@inproceedings{22003,
  abstract     = {Let G be a finite, connected metric graph and let X be a subset of G. If X is sufficiently dense in G, we show that the Gromov-Hausdorff distance matches the Hausdorff distance, namely d_GH(G,X) = d_H(G,X). When the metric graph is the circle G = S¹ with circumference 2π, a recent study established the equality d_GH(S¹,X) = d_H(S¹,X) whenever d_GH(S¹,X) < π/6. Our results relax this hypothesis to d_GH(S¹,X) < π/3, and furthermore, we show that the constant π/3 is the best possible. We lower bound the Gromov-Hausdorff distance d_GH(G,X) by the Hausdorff distance d_H(G,X) via a simple topological obstruction: the existence of a possibly discontinuous function f: G → X with too small distortion contradicts the connectedness of G.},
  author       = {Adams, Henry and Majhi, Sushovan and Manin, Fedor and Virk, Ziga and Zava, Nicolò},
  booktitle    = {42nd International Symposium on Computational Geometry},
  isbn         = {9783959774185},
  issn         = {1868-8969},
  keywords     = {Gromov–Hausdorff distance, distortion, connectedness, Borsuk–Ulam theorem},
  location     = {New Brunswick, NJ, United States},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Lower bounding the Gromov–Hausdorff distance in metric graphs}},
  doi          = {10.4230/LIPIcs.SoCG.2026.3},
  volume       = {367},
  year         = {2026},
}

@article{20867,
  abstract     = {We discuss the embeddability of subspaces of the Gromov–Hausdorff space, which consists of isometry classes of compact metric spaces endowed with the Gromov–Hausdorff distance, into Hilbert spaces. These embeddings are particularly valuable for applications to topological data analysis. We prove that its subspace consisting of metric spaces with at most n points has asymptotic dimension n(n−1)∕2. Thus, there exists a coarse embedding of that space into a Hilbert space. On the contrary, if the number of points is not bounded, then the subspace cannot be coarsely embedded into any uniformly convex Banach space and so, in particular, into any Hilbert space. Furthermore, we prove that, even if we restrict to finite metric spaces whose diameter is bounded by some constant, the subspace still cannot be bi-Lipschitz embedded into any finite-dimensional Hilbert space. We obtain both nonembeddability results by finding obstructions to coarse and bi-Lipschitz embeddings in families of isometry classes of finite subsets of the real line endowed with the Euclidean–Hausdorff distance.},
  author       = {Zava, Nicolò},
  issn         = {1472-2739},
  journal      = {Algebraic & Geometric Topology},
  number       = {8},
  pages        = {5153--5174},
  publisher    = {Mathematical Sciences Publishers},
  title        = {{Coarse and bi-Lipschitz embeddability of subspaces of the Gromov–Hausdorff space into Hilbert spaces}},
  doi          = {10.2140/agt.2025.25.5153},
  volume       = {25},
  year         = {2025},
}

@article{12764,
  abstract     = {We study a new discretization of the Gaussian curvature for polyhedral surfaces. This discrete Gaussian curvature is defined on each conical singularity of a polyhedral surface as the quotient of the angle defect and the area of the Voronoi cell corresponding to the singularity. We divide polyhedral surfaces into discrete conformal classes using a generalization of discrete conformal equivalence pioneered by Feng Luo. We subsequently show that, in every discrete conformal class, there exists a polyhedral surface with constant discrete Gaussian curvature. We also provide explicit examples to demonstrate that this surface is in general not unique.},
  author       = {Kourimska, Hana},
  issn         = {1432-0444},
  journal      = {Discrete and Computational Geometry},
  pages        = {123--153},
  publisher    = {Springer Nature},
  title        = {{Discrete yamabe problem for polyhedral surfaces}},
  doi          = {10.1007/s00454-023-00484-2},
  volume       = {70},
  year         = {2023},
}

@article{14557,
  abstract     = {Motivated by a problem posed in [10], we investigate the closure operators of the category SLatt of join semilattices and its subcategory SLattO of join semilattices with bottom element. In particular, we show that there are only finitely many closure operators of both categories, and provide a complete classification. We use this result to deduce the known fact that epimorphisms of SLatt and SLattO are surjective. We complement the paper with two different proofs of this result using either generators or Isbell’s zigzag theorem.},
  author       = {Dikranjan, D. and Giordano Bruno, A. and Zava, Nicolò},
  issn         = {1727-933X},
  journal      = {Quaestiones Mathematicae},
  number       = {S1},
  pages        = {191--221},
  publisher    = {Taylor & Francis},
  title        = {{Epimorphisms and closure operators of categories of semilattices}},
  doi          = {10.2989/16073606.2023.2247731},
  volume       = {46},
  year         = {2023},
}