEPHEN; MANCE, BILL; ROTH, SAMUEL (August 2022, Ergodic Theory and Dynamical Systems)

   Abstract We consider the complexity of special $$\alpha $$ -limit sets, a kind of backward limit set for non-invertible dynamical systems. We show that these sets are always analytic, but not necessarily Borel, even in the case of a surjective map on the unit square. This answers a question posed by Kolyada, Misiurewicz, and Snoha. 

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