Research

My broad area of interest is Geometry and Topology and my research combines themes from Riemannian geometry, algebraic topology, and geometric group theory.

The focus of my recent research concerns large-scale properties of metric spaces. I am interested in \(\delta\)-hyperbolic spaces and their applications to complex networks geometry and geometrical data analysis. Introduced by Gromov in 1987, \(\delta\)-hyperbolic spaces generalize the metric properties of the classical hyperbolic geometry and that of trees. These spaces play an instrumental role in geometric group theory and in the geometry of negatively curved spaces. More recently the concept of \(\delta\)-hyperbolicity has emerged in discrete mathematics, algorithms and networking. For example, it has been shown empirically that, from a metric point of view, many real-world graphs are tree-like or they have small hyperbolicity. Examples include Internet application networks, web networks, social networks, and biological networks. It has also been shown that the Internet topology embeds with better accuracy (less distortion) into a hyperbolic space than into a Euclidean space of comparable dimension.

One of the fundamental questions in studying metric spaces is the isometric embedding problem. That is, finding conditions on metric spaces \((X,d)\) to isometrically embed in spaces with additional structure or properties, such as the Euclidean space, or certain normed linear spaces. In general, it is rare to find isometric embeddings between two spaces of interest and one often allows the embedding to alter the distances in some controlled way (distortion). In [1], we introduce the quasi-hyperbolicity constant of a metric space, a rough isometry invariant that measures how a metric space deviates from being Gromov hyperbolic. This number, for unbounded spaces, lies in the closed interval \([1,2]\). The quasi-hyperbolicity constant of an unbounded Gromov hyperbolic space is equal to one. For a CAT\((0)\)-space, it is bounded from above by \(\sqrt{2}\). The quasi-hyperbolicity constant of a Banach space that is at least two dimensional is bounded from below by \(\sqrt{2}\), and for a non-trivial \(L_p\)-space it is exactly \(\max\{2^{1/p},2^{1-1/p}\}\). If \(0 < \alpha < 1\) then the quasi-hyperbolicity constant of the \(\alpha\)-snowflake of any metric space is bounded from above by \(2^\alpha\). We give an exact calculation in the case of the \(\alpha\)-snowflake of the Euclidean real line. The aim of this line of research is to further investigate applications of the newly defined quasi-hyperbolicity constant to the analysis of the underlying geometry of large-scale networks and non-Euclidean random geometric graphs.

A different approach to studying metric spaces is to fix the point set \(X\) and define a new metric \(d_\varphi=\varphi\circ d\) on \(X\) obtained from the old metric by composing it with a real valued function \(\varphi\). A function \(\varphi\) with the property that for every metric space \((X,d)\) the composition \(d_\varphi=d\circ\varphi\) is also a metric on \(X\) is called a metric transform. A central question concerning metric transforms is whether there exist such functions \(\varphi\) for which the transformed metric space \((X,d_\varphi)\) has certain specified properties or preserves some of the characteristics of the original metric space \((X,d)\). In [2], we obtained the following metric transform rigidity of roughly geodesic Gromov hyperbolic spaces: If \((X,d)\) is any metric space containing a rough geodesic ray and \(\varphi\) is an approximately nondecreasing, unbounded metric transform such that the transformed space \((X,d_\varphi)\) is Gromov hyperbolic and roughly geodesic then \(\varphi\) is an approximate dilation and the original space \((X,d)\) is Gromov hyperbolic and roughly geodesic.

Another direction of my research is concerned with the geometry and topology of orbifolds, and more specifically with the existence of closed geodesics on compact developable orbifolds. This project involves the study of discrete group actions on manifolds and reveals some interesting connections between the algebraic properties of such groups and the geometry of the quotient space. By showing that a compact developable orbifold \(\mathcal{Q}\) admits a closed geodesic of positive length whenever the orbifold fundamental group contains a hyperbolic element, the existence problem reduces to compact orbifolds \(\mathcal{Q}\) with \(\pi_1^{orb}(\mathcal{Q})\) an infinite group of finite exponent and with finitely many conjugacy classes. A question that arises is whether such an infinite torsion group can act geometrically on a simply connected complete Riemannian manifold. While ruling out such actions in the general case is extremely difficult, it can be shown that such actions cannot occur if the manifold is assumed to carry a metric of nonpositive or nonnegative sectional curvature; and as a consequence one obtains that any compact orbifold of nonpositive or nonnegative curvature admits a closed geodesic of positive length (see [4]). By taking the opposite approach and, by showing that any compact orbifold of dimension \(3\), \(5\) or \(7\) admits a closed geodesic of positive length, it follows that infinite torsion groups cannot act geometrically on simply connected complete Riemannian manifolds without closed geodesics and having dimension \(3\), \(5\) or \(7\) (see [3]).

Last modified: 2020-01-04.