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1. Local-global principles in circle packings

   https://doi.org/10.1112/S0010437X19007139  

   Fuchs, Elena ; Stange, Katherine E. ; Zhang, Xin ( June 2019 , Compositio Mathematica)

   We generalize work by Bourgain and Kontorovich [ On the local-global conjecture for integral Apollonian gaskets , Invent. Math. 196 (2014), 589–650] and Zhang [ On the local-global principle for integral Apollonian 3-circle packings , J. Reine Angew. Math. 737 , (2018), 71–110], proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group ${\mathcal{A}}\leqslant \text{PSL}_{2}(K)$ satisfying certain conditions, where $K$ is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof is that ${\mathcal{A}}$ possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in $\operatorname{PSL}_{2}({\mathcal{O}}_{K})$ containing a Zariski dense subgroup of $\operatorname{PSL}_{2}(\mathbb{Z})$ . 

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2. Manufacturing of a microlens array mold by a two-step method combining microindentation and precision polishing

   https://doi.org/10.1364/AO.397448  

   Zhang, Lin ; Yi, Allen Y. ( August 2020 , Applied Optics)

   A novel two-step method for manufacturing microlens array molds by combining microindentation and precision polishing is proposed. Compared with conventional manufacturing methods, such as single-point diamond turning, this two-step method, as an alternative method, presents great advantages on cost and flexibility on spherical microlens array mold fabrication. Various curvatures of radii and arrangements for microlens array molds can be fabricated in the same way. In this paper, a hexagonal microlens array with 1.58 mm curvature radius was demonstrated to prove the feasibility of the proposed method. First, a large number of precise steel balls were organized in hexagonal arrangement and pressed into the mold’s surface to generate multiple microdimples. Second, the pileups around the microdimples were removed from the mold surface by precision polishing. The geometrical accuracy and surface quality were investigated by an optical surface profiler. The measurement indicated that, compared with the initial surface, the surface inside the dimple had significantly higher hardness and better surface quality than that of the steel balls. Then the microlens array on the mold was further replicated to poly(methyl methacrylate) substrates by a precision compression molding process. The experimental results showed that the fabricated mold and the polymer replicas have high fidelity, great uniformity, and good surface roughness. The proposed two-step, low-cost mold fabrication method can produce highly uniform microlens arrays and is therefore suitable for high-volume fabrication of precise optical elements such as integrated light-emitting diodes and other similar micro-optics.

    

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3. Geometry and arithmetic of crystallographic sphere packings

   https://doi.org/10.1073/pnas.1721104116  

   Kontorovich, Alex ; Nakamura, Kei ( December 2018 , Proceedings of the National Academy of Sciences)

   We introduce the notion of a “crystallographic sphere packing,” defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit an infinite family of conformally inequivalent crystallographic packings with all radii being reciprocals of integers. We then prove a result in the opposite direction: the “superintegral” ones exist only in finitely many “commensurability classes,” all in, at most, 20 dimensions.

    

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4. Quantum liquid from strange frustration in the trimer magnet Ba4Ir3O10

   https://doi.org/10.1038/s41535-020-0232-6  

   Cao, Gang ; Zheng, Hao ; Zhao, Hengdi ; Ni, Yifei ; Pocs, Christopher A. ; Zhang, Yu ; Ye, Feng ; Hoffmann, Christina ; Wang, Xiaoping ; Lee, Minhyea ; et al ( May 2020 , npj Quantum Materials)

   Quantum spin systems such as magnetic insulators usually show magnetic order, but such classical states can give way toquantum liquids with exotic entanglementthrough two known mechanisms of frustration: geometric frustration in lattices with triangle motifs, and spin-orbit-coupling frustration in the exactly solvable quantum liquid of Kitaev’s honeycomb lattice. Here we present the experimental observation of a new kind of frustrated quantum liquid arising in an unlikely place: the magnetic insulator Ba₄Ir₃O₁₀where Ir₃O₁₂trimers form an unfrustrated square lattice. The crystal structure shows no apparent spin chains. Experimentally we find a quantum liquid state persisting down to 0.2 K that is stabilized by strong antiferromagnetic interaction with Curie–Weiss temperature ranging from −766 to −169 K due to magnetic anisotropy. The anisotropy-averaged frustration parameter is 2000, seldom seen in iridates. Heat capacity and thermal conductivity are both linear at low temperatures, a familiar feature in metals but here in an insulator pointing to an exotic quantum liquid state; a mere 2% Sr substitution for Ba produces long-range order at 130 K and destroys the linear-T features. Although the Ir⁴⁺(5d⁵) ions in Ba₄Ir₃O₁₀appear to form Ir₃O₁₂trimers of face-sharing IrO₆octahedra, we propose that intra-trimer exchange is reduced and the lattice recombines into an array of coupled 1D chains with additional spins. An extreme limit of decoupled 1D chains can explain most but not all of the striking experimental observations, indicating that the inter-chain coupling plays an important role in the frustration mechanism leading to this quantum liquid.

    

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5. The Apollonian Staircase

   https://doi.org/10.1093/imrn/rnad007  

   Rickards, James ( March 2023 , International Mathematics Research Notices)

   Abstract A circle of curvature $n\in \mathbb{Z}^+$ is a part of finitely many primitive integral Apollonian circle packings. Each such packing has a circle of minimal curvature $-c\leq 0$, and we study the distribution of $c/n$ across all primitive integral packings containing a circle of curvature $n$. As $n\rightarrow \infty $, the distribution is shown to tend towards a picture we name the Apollonian staircase. A consequence of the staircase is that if we choose a random circle packing containing a circle $C$ of curvature $n$, then the probability that $C$ is tangent to the outermost circle tends towards $3/\pi $. These results are found by using positive semidefinite quadratic forms to make $\mathbb{P}^1(\mathbb{C})$ a parameter space for (not necessarily integral) circle packings. Finally, we examine an aspect of the integral theory known as spikes. When $n$ is prime, the distribution of $c/n$ is extremely smooth, whereas when $n$ is composite, there are certain spikes that correspond to prime divisors of $n$ that are at most $\sqrt{n}$. 

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