arXiv subjects

Henry Adams

Publications and source records attributed to Henry Adams.

At least 19 recordsLinked to original sources

arXiv2026

Gromov--Hausdorff Distance Between Euclidean Unit Balls

What is the Gromov--Hausdorff distance between Euclidean unit balls of different dimensions, denoted by $d_\gh(B^m,B^n)$, for $m>n$? Note that the lower bound coming from the stability of persistent homology is zero, since all balls possess identical (trivial) persistent homology. To establish non-trivial lower bounds, we exploit the Borsuk--Ulam theorem. For any $n \ge 1$, we prove that $d_\gh(B^m,B^n)\ge \frac{\sqrt{n+1}}{\sqrt{n+1}+\sqrt{n}} > \frac{1}{2}$ for $m>n$, and that $d_\gh(B^m,B^n)\to 1$ as $m\to \infty$. Finally, we prove that $d_\gh(B^m,B^n)<1$ for all finite $m>n\geq 1$.

math.MG↗

arXiv2026

Tight upper bound on $d_{GH}(S^1,S^{2k+1})$: GPT's short proof

We describe GPT-5.6 Sol Pro's short proof that $2\cdot d_{GH}(S^1,S^{2k+1}) \le \tfrac{2\pi k}{2k+1}$. This is the odd-dimensional case of a tight upper bound by Harrison and Jeffs on the Gromov--Hausdorff distance between the circle and spheres.

math.MG↗

arXiv2026

Gromov-Hausdorff distance and Jung constant of finite-dimensional normed spaces

For a finite-dimensional normed space $V$ and a subset $X$ with finite Hausdorff distance from $V$, we prove that the Gromov--Hausdorff distance between $X$ and $V$ is at least the Hausdorff distance between $X$ and $V$, divided by twice the relative Jung constant of $V$. If $V$ furthermore satisfies a certain intersection property, we show a stronger result where the relative Jung constant can be replaced with its absolute version. Key words: Normed spaces, Jung constant, Hausdorff distance, Gromov--Hausdorff distance.

math.MG↗

arXiv2025

Vietoris thickenings and complexes of manifolds are homotopy equivalent

We show that if $X$ is a finite-dimensional Polish metric space, then the natural bijection $\mathrm{VR}(X;r)\to \mathrm{VR^m}(X;r)$ from the (open) Vietoris-Rips complex to the Vietoris-Rips metric thickening is a homotopy equivalence. This occurs, for example, if $X$ is a Riemannian manifold. The same is true for the map $\mathrm{\check{C}}(X;r)$ to $\mathrm{\check{C}}^\mathrm{m}(X;r)$ from the \v{C}ech complex to the \v{C}ech metric thickening, and more generally, for the natural bijection $\mathrm{V}(\mathcal W)\to \mathrm{V^m}(\mathcal W)$ from the Vietoris complex to the Vietoris metric thickening of any uniformly bounded cover $\mathcal W$ of a finite dimensional Polish metric space. We also show that if $X$ is a compact metrizable space, then $\mathrm{V^m}(\mathcal W)$ is strongly locally contractible.

math.GT↗

arXiv2025

An optimal Brouwer's fixed point theorem for discontinuous functions

Brouwer's fixed point theorem states that any continuous function from a closed $n$-dimensional ball to itself has a fixed point. In 1961, Klee showed that if such a function has discontinuities that are bounded, then it has a point that is close to being fixed. We improve upon Klee's results in any finite-dimensional Euclidean space, and prove that our bounds are the best possible.

math.MG↗

arXiv2025

Vietoris--Rips complexes of ellipses at larger scales

For $X$ a metric space and $r>0$, the Vietoris--Rips simplicial complex $\mathrm{VR}(X;r)$ has $X$ as its vertex set, and a finite subset $\sigma \subseteq X$ as a simplex whenever the diameter of $\sigma$ is less than $r$. In ``On Vietoris--Rips complexes of ellipses'', the authors studied the homotopy types of Vietoris--Rips complexes of ellipses $E_a=\{(x,y)\in \mathbb{R}^2~|~(x/a)^2+y^2=1\}$ of small eccentricity, meaning $1<a< \sqrt{2}$, at small scales $r < \frac{4\sqrt{3}a^2}{3a^2+1}$. In this paper, we further investigate the homotopy types that appear at larger scales. In particular, we identify the scale parameters $r$, as a function of the eccentricity $a$, for which the Vietoris--Rips complex $\mathrm{VR}(E_a;r)$ is homotopy equivalent to a $3$-sphere, to a wedge sum of $4$-spheres, or to a $5$-sphere.

math.MG↗

arXiv2025

Quantifying discontinuity

Given a compact space $X$ that does not admit an embedding (an injective continuous function) into $\mathbb{R}^d$, we study the ''degree'' of discontinuity that any injective function $X \to \mathbb{R}^d$ must have. To this end, we define a scale invariant modulus of discontinuity and obtain general lower bounds, thus obtaining quantified nonembeddability results of Haefliger--Weber type. Moreover, we establish analogous lower bounds for simplicial complexes that do not admit an almost $r$-embedding in $\mathbb{R}^d$, thus obtaining a quantified version of the topological Tverberg theorem.

math.MG↗

arXiv2025

Anti-Vietoris--Rips metric thickenings and Borsuk graphs

For $X$ a metric space and $r\ge 0$, the anti-Vietoris-Rips metric thickening $\mathrm{AVR^m}(X;r)$ is the space of all finitely supported probability measures on $X$ whose support has spread at least $r$, equipped with an optimal transport topology. We study the anti-Vietoris-Rips metric thickenings of spheres. We have a homeomorphism $\mathrm{AVR^m}(S^n;r) \cong S^n$ for $r > \pi$, a homotopy equivalence $\mathrm{AVR^m}(S^n;r) \simeq \mathbb{RP}^{n}$ for $\frac{2\pi}{3} < r \le \pi$, and contractibility $\mathrm{AVR^m}(S^n;r) \simeq *$ for $r=0$. For an $n$-dimensional compact Riemannian manifold $M$, we show that the covering dimension of $\mathrm{AVR^m}(M;r)$ is at most $(n+1)p-1$, where $p$ is the packing number of $M$ at scale $r$. Hence the $k$-dimensional \v{C}ech cohomology of $\mathrm{AVR^m}(M;r)$ vanishes in all dimensions $k\geq (n+1)p$. We prove more about the topology of $\mathrm{AVR^m}(S^n;\frac{2\pi}{3})$, which has vanishing cohomology in dimensions $2n+2$ and higher. We explore connections to chromatic numbers of Borsuk graphs, and in particular we prove that for $k>n$, no graph homomorphism $\mathrm{Bor}(S^k;r) \to \mathrm{Bor}(S^n;\alpha)$ exists when $\alpha > \frac{2\pi}{3}$.

math.AT↗

arXiv2025

Vietoris-Rips complexes of torus grids

We study the topology of Vietoris--Rips complexes of finite grids on the torus. Let $T_{n,n}$ be the grid of $n\times n$ points on the flat torus $S^1\times S^1$, equipped with the $l^1$ metric. Let $\mathrm{VR}(T_{n,n};k)$ be the Vietoris--Rips simplicial complex of this torus grid at scale $k\ge 0$. For $n\ge 7$ and small scales $2\le k\le \frac{n-1}{3}$, the complex $\mathrm{VR}(T_{n,n};k)$ is homotopy equivalent to the torus. For large scales $k\ge 2\lfloor\frac{n}{2}\rfloor$, the complex $\mathrm{VR}(T_{n,n};k)$ is a simplex and hence contractible. Interesting topology arises over intermediate scales $\frac{n-1}{3}<k<2\lfloor\frac{n}{2}\rfloor$. For example, we prove that $\mathrm{VR}(T_{2n,2n};2n-1)\cong S^{2n^2-1}$ for $n\ge 2$, that $\mathrm{VR}(T_{3n,3n};n)\simeq\vee^{6n^2-1}S^2$ for $n\ge 2$, and that $\mathrm{VR}(T_{3n-1,3n-1};n)\simeq \bigvee_{6n-3} S^2\vee \bigvee_{6n-2}S^3$ for $n\geq 3$. Based on homology computations, we conjecture that $\mathrm{VR}(T_{n,n};k)$ is homotopy equivalent to a $3$-sphere for a countable family of $(n,k)$ pairs, and we prove this for $(n,k)=(7,4)$.

math.AT↗

arXiv2025

Homotopy connectivity of \v{C}ech complexes of spheres

Let $S^n$ be the $n$-sphere with the geodesic metric and of diameter $\pi$. The intrinsic \v{C}ech complex of $S^n$ at scale $r$ is the nerve of all open balls of radius $r$ in $S^n$. In this paper, we show how to control the homotopy connectivity of \v{C}ech complexes of spheres at each scale between $0$ and $\pi$ in terms of coverings of spheres. Our upper bound on the connectivity, which is sharp in the case $n=1$, comes from the chromatic numbers of Borsuk graphs of spheres. Our lower bound is obtained using the conicity (in the sense of Barmak) of \v{C}ech complexes of the sufficiently dense, finite subsets of $S^n$. Our bounds imply the new result that for $n\ge 1$, the homotopy type of the \v{C}ech complex of $S^n$ at scale $r$ changes infinitely many times as $r$ varies over $(0,\pi)$; we conjecture only countably many times. Additionally, we lower bound the homological dimension of \v{C}ech complexes of finite subsets of $S^n$ in terms of their packings.

math.AT↗

arXiv2024

Sim-to-real supervised domain adaptation for radioisotope identification

Machine learning has the potential to improve the speed and reliability of radioisotope identification using gamma spectroscopy. However, meticulously labeling an experimental dataset for training is often prohibitively expensive, while training models purely on synthetic data is risky due to the domain gap between simulated and experimental measurements. In this research, we demonstrate that supervised domain adaptation can substantially improve the performance of radioisotope identification models by transferring knowledge between synthetic and experimental data domains. We consider two domain adaptation scenarios: (1) a simulation-to-simulation adaptation, where we perform multi-label proportion estimation using simulated high-purity germanium detectors, and (2) a simulation-to-experimental adaptation, where we perform multi-class, single-label classification using measured spectra from handheld lanthanum bromide (LaBr) and sodium iodide (NaI) detectors. We begin by pretraining a spectral classifier on synthetic data using a custom transformer-based neural network. After subsequent fine-tuning on just 64 labeled experimental spectra, we achieve a test accuracy of 96% in the sim-to-real scenario with a LaBr detector, far surpassing a synthetic-only baseline model (75%) and a model trained from scratch (80%) on the same 64 spectra. Furthermore, we demonstrate that domain-adapted models learn more human-interpretable features than experiment-only baseline models. Overall, our results highlight the potential for supervised domain adaptation techniques to bridge the sim-to-real gap in radioisotope identification, enabling the development of accurate and explainable classifiers even in real-world scenarios where access to experimental data is limited.

cs.LG↗

arXiv2024

Lower Bounding the Gromov--Hausdorff distance in Metric Graphs

Let $G$ be a finite, connected metric graph and let $X\subseteq G$ be a subset. 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 $G=S^1$ with circumference $2\pi$, a recent study established the equality $d_\gh(S^1,X)=d_\h(S^1,X)$ whenever $d_\gh(S^1,X)<\frac{\pi}{6}$. Our results relax this hypothesis to $d_\gh(S^1,X)<\frac{\pi}{3}$, and furthermore, we show that the constant $\frac{\pi}{3}$ 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 $f\colon G \to X$ with too small distortion contradicts the connectedness of $G$.

math.MG↗

arXiv2024

Gromov-Hausdorff distances between quotient metric spaces

The Hausdorff distance measures how far apart two sets are in a common metric space. By contrast, the Gromov-Hausdorff distance provides a notion of distance between two abstract metric spaces. How do these distances behave for quotients of spaces under group actions? Suppose a group $G$ acts by isometries on two metric spaces $X$ and $Y$. In this article, we study how the Hausdorff and Gromov-Hausdorff distances between $X$ and $Y$ and their quotient spaces $X/G$ and $Y/G$ are related. For the Hausdorff distance, we show that if $X$ and $Y$ are $G$-invariant subsets of a common metric space, then we have $d_{\mathrm{H}}(X,Y)=d_{\mathrm{H}}(X/G,Y/G)$. However, the Gromov-Hausdorff distance does not preserve this relationship: we show how to make the ratio $\frac{d_{\mathrm{GH}}(X/G,Y/G)}{d_{\mathrm{GH}}(X,Y)}$ both arbitrarily large and arbitrarily small, even if $X$ is an arbitrarily dense $G$-invariant subset of $Y$.

math.MG↗

arXiv2024

Persistent equivariant cohomology

This article has two goals. First, we hope to give an accessible introduction to persistent equivariant cohomology. Given a topological group $G$ acting on a filtered space, persistent Borel equivariant cohomology measures not only the shape of the filtration, but also attributes of the group action on the filtration, including in particular its fixed points. Second, we give an explicit description of the persistent equivariant cohomology of the circle action on the Vietoris-Rips metric thickenings of the circle, using the Serre spectral sequence and the Gysin homomorphism. Indeed, if $\frac{2\pi k}{2k+1} \le r < \frac{2\pi(k+1)}{2k+3}$, then $H^*_{S^1}(\mathrm{VR}^\mathrm{m}(S^1;r))\cong \mathbb{Z}[u]/(1\cdot3\cdot5\cdot\ldots \cdot (2k+1)\, u^{k+1})$ where $\mathrm{deg}(u)=2$.

math.AT↗

arXiv2024

The connectivity of Vietoris-Rips complexes of spheres

We survey what is known and unknown about Vietoris-Rips complexes and thickenings of spheres. Afterwards, we show how to control the homotopy connectivity of Vietoris-Rips complexes of spheres in terms of coverings of spheres and projective spaces. Let $S^n$ be the $n$-sphere with the geodesic metric, and of diameter $\pi$, and let $\delta > 0$. Suppose that the first nontrivial homotopy group of the Vietoris-Rips complex $\mathrm{VR}(S^n;\pi-\delta)$ of the $n$-sphere at scale $\pi-\delta$ occurs in dimension $k$, i.e., suppose that the connectivity is $k-1$. Then $\mathrm{cov}_{S^n}(2k+2) \le \delta < 2\cdot \mathrm{cov}_{\mathbb{R}P^n}(k)$. In other words, there exist $2k+2$ balls of radius $\delta$ that cover $S^n$, and no set of $k$ balls of radius $\frac{\delta}{2}$ cover the projective space $\mathbb{R}P^n$. As a corollary, the homotopy type of $\mathrm{VR}(S^n;r)$ changes infinitely many times as the scale $r$ increases.

math.AT↗

arXiv2023

Time-Varying Spaces and Mobile Sensor Networks

Consider a mobile sensor network, in which each sensor covers a ball. Sensors do not know their locations, but can detect if the covered balls overlap. An intruder cannot pass undetected between overlapping sensors. An evasion path exists if it is possible for an intruder to move in the domain without ever entering a covered region. We examine two time-varying topological spaces arising from such mobile sensor networks. These examples were constructed by Adams and Carlsson to show that the time-varying homotopy type of the covered region does not determine whether an evasion path exists or not. One of these spaces has an evasion path and the other does not, which means that the uncovered regions are not time-varying homotopy equivalent. We show that the covered regions of these spaces are not time-varying homeomorphic, even though they are time-varying homotopy equivalent. We then elaborate on this time-varying homotopy equivalence between the covered regions.

math.AT↗

arXiv2023

Hausdorff vs Gromov-Hausdorff distances

Let $M$ be a closed Riemannian manifold and let $X\subseteq M$. If the sample $X$ is sufficiently dense relative to the curvature of $M$, then the Gromov-Hausdorff distance between $X$ and $M$ is bounded from below by half their Hausdorff distance, namely $d_{GH}(X,M) \ge \frac{1}{2} d_H(X,M)$. The constant $\frac{1}{2}$ can be improved depending on the dimension and curvature of the manifold $M$, and obtains the optimal value $1$ in the case of the unit circle, meaning that if $X\subseteq S^1$ satisfies $d_{GH}(X,S^1)<\tfrac{\pi}{6}$, then $d_{GH}(X,S^1)=d_H(X,S^1)$. We also provide versions lower bounding the Gromov-Hausdorff distance $d_{GH}(X,Y)$ between two subsets $X,Y\subseteq M$. Our proofs convert discontinuous functions between metric spaces into simplicial maps between \v{C}ech or Vietoris-Rips complexes. We then produce topological obstructions to the existence of certain maps using the nerve lemma and the fundamental class of the manifold, thus lower bounding the Gromov-Hausdorff distance.

math.MG↗

arXiv2023

Lower bounds on the homology of Vietoris-Rips complexes of hypercube graphs

We provide novel lower bounds on the Betti numbers of Vietoris-Rips complexes of hypercube graphs of all dimensions, and at all scales. In more detail, let $Q_n$ be the vertex set of $2^n$ vertices in the $n$-dimensional hypercube graph, equipped with the shortest path metric. Let $VR(Q_n;r)$ be its Vietoris--Rips complex at scale parameter $r \ge 0$, which has $Q_n$ as its vertex set, and all subsets of diameter at most $r$ as its simplices. For integers $r<r'$ the inclusion $VR(Q_n;r)\hookrightarrow VR(Q_n;r')$ is nullhomotopic, meaning no persistent homology bars have length longer than one, and we therefore focus attention on the individual spaces $VR(Q_n;r)$. We provide lower bounds on the ranks of homology groups of $VR(Q_n;r)$. For example, using cross-polytopal generators, we prove that the rank of $H_{2^r-1}(VR(Q_n;r))$ is at least $2^{n-(r+1)}\binom{n}{r+1}$. We also prove a version of \emph{homology propagation}: if $q\ge 1$ and if $p$ is the smallest integer for which $rank H_q(VR(Q_p;r))\neq 0$, then $rank H_q(VR(Q_n;r)) \ge \sum_{i=p}^n 2^{i-p} \binom{i-1}{p-1} \cdot rank H_q(VR(Q_p;r))$ for all $n \ge p$. When $r\le 3$, this result and variants thereof provide tight lower bounds on the rank of $H_q(VR(Q_n;r))$ for all $n$, and for each $r \ge 4$ we produce novel lower bounds on the ranks of homology groups. Furthermore, we show that for each $r\ge 2$, the homology groups of $VR(Q_n;r)$ for $n \ge 2r+1$ contain propagated homology not induced by the initial cross-polytopal generators.

math.CO↗