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1. Bi-Lipschitz embeddings of quasiconformal trees

   https://doi.org/10.1090/proc/16252  

   David, Guy; Eriksson-Bique, Sylvester; Vellis, Vyron (May 2023, Proceedings of the American Mathematical Society)

   A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. In this paper we show that every quasiconformal tree bi-Lipschitz embeds in some Euclidean space, with the ambient dimension and the bi-Lipschitz constant depending only on the doubling and bounded turning constants of the tree. This answers Question 1.6 of David and Vellis [Illinois J. Math. 66 (2022), pp. 189–244]. 

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2. Uniformization of Cantor sets with bounded geometry

   https://doi.org/10.1090/ecgd/360  

   Vellis, Vyron (August 2021, Conformal Geometry and Dynamics of the American Mathematical Society)

   In this note we provide a quasisymmetric taming of uniformly perfect and uniformly disconnected sets that generalizes a result of MacManus [Rev. Mat. Iberoamericana 15 (1999), pp. 267–277] from 2 to higher dimensions. In particular, we show that a compact subset of R n \mathbb {R}^n is uniformly perfect and uniformly disconnected if and only if it is ambiently quasiconformal to the standard Cantor set C \mathcal {C} in R n + 1 \mathbb {R}^{n+1} . 

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3. COVER TIME FOR THE FROG MODEL ON TREES

   https://doi.org/10.1017/fms.2019.37  

   HOFFMAN, CHRISTOPHER; JOHNSON, TOBIAS; JUNGE, MATTHEW (January 2019, Forum of Mathematics, Sigma)

   The frog model is a branching random walk on a graph in which particles branch only at unvisited sites. Consider an initial particle density of $$\unicode[STIX]{x1D707}$$ on the full $$d$$ -ary tree of height $$n$$ . If $$\unicode[STIX]{x1D707}=\unicode[STIX]{x1D6FA}(d^{2})$$ , all of the vertices are visited in time $$\unicode[STIX]{x1D6E9}(n\log n)$$ with high probability. Conversely, if $$\unicode[STIX]{x1D707}=O(d)$$ the cover time is $$\exp (\unicode[STIX]{x1D6E9}(\sqrt{n}))$$ with high probability. 

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4. Uniformly Bounded Arrays and Mutually Algebraic Structures

   https://doi.org/doi:10.1215/00294527-2020-0004  

   Laskowski, Michael C.; Terry, Caroline A. (January 2020, Notre Dame journal of formal logic)

   We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure M. We prove that if is a complete L-theory, then T is mutually algebraic if and only if there is some model M of T for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory T is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model M of T. 

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5. Branch-Pipe: Improving Graph Skeletonization around Branch Points in 3D Point Clouds

   https://doi.org/10.3390/rs13193802  

   Ziamtsov, Illia; Faizi, Kian; Navlakha, Saket (September 2021, Remote sensing)

   Modern plant phenotyping requires tools that are robust to noise and missing data, while being able to efficiently process large numbers of plants. Here, we studied the skeletonization of plant architectures from 3D point clouds, which is critical for many downstream tasks, including analyses of plant shape, morphology, and branching angles. Specifically, we developed an algorithm to improve skeletonization at branch points (forks) by leveraging the geometric properties of cylinders around branch points. We tested this algorithm on a diverse set of high-resolution 3D point clouds of tomato and tobacco plants, grown in five environments and across multiple developmental timepoints. Compared to existing methods for 3D skeletonization, our method efficiently and more accurately estimated branching angles even in areas with noisy, missing, or non-uniformly sampled data. Our method is also applicable to inorganic datasets, such as scans of industrial pipes or urban scenes containing networks of complex cylindrical shapes. 

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