CyberSec Research

The complement of the union of a collection of disjoint open disks in the $2$-sphere is called a Schottky set. We prove that a subset $S$ of the $2$-sphere is quasiconformally equivalent to a Schottky set if and only if every pair of complementary components of $S$ can be mapped to a pair of open disks with a uniformly quasiconformal homeomorphism of the sphere. Our theorem applies to Sierpi\'nski carpets and gaskets, yielding for the first time a general quasiconformal uniformization result for gaskets. Moreover, it contains Bonk's uniformization result for carpets as a special case and does not rely on the condition of uniform relative separation that is used in relevant works.

We prove Kalai's full flag conjecture for the class of locally anti-blocking polytopes, and show that there is equality if and only if the polytope is a (generalized) Hanner polytope.

We provide a foundation for an approach to the open problem of bilipschitz extendability of mappings defined on a Euclidean separated net. In particular, this allows for the complete positive solution of the problem in dimension two. Along the way, we develop a set of tools for bilipschitz extensions of mappings between subsets of Euclidean spaces.

The domain-independent universal Normalized Information Distance based on Kolmogorov complexity has been (in approximate form) successfully applied to a variety of difficult clustering problems. In this paper we investigate theoretical properties of the un-normalized algorithmic information distance $d_K$. The main question we are asking in this work is what properties this curious distance has, besides being a metric. We show that many (in)finite-dimensional spaces can(not) be isometrically scale-embedded into the space of finite strings with metric $d_K$. We also show that $d_K$ is not an Euclidean distance, but any finite set of points in Euclidean space can be scale-embedded into $(\{0,1\}^*,d_K)$. A major contribution is the development of the necessary framework and tools for finding more (interesting) properties of $d_K$ in future, and to state several open problems.

We introduce two definitions with the purpose of quantifying the concept of a $C^{2,\alpha}$ surface for $0 < \alpha < 1$. The intrinsic definition is given in terms of the $\alpha$-H\"{o}lder norm of the Gauss curvature function. The extrinsic one relies on the existence of a smooth local representation of the Riemannian metric. We show that these definitions are equivalent up to a constant depending on $\alpha$.

We show that every real-valued Lipschitz function on a subset of a metric space can be extended to the whole space while preserving the slope and, up to a small error, the global Lipschitz constant. This answers a question posed by Di Marino, Gigli, and Pratelli, who established the analogous property for the asymptotic Lipschitz constant. We also prove the same result for the ascending slope and for the descending slope.

We address an old open question in convex geometry that dates back to the work of Minkowski: what are the equality cases of the monotonicity of mixed volumes? The problem is equivalent to that of providing a geometric characterization of the support of mixed area measures. A conjectural characterization was put forward by Schneider (1985), but has been verified to date only for special classes of convex bodies. In this paper we resolve one direction of Schneider's conjecture for arbitrary convex bodies in $\mathbb{R}^n$, and resolve the full conjecture in $\mathbb{R}^3$. Among the implications of these results is a mixed counterpart of the classical fact, due to Monge, Hartman--Nirenberg, and Pogorelov, that a surface with vanishing Gaussian curvature is a ruled surface.

We study the metric compactification of a Kobayashi hyperbolic complex manifold \(\mathcal{X} \) equipped with the Kobayashi distance \( \mathsf{k}_{\mathcal{X}} \). We show that this compactification is genuine -- i.e., \( \mathcal{X} \) embeds as a dense open subset -- even without completeness of \( \mathsf{k}_{\mathcal{X}} \), and that it becomes a \emph{good compactification} in the sense of Bharali--Zimmer when \((\mathcal{X}, \mathsf{k}_{\mathcal{X}}) \) is complete. As an application, we obtain a criterion for the continuous extension of quasi-isometric embeddings from \( (\mathcal{X}, \mathsf{k}_{\mathcal{X}}) \) into visibility domains of complex manifolds. For a Kobayashi hyperbolic domain \( \Omega \subsetneq \mathcal{X} \), to each boundary point of \( \Omega \) in the end compactification, we associate a fiber of metric boundary points. This allows the small and big horospheres of Abate to be expressed as the intersection and union of horoballs centered at metric boundary points. We use this to formulate a Wolff-type lemma in terms of horoballs and prove a Denjoy--Wolff theorem for complete hyperbolic domains satisfying a boundary divergence condition for \(\mathsf{k}_{\mathcal{X}} \). Finally, we present necessary and sufficient conditions under which the identity map extends continuously between the metric and end compactifications.

Chromatic metric pairs consist of a metric space and a coloring function partitioning a subset thereof into various colors. It is a natural extension of the notion of chromatic point sets studied in chromatic topological data analysis. A useful tool in the field is the six-pack, a collection of six persistence diagrams, summarizing homological information about how the colored subsets interact. We introduce a suitable generalization of the Gromov-Hausdorff distance to compare chromatic metric pairs. We show some basic properties and validate this definition by obtaining the stability of the six-pack with respect to that distance. We conclude by discussing its restriction to metric pairs and its role in the stability of the \v{C}ech persistence diagrams.

In this paper, we give a short and self-contained proof to a 1991 conjecture by Moore concerning the structure of certain finite-dimensional Gromov--Hausdorff limits, in the ANR setting. As a consequence, one easily characterizes finite dimensional limits of PL-able or Riemannian $n$-manifolds with a uniform contractibility function. For example, one can define for any compact connected metric space that is a resolvable ANR homology manifold of covering dimension at least 5, an obstruction, which vanishes if and only if the homology manifold can be approximated in the Gromov--Hausdorff sense by PL-manifolds of the same dimension and with a uniform contractibility function. Further, it provides short proofs to certain well known results by reducing them to problems in Bing topology. We also give another proof using more classical arguments that yield more structural information. We give several applications to the theory of homology manifolds, Alexandrov spaces, Wasserstein spaces and a generalized form of the diffeomorphism stability conjecture.

We put Darboux's porism on folding of quadrilaterals, as well as closely related Bottema's zigzag porism, in the context of Arnold-Liouville integrability.

Given a probability measure on the unit disk, we study the problem of deciding whether, for some threshold probability, this measure is supported near a real algebraic variety of given dimension and bounded degree. We call this "testing the variety hypothesis". We prove an upper bound on the so-called "sample complexity" of this problem and show how it can be reduced to a semialgebraic decision problem. This is done by studying in a quantitative way the Hausdorff geometry of the space of real algebraic varieties of a given dimension and degree.

We prove that for any log-concave random vector $X$ in $\mathbb{R}^n$ with mean zero and identity covariance, $$ \mathbb{E} (|X| - \sqrt{n})^2 \leq C $$ where $C > 0$ is a universal constant. Thus, most of the mass of the random vector $X$ is concentrated in a thin spherical shell, whose width is only $C / \sqrt{n}$ times its radius. This confirms the thin-shell conjecture in high dimensional convex geometry. Our method relies on the construction of a certain coupling between log-affine perturbations of the law of $X$ related to Eldan's stochastic localization and to the theory of non-linear filtering. A crucial ingredient is a recent breakthrough technique by Guan that was previously used in our proof of Bourgain's slicing conjecture, which is known to be implied by the thin-shell conjecture.

We prove that there does not exist a piecewise affine isometric embedding of a flat torus into R^3 whose image has 7 vertices.

In this paper, we prove that for any $X\subset \R^n$ that is a $d$-dimensional definable set in an o-minimal structure and is Lipschitz regular at any $p\in X$, the following items are equivalent: (i) $C_3(X,p)$ is a $d$-dimensional linear subspace for any $p\in X$, and depends continuously on $p$; (ii) $C_4(X,p)=C_3(X,p)$ for any $p\in X$; (iii) $C_4(X,p)$ is a $d$-dimensional linear subspace for any $p\in X$; (iv) $C_5(X,p)$ is a $d-$dimensional linear subspace for any $p\in X$; (v) $X$ is $C^1$ smooth.

This paper investigates the failure of certain metric measure spaces to be infinitesimally Hilbertian or quasi-Riemannian manifolds, by constructing examples arising from a manifold $M$ endowed with a Riemannian metric $g$ that is possibly discontinuous, with $g, g^{-1} \in L^\infty_{\mathrm{loc}} $ and $ g \in W^{1,p}_{\mathrm{loc}}$ for $ p < \mathrm{dim} M - 1 $.

It is a standard assumption that datasets in high dimension have an internal structure which means that they in fact lie on, or near, subsets of a lower dimension. In many instances it is important to understand the real dimension of the data, hence the complexity of the dataset at hand. A great variety of dimension estimators have been developed to find the intrinsic dimension of the data but there is little guidance on how to reliably use these estimators. This survey reviews a wide range of dimension estimation methods, categorising them by the geometric information they exploit: tangential estimators which detect a local affine structure; parametric estimators which rely on dimension-dependent probability distributions; and estimators which use topological or metric invariants. The paper evaluates the performance of these methods, as well as investigating varying responses to curvature and noise. Key issues addressed include robustness to hyperparameter selection, sample size requirements, accuracy in high dimensions, precision, and performance on non-linear geometries. In identifying the best hyperparameters for benchmark datasets, overfitting is frequent, indicating that many estimators may not generalise well beyond the datasets on which they have been tested.

In this paper, we present a new method for computing the f-vector of a marked order polytope. Namely, given an arbitrary (polyhedral) subdivision of an arbitrary convex polytope, we construct a cochain complex (over the two-element field Z_2) such that the dimensions of its cohomology groups equal the components of the f-vector of the original polytope. In the case of a marked order polytope and its well-known cubosimplicial subdivision, this cochain complex can be described purely combinatorially -- which yields the said computation of the f-vector. Of independent interest may be our combinatorial description of the said cubosimplicial subdivision (which was originally constructed geometrically).