@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{14362,
  abstract     = {Motivated by recent applications to entropy theory in dynamical systems, we generalise notions introduced by Matthews and define weakly weighted and componentwise weakly weighted (generalised) quasi-metrics. We then systematise and extend to full generality the correspondences between these objects and other structures arising in theoretical computer science and dynamics. In particular, we study the correspondences with weak partial metrics and, if the underlying space is a semilattice, with invariant (generalised) quasi-metrics satisfying the descending path condition, and with strictly monotone semi(-co-)valuations.
We conclude discussing, for endomorphisms of generalised quasi-metric semilattices, a generalisation of both the known intrinsic semilattice entropy and the semigroup entropy.},
  author       = {Castellano, Ilaria and Giordano Bruno, Anna and Zava, Nicolò},
  issn         = {0304-3975},
  journal      = {Theoretical Computer Science},
  publisher    = {Elsevier},
  title        = {{Weakly weighted generalised quasi-metric spaces and semilattices}},
  doi          = {10.1016/j.tcs.2023.114129},
  volume       = {977},
  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},
}

@article{10413,
  abstract     = {Motivated by the recent introduction of the intrinsic semilattice entropy, we study generalized quasi-metric semilattices and their categories. We investigate the relationship between these objects and generalized semivaluations, extending Nakamura and Schellekens' approach. Finally, we use this correspondence to compare the intrinsic semilattice entropy and the semigroup entropy induced in particular situations, like sets, torsion abelian groups and vector spaces.},
  author       = {Dikranjan, Dikran and Giordano Bruno, Anna and Künzi, Hans Peter and Zava, Nicolò and Toller, Daniele},
  issn         = {0166-8641},
  journal      = {Topology and its Applications},
  publisher    = {Elsevier},
  title        = {{Generalized quasi-metric semilattices}},
  doi          = {10.1016/j.topol.2021.107916},
  volume       = {309},
  year         = {2022},
}

@article{10608,
  abstract     = {We consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.},
  author       = {Weighill, Thomas and Yamauchi, Takamitsu and Zava, Nicolò},
  issn         = {2199-6768},
  journal      = {European Journal of Mathematics},
  number       = {1},
  pages        = {335--355},
  publisher    = {Springer Nature},
  title        = {{Coarse infinite-dimensionality of hyperspaces of finite subsets}},
  doi          = {10.1007/s40879-021-00515-3},
  volume       = {8},
  year         = {2022},
}