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1. Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile

   https://doi.org/10.1007/s00454-022-00426-4  

   Greenfeld, Rachel; Tao, Terence (January 2023, Discrete & Computational Geometry)

   Abstract We construct an example of a group$$G = \mathbb {Z}^2 \times G_0$$ $G=Z^2\times G_0$for a finite abelian group $$G_0$$ $G_0$, a subsetEof $$G_0$$ $G_0$, and two finite subsets$$F_1,F_2$$ $F_1,F_2$of G, such that it is undecidable in ZFC whether$$\mathbb {Z}^2\times E$$ $Z^2\times E$can be tiled by translations of$$F_1,F_2$$ $F_1,F_2$. In particular, this implies that this tiling problem isaperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings ofEby the tiles$$F_1,F_2$$ $F_1,F_2$, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $$\mathbb {Z}^2$$ $Z^2$). A similar construction also applies for$$G=\mathbb {Z}^d$$ $G=Z^d$for sufficiently large d. If one allows the group$$G_0$$ $G_0$to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles. 

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2. On Harmonic Hilbert Spaces on Compact Abelian Groups

   https://doi.org/10.1007/s00041-023-09992-4  

   Das, Suddhasattwa; Giannakis, Dimitrios (February 2023, Journal of Fourier Analysis and Applications)

   Abstract Harmonic Hilbert spaces on locally compact abelian groups are reproducing kernel Hilbert spaces (RKHSs) of continuous functions constructed by Fourier transform of weighted$$L^2$$ $L^2$spaces on the dual group. It is known that for suitably chosen subadditive weights, every such space is a Banach algebra with respect to pointwise multiplication of functions. In this paper, we study RKHSs associated with subconvolutive functions on the dual group. Sufficient conditions are established for these spaces to be symmetric Banach$$^*$$ ${}^{\ast }$-algebras with respect to pointwise multiplication and complex conjugation of functions (here referred to as RKHAs). In addition, we study aspects of the spectra and state spaces of RKHAs. Sufficient conditions are established for an RKHA on a compact abelian groupGto have the same spectrum as the$$C^*$$ $C^{\ast }$-algebra of continuous functions onG. We also consider one-parameter families of RKHSs associated with semigroups of self-adjoint Markov operators on$$L^2(G)$$ $L^2\left(G\right)$, and show that in this setting subconvolutivity is a necessary and sufficient condition for these spaces to have RKHA structure. Finally, we establish embedding relationships between RKHAs and a class of Fourier–Wermer algebras that includes spaces of dominating mixed smoothness used in high-dimensional function approximation. 

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3. 3D Mirror Symmetry for Instanton Moduli Spaces

   https://doi.org/10.1007/s00220-023-04831-5  

   Koroteev, Peter; Zeitlin, Anton M. (September 2023, Communications in Mathematical Physics)

   Abstract We prove that the Hilbert scheme ofkpoints on$${\mathbb {C}}^2$$ $C^2$($$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^k\left[C^2\right]$) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$${\mathbb {C}}^\times _\hbar $$ $C_{ħ}^{\times }$-action. First, we find a two-parameter family$$X_{k,l}$$ $X_{k,l}$of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^k\left[C^2\right]$is obtained via direct limit$$l\longrightarrow \infty $$ $l⟶\infty$and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$\hbar $$ $ħ$-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-Nsheaves on$${\mathbb {P}}^2$$ $P^2$with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual. 

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4. Notes on gauging noninvertible symmetries. Part I. Multiplicity-free cases

   https://doi.org/10.1007/JHEP02(2024)154  

   Perez-Lona, A; Robbins, D; Sharpe, E; Vandermeulen, T; Yu, X (February 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> In this paper we discuss gauging noninvertible zero-form symmetries in two dimensions. We specialize to certain gaugeable cases, specifically, fusion categories of the form$$ \textrm{Rep}\left(\mathcal{H}\right) $$ $\mathrm{Rep}\left(H\right)$for$$ \mathcal{H} $$ $H$a suitable Hopf algebra (which includes the special case Rep(G) forGa finite group). We also specialize to the case that the fusion category is multiplicity-free. We discuss how to construct a modular-invariant partition function from a choice of Frobenius algebra structure on$$ {\mathcal{H}}^{\ast } $$ $H^{\ast }$. We discuss how ordinaryGorbifolds for finite groupsGare a special case of the construction, corresponding to the fusion category Vec(G) = Rep(ℂ[G]^*). For the cases Rep(S₃), Rep(D₄), and Rep(Q₈), we construct the crossing kernels for general intertwiner maps. We explicitly compute partition functions in the examples of Rep(S₃), Rep(D₄), Rep(Q₈), and$$ \textrm{Rep}\left({\mathcal{H}}_8\right) $$ $\mathrm{Rep}\left(H_8\right)$, and discuss applications inc= 1 CFTs. We also discuss decomposition in the special case that the entire noninvertible symmetry group acts trivially. 

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5. L-Systems and the Lovász Number

   https://doi.org/10.1007/s00493-025-00136-4  

   Linz, William (April 2025, Combinatorica)

   Abstract Given integers$$n> k > 0$$ $n>k>0$, and a set of integers$$L \subset [0, k-1]$$ $L\subset [0,k-1]$, anL-systemis a family of sets$$\mathcal {F}\subset \left( {\begin{array}{c}[n]\\ k\end{array}}\right) $$ $F\subset \left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$such that$$|F \cap F'| \in L$$ $|F\cap F^{\prime }|\in L$for distinct$$F, F'\in \mathcal {F}$$ $F,F^{\prime }\in F$.L-systems correspond to independent sets in a certain generalized Johnson graphG(n, k, L), so that the maximum size of anL-system is equivalent to finding the independence number of the graphG(n, k, L). TheLovász number$$\vartheta (G)$$ $\vartheta \left(G\right)$is a semidefinite programming approximation of the independence number$$\alpha $$ $\alpha$of a graphG. In this paper, we determine the leading order term of$$\vartheta (G(n, k, L))$$ $\vartheta \left(G\right(n,k,L\left)\right)$of any generalized Johnson graph withkandLfixed and$$n\rightarrow \infty $$ $n\to \infty$. As an application of this theorem, we give an explicit construction of a graphGonnvertices with a large gap between the Lovász number and the Shannon capacityc(G). Specifically, we prove that for any$$\epsilon > 0$$ $ϵ>0$, for infinitely manynthere is a generalized Johnson graphGonnvertices which has ratio$$\vartheta (G)/c(G) = \Omega (n^{1-\epsilon })$$ $\vartheta \left(G\right)/c\left(G\right)=\Omega \left(n^{1-ϵ}\right)$, which improves on all known constructions. The graphGa fortiorialso has ratio$$\vartheta (G)/\alpha (G) = \Omega (n^{1-\epsilon })$$ $\vartheta \left(G\right)/\alpha \left(G\right)=\Omega \left(n^{1-ϵ}\right)$, which greatly improves on the best known explicit construction. 

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