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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. Polyhedral approximation and uniformization for non-length surfaces

   https://doi.org/10.4171/JEMS/1538  

   Ntalampekos, Dimitrios; Romney, Matthew (September 2024, Journal of the European Mathematical Society)

   We prove that any metric surface (that is, metric space homeomorphic to a 2-manifold with boundary) with locally finite Hausdorff 2-measure is the Gromov–Hausdorff limit of polyhedral surfaces with controlled geometry. We use this result, together with the classical uniformization theorem, to prove that any metric surface homeomorphic to the 2-sphere with finite Hausdorff 2-measure admits a weakly quasiconformal parametrization by the Riemann sphere, answering a question of Rajala–Wenger. These results have been previously established by the authors under the assumption that the metric surface carries a length metric. As a corollary, we obtain new proofs of the uniformization theorems of Bonk–Kleiner for quasispheres and of Rajala for reciprocal surfaces. Another corollary is a simplification of the definition of a reciprocal surface, which answers a question of Rajala concerning minimal hypotheses under which a metric surface is quasiconformally equivalent to a Euclidean domain. 

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4. 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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5. On Matoušek-like embedding obstructions of countably branching graphs

   https://doi.org/10.1016/j.jmaa.2024.128896  

   Malthaner, Ryan (March 2025, Journal of Mathematical Analysis and Applications)

   In this paper we present new proofs of the non-embeddability of countably branching trees into Banach spaces satisfying property beta_p and of countably branching diamonds into Banach spaces which are l_p-asymptotic midpoint uniformly convex (p-AMUC) for p>1. These proofs are entirely metric in nature and are inspired by previous work of Jiří Matoušek. In addition, using this metric method, we succeed in extending these results to metric spaces satisfying certain embedding obstruction inequalities. Finally, we give Tessera-type lower bounds on the compression for a class of Lipschitz embeddings of the countably branching trees into Banach spaces containing l_p-asymptotic models for p>=1. 

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