Gromov-Hausdorff Distance Research Articles

On turbulent relations

This paper extends the theory of turbulence of Hjorth to certain classes of equivalence relations that cannot be induced by Polish actions. The results are applied to analyze the quasi-isometry relation and finite Gromov–Hausdorff distance relation in the space of isometry classes of pointed proper metric spaces, called the Gromov space.

Nonreduction of Relations in the Gromov Space to Polish Actions

We show that in the Gromov space of isometry classes of pointed proper metric spaces, the equivalence relations defined by existence of coarse quasi-isometries or being at finite Gromov–Hausdorff distance cannot be reduced to the equivalence relation defined by any Polish action.

Modular Invariance of Conformal Field Theory on S^{1}×S^{3} and Circle Fibrations.

I conjecture a high-temperature-low-temperature duality for conformal field theories defined on circle fibrations like S^{3} and its lens space family. The duality is an exchange between the thermal circle and the fiber circle in the limit where both are small. The conjecture is motivated by the fact that π_{1}(S^{3}/Z_{p→∞})=Z=π_{1}(S^{1}×S^{2}) and the Gromov-Hausdorff distance between S^{3}/Z_{p→∞} and S^{1}/Z_{p→∞}×S^{2} vanishes. Several checks of the conjecture are provided: free fields, N=1 theories in four dimensions (which shows that the Di Pietro-Komargodski supersymmetric Cardy formula and its generalizations are given exactly by a supersymmetric Casimir energy), N=4 super Yang-Mills at strong coupling, and the six-dimensional N=(2,0) theory. For all examples considered, the duality is powerful enough to control the high-temperature asymptotics on the unlensed S^{3}, relating it to the Casimir energy on a highly lensed S^{3}. Such large-order quotients are more generally useful for studying quantum field theory on curved spacetimes.

Metric Scott analysis

We develop an analogue of the classical Scott analysis for metric structures and infinitary continuous logic. Among our results are the existence of Scott sentences for metric structures and a version of the López-Escobar theorem. We also derive some descriptive set theoretic consequences: most notably, that isomorphism on a class of separable structures is a Borel equivalence relation iff their Scott rank is uniformly bounded below ω1. Finally, we apply our methods to study the Gromov–Hausdorff distance between metric spaces and the Kadets distance between Banach spaces, showing that the set of spaces with distance 0 to a fixed space is a Borel set.

Computational Aspects of the Gromov–Hausdorff Distance and its Application in Non-rigid Shape Matching

The Gromov---Hausdorff distance of two compact metric spaces is a measure for how far the spaces are from being isometric and has been extensively studied in the field of metric geometry. In recent years, a lot of attention has been devoted to computational aspects of the Gromov---Hausdorff distance. One of the most prominent applications is the problem of shape matching, where the goal is to decide whether two shapes given as polygonal meshes are equivalent under certain invariances. Therefore, many methods have been proposed which heuristically estimate the Gromov---Hausdorff distance of metric spaces induced by the shapes. However, the computational complexity of computing the Gromov---Hausdorff distance has not yet been thoroughly investigated. We show that--under standard complexity theoretic assumptions--determining the Gromov---Hausdorff distance of two finite metric spaces cannot be approximated within any reasonable bound in polynomial time. Furthermore, we discover attributes of metric spaces which have a major impact on the complexity of an instance. This enables us to develop an approximation algorithm which is fixed parameter tractable with respect to corresponding parameters.

Topological stability from Gromov-Hausdorff viewpoint

We combine the classical Gromov-Hausdorff metric [ 5 ] with the \begin{document}$C^0$\end{document} distance to obtain the \begin{document}$C^0$\end{document} -Gromov-Hausdorff distance between maps of possibly different metric spaces. The latter is then combined with Walters's topological stability [ 11 ] to obtain the notion of topologically GH-stable homeomorphism. We prove that there are topologically stable homeomorphism which are not topologically GH-stable. Also that every topological GH-stable circle homeomorphism is topologically stable. Afterwards, we prove that every expansive homeomorphism with the pseudo-orbit tracing property of a compact metric space is topologically GH-stable. This is related to Walters's stability theorem [ 11 ]. Finally, we extend the topological GH-stability to continuous maps and prove the constant maps on compact homogeneous manifolds are topologically GH-stable.

A Gromov–Hausdorff distance between von Neumann algebras and an application to free quantum fields

A distance between von Neumann algebras is introduced, depending on a further norm inducing the w⁎-topology on bounded sets. Such notion is related both with the Gromov–Hausdorff distance for quantum metric spaces of Rieffel [24] and with the Effros–Maréchal topology [10,19] on the von Neumann algebras acting on a Hilbert space. This construction is tested on the local algebras of free quantum fields endowed with norms related with the Buchholz–Wichmann nuclearity condition [3], showing the continuity of such algebras w.r.t. the mass parameter.

Emergent space-time via a geometric renormalization method

We present a purely geometric renormalization scheme for metric spaces (including uncolored graphs), which consists of a coarse graining and a rescaling operation on such spaces. The coarse graining is based on the concept of quasi-isometry, which yields a sequence of discrete coarse grained spaces each having a continuum limit under the rescaling operation. We provide criteria under which such sequences do converge within a superspace of metric spaces, or may constitute the basin of attraction of a common continuum limit, which hopefully, may represent our space-time continuum. We discuss some of the properties of these coarse grained spaces as well as their continuum limits, such as scale invariance and metric similarity, and show that different layers of spacetime can carry different distance functions while being homeomorphic. Important tools in this analysis are the Gromov-Hausdorff distance functional for general metric spaces and the growth degree of graphs or networks. The whole construction is in the spirit of the Wilsonian renormalization group. Furthermore we introduce a physically relevant notion of dimension on the spaces of interest in our analysis, which e.g. for regular lattices reduces to the ordinary lattice dimension. We show that this dimension is stable under the proposed coarse graining procedure as long as the latter is sufficiently local, i.e. quasi-isometric, and discuss the conditions under which this dimension is an integer. We comment on the possibility that the limit space may turn out to be fractal in case the dimension is non-integer. At the end of the paper we briefly mention the possibility that our network carries a translocal far-order which leads to the concept of wormhole spaces and a scale dependent dimension if the coarse graining procedure is no longer local.

On the scaling limit of finite vertex transitive graphs with large diameter

Let $(X_n)$ be an unbounded sequence of finite, connected, vertex transitive graphs such that $ |X_n | = o(diam(X_n)^q)$ for some $q>0$. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence $(X_n)$ converges in the Gromov Hausdorff distance to a torus of dimension $<q$, equipped with some invariant Finsler metric. The proof relies on a recent quantitative version of Gromov's theorem on groups with polynomial growth obtained by Breuillard, Green and Tao. If $X_n$ is only roughly transitive and $|X_n| = o\bigl({diam(X_n)^{\delta}}\bigr)$ for $\delta > 1$ sufficiently small, we prove, this time by elementary means, that $(X_n)$ converges to a circle.

A note on the perturbations of compact quantum metric spaces

In this short note, we consider the perturbation of compact quantum metric spaces. We first show that for two compact quantum metric spaces (A, P) and (B, Q) for which A and B are subspaces of an order-unit space C and P and Q are Lip-norms on A and B respectively, the quantum Gromov–Hausdorff distance between (A, P) and (B, Q) is small under certain conditions. Then some other perturbation results on compact quantum metric spaces derived from spectral triples are also given.

Gromov–Hausdorff distance for pointed metric spaces

We establish an embedding theorem for the Gromov–Hausdorff limit of a sequence of proper pointed metric spaces.

Relative interleavings and applications to sensor networks

In this paper, we prove several results on interleavings for persistent relative homology of sub-level sets, \({\check{\mathrm{C}}\hbox {ech}}\) complexes and Rips complexes. To prove the relative interleavings for \({\check{\mathrm{C}}\hbox {ech}}\) complexes and Rips complexes, we define a relative correspondence which is related to the Gromov-Hausdorff distance. We also apply the relative Rips interleaving to a coverage problem in sensor networks, and show that the interleaving captures some features about the sensors after perturbation from the information of the unperturbed system.

Volume comparison without curvature-assumption

Let \(\, X^n ,\ M^n\,\) be any pair of compact connected Riemannian manifolds; the only assumption on \(\,M^n\,\) is that the Gromov–Hausdorff distance \(\,\varepsilon \,\) between \(\,M^n\,\) and \(\,X^n\,\) is smaller than the (normalized) injectivity radius of \(\, X^n\,\). Using a transport of measures and a new notion of barycentre, we construct a \(\,C^1\,\) Gromov–Hausdorff \(\,C_0\cdot \varepsilon ^{2/3}\)-approximation \(\,H : M^n \rightarrow X^n\,\), whose energy and Jacobian determinant are sharply bounded from above by \(\, 1 + C_1\cdot \varepsilon ^{1/3}\,\) (the constants \(\,C_i\,\) are explicit), implying a sharp lower bound for the ratio between the volumes of \(\,M^n\,\) and \(\,X^n\,\). This result extends to the non compact case, when \(\,X^n\,\) and \(\,M^n\,\) are only supposed to be quasi-isometric.

A compactness theorem for the dual Gromov-Hausdorff propinquity

We prove a compactness theorem for the dual Gromov-Hausdorff propinquity as a noncommutative analogue of the Gromov compactness theorem for the Gromov-Hausdorff distance. Our theorem is valid for subclasses of quasi-Leibniz quantum compact metric spaces of the closure of finite dimensional quasi-Leibniz quantum compact metric spaces for the dual propinquity. While finding characterizations of this class proves delicate, we show that all nuclear, quasi-diagonal quasi-Leibniz quantum compact metric spaces are limits of finite dimensional quasi-Leibniz quantum compact metric spaces. This result involves a mild extension of the definition of the dual propinquity to quasi-Leibniz quantum compact metric spaces, which is presented in the first part of this paper.

The quantum Gromov-Hausdorff propinquity

We introduce the quantum Gromov-Hausdorff propinquity, a new distance between quantum compact metric spaces, which extends the Gromov-Hausdorff distance to noncommutative geometry and strengthens Rieffel’s quantum Gromov-Hausdorff distance and Rieffel’s proximity by making *-isomorphism a necessary condition for distance zero, while being well adapted to Leibniz seminorms. This work offers a natural solution to the long-standing problem of finding a framework for the development of a theory of Leibniz Lip-norms over C*-algebras.

The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs

One major open conjecture in the area of critical random graphs, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade [23, 24, 28, 63] is as follows: for a wide array of random graph models with degree exponent $\tau\in (3,4)$, distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $n^{(\tau-3)/(\tau-1)}$. In this paper we study the metric space structure of maximal components of the multiplicative coalescent, in the regime where the sizes converge to excursions of L\'evy processes without replacement [10], yielding a completely new class of limiting random metric spaces. A by-product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent $\tau\in (3,4)$ where edges are rescaled by $n^{-(\tau-3)/(\tau-1)}$ yielding the first rigorous proof of the above conjecture. The limits in this case are compact tree-like random fractals with finite fractal dimensions and with a dense collection of hubs (infinite degree vertices) a finite number of which are identified with leaves to form shortcuts. It is generally believed that dynamic versions of a number of fundamental random graph models, as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent. In work in progress, the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent $\tau\in (3,4)$. Our proof makes crucial use of inhomogeneous continuum random trees (ICRT), which have previously arisen in the study of the entrance boundary of the additive coalescent. We show that tilted versions of the same objects using the associated mass measure, describe connectivity properties of the multiplicative coalescent. Since convergence of height processes of corresponding approximating p-trees is not known, we use general methodology in [14] and develop novel techniques relying on first showing convergence in the Gromov-weak topology and then extending this to Gromov-Hausdorff-Prokhorov convergence by proving a global lower mass-bound. Keywords: Multiplicative coalescent, p-trees, inhomogeneous continuum random trees, critical random graphs, Gromov-Hausdorff distance, Gromov-weak topology

Uniform approximation of metrics by graphs

We say that a metric graph is uniformly bounded if the degrees of all vertices are uniformly bounded and the lengths of edges are pinched between two positive constants; a metric space is approximable by a uniform graph if there is one within a finite Gromov-Hausdorff distance. We show that the Euclidean plane and Gromov hyperbolic geodesic spaces with bounded geometry are approximable by uniform graphs, and pose a number of open problems.

ON THE GEOMETRY OF LORENTZ SPACES AS A LIMIT SPACE

Abstract. In this paper, we prove that there is no branch point in theLorentz space (M,d) which is the limit space of a sequence {(M α ,d α )}ofcompact globally hyperbolic interpolating spacetimes with C α± -propertiesand curvature bounded below. Using this, we also obtain that every max-imal timelike geodesic in the limit space (M,d) can be expressed as thelimit curve of a sequence of maximal timelike geodesics in {(M α ,d α )}.Finally, we show that the limit space (M,d) satisfies a timelike trian-gle comparison property which is analogous to the case of Alexandrovcurvature bounds in length spaces. 1. IntroductionLorentzian Gromov-Hausdorff theory was first introduced by J. Noldus in[3] and the notion of Gromov-Hausdorff (GH) distance d GH ((M,g),(N,h))between two compact, globally hyperbolic (CGH),interpolating spacetimes(M,d) and (N,h) was defined in a similar way as in the Riemannian case.Specifically, we say (M,d) and (N,h) ǫ-close if and only if there exist map-pings ψ: M→ N, ξ: N→ Msuch that| d

Gromov–Hausdorff convergence of non-Archimedean fuzzy metric spaces

We introduce the notion of the Gromov–Hausdorff fuzzy distance between two non-Archimedean fuzzy metric spaces (in the sense of Kramosil and Michalek). Basic properties involving convergence and the fuzzy version of the completeness theorem are presented. We show that the topological properties induced by the classic Gromov–Hausdorff distance on metric spaces can be deduced from our approach.

The dual Gromov–Hausdorff propinquity

Motivated by the quest for an analogue of the Gromov–Hausdorff distance in noncommutative geometry which is well-behaved with respect to C⁎-algebraic structures, we propose a complete metric on the class of Leibniz quantum compact metric spaces, named the dual Gromov–Hausdorff propinquity. This metric resolves several important issues raised by recent research in noncommutative metric geometry: it makes *-isomorphism a necessary condition for distance zero, it is well-adapted to Leibniz seminorms, and — very importantly — is complete, unlike the quantum propinquity which we introduced earlier. Thus our new metric provides a natural tool for noncommutative metric geometry, designed to allow for the generalizations of techniques from metric geometry to C*-algebra theory.