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1. Extension Theory for Braided-Enriched Fusion Categories

   https://doi.org/10.1093/imrn/rnab133  

   Jones, Corey; Morrison, Scott; Penneys, David; Plavnik, Julia (July 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract For a braided fusion category $$\mathcal{V}$$, a $$\mathcal{V}$$-fusion category is a fusion category $$\mathcal{C}$$ equipped with a braided monoidal functor $$\mathcal{F}:\mathcal{V} \to Z(\mathcal{C})$$. Given a fixed $$\mathcal{V}$$-fusion category $$(\mathcal{C}, \mathcal{F})$$ and a fixed $$G$$-graded extension $$\mathcal{C}\subseteq \mathcal{D}$$ as an ordinary fusion category, we characterize the enrichments $$\widetilde{\mathcal{F}}:\mathcal{V} \to Z(\mathcal{D})$$ of $$\mathcal{D}$$ that are compatible with the enrichment of $$\mathcal{C}$$. We show that G-crossed extensions of a braided fusion category $$\mathcal{C}$$ are G-extensions of the canonical enrichment of $$\mathcal{C}$$ over itself. As an application, we parameterize the set of $$G$$-crossed braidings on a fixed $$G$$-graded fusion category in terms of certain subcategories of its center, extending Nikshych’s classification of the braidings on a fusion category. 

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2. Variation of canonical height for\break Fatou points on ℙ ¹

   https://doi.org/10.1515/crelle-2022-0078  

   DeMarco, Laura; Mavraki, Niki Myrto (December 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let f : ℙ 1 → ℙ 1 {f:\mathbb{P}^{1}\to\mathbb{P}^{1}} be a map of degree > 1 {>1} defined over a function field k = K ⁢ ( X ) {k=K(X)} , where K is a number field and X is a projective curve over K . For each point a ∈ ℙ 1 ⁢ ( k ) {a\in\mathbb{P}^{1}(k)} satisfying a dynamical stability condition, we prove that the Call–Silverman canonical height for specialization f t {f_{t}} at point a t {a_{t}} , for t ∈ X ⁢ ( ℚ ¯ ) {t\in X(\overline{\mathbb{Q}})} outside a finite set, induces a Weil height on the curve X ; i.e., we prove the existence of a ℚ {\mathbb{Q}} -divisor D = D f , a {D=D_{f,a}} on X so that the function t ↦ h ^ f t ⁢ ( a t ) - h D ⁢ ( t ) {t\mapsto\hat{h}_{f_{t}}(a_{t})-h_{D}(t)} is bounded on X ⁢ ( ℚ ¯ ) {X(\overline{\mathbb{Q}})} for any choice of Weil height associated to D . We also prove a local version, that the local canonical heights t ↦ λ ^ f t , v ⁢ ( a t ) {t\mapsto\hat{\lambda}_{f_{t},v}(a_{t})} differ from a Weil function for D by a continuous function on X ⁢ ( ℂ v ) {X(\mathbb{C}_{v})} , at each place v of the number field K . These results were known for polynomial maps f and all points a ∈ ℙ 1 ⁢ ( k ) {a\in\mathbb{P}^{1}(k)} without the stability hypothesis,[21, 14],and for maps f that are quotients of endomorphisms of elliptic curves E over k and all points a ∈ ℙ 1 ⁢ ( k ) {a\in\mathbb{P}^{1}(k)} . [32, 29].Finally, we characterize our stability condition in terms of the geometry of the induced map f ~ : X × ℙ 1 ⇢ X × ℙ 1 {\tilde{f}:X\times\mathbb{P}^{1}\dashrightarrow X\times\mathbb{P}^{1}} over K ; and we prove the existence of relative Néron models for the pair ( f , a ) {(f,a)} , when a is a Fatou point at a place γ of k , where the local canonical height λ ^ f , γ ⁢ ( a ) {\hat{\lambda}_{f,\gamma}(a)} can be computed as an intersection number. 

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3. Failed zero forcing and critical sets on directed graphs

   Adams, Alyssa; Jacob, Bonnie (October 2021, The Australasian journal of combinatorics)

   Let $$D$$ be a simple digraph (directed graph) with vertex set $V(D)$ and arc set $A(D)$ where $n=|V(D)|$, and each arc is an ordered pair of distinct vertices. If $$(v,u) \in A(D)$$, then $$u$$ is considered an \emph{out-neighbor} of $$v$$ in $$D$$. Initially, we designate each vertex to be either filled or empty. Then, the following color change rule (CCR) is applied: if a filled vertex $$v$$ has exactly one empty out-neighbor $$u$$, then $$u$$ will be filled. If all vertices in $V(D)$$ are eventually filled under repeated applications of the CCR, then the initial set is called a \emph{zero forcing set} (ZFS); if not, it is a \emph{failed zero forcing set} (FZFS). We introduce the \emph{failed zero forcing number} $$\F(D)$$ on a digraph, which is the maximum cardinality of any FZFS. The \emph{zero forcing number}, $$\Z(D)$, is the minimum cardinality of any ZFS. We characterize digraphs that have $$\F(D)<\Z(D)$$ and determine $$\F(D)$$ for several classes of digraphs including de Bruijn and Kautz digraphs. We also characterize digraphs with $$\F(D)=n-1$$, $$\F(D)=n-2$$, and $$\F(D)=0$$, which leads to a characterization of digraphs in which any vertex is a ZFS. Finally, we show that for any integer $$n \geq 3$$ and any non-negative integer $$k$$ with $k 

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4. 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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5. Doubly isogenous genus-2 curves with 𝐷₄-action

   https://doi.org/10.1090/mcom/3891  

   Arul, Vishal; Booher, Jeremy; Groen, Steven; Howe, Everett; Li, Wanlin; Matei, Vlad; Pries, Rachel; Springer, Caleb (January 2024, Mathematics of Computation)

   Abstract. We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose C and C ′ are curves over a finite field K, with K-rational base points P and P ′ , and let D and D ′ be the pullbacks (via the Abel–Jacobi map) of the multiplication-by-2 maps on their Jacobians. We say that (C, P) and (C ′ , P ′ ) are doubly isogenous if Jac(C) and Jac(C ′ ) are isogenous over K and Jac(D) and Jac(D ′ ) are isogenous over K. For curves of genus 2 whose automorphism groups contain the dihedral group of order eight, we show that the number of pairs of doubly isogenous curves is larger than na¨ıve heuristics predict, and we provide an explanation for this phenomenon. 

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