Related papers: Hierarchically hyperbolic spaces II: Combination t…
Clustering is a fundamental unsupervised learning task for uncovering patterns in data. While Gaussian Blurring Mean Shift (GBMS) has proven effective for identifying arbitrarily shaped clusters in Euclidean space, it struggles with…
We establish necessary and sufficient conditions for determining when a flat manifold can occur as a cusp cross-section within a given commensurability class of cusped arithmetic hyperbolic manifolds. This reduces the problem of identifying…
Hierarchical Topic Models (HTMs) are useful for discovering topic hierarchies in a collection of documents. However, traditional HTMs often produce hierarchies where lowerlevel topics are unrelated and not specific enough to their…
Hyperbolic manifolds for visual representation learning allow for effective learning of semantic class hierarchies by naturally embedding tree-like structures with low distortion within a low-dimensional representation space. The highly…
The space of directions is a notion of boundary associated to an arbitrary totally disconnected locally compact group. We explicitly calculate the space of directions of a group acting vertex transitively with compact open vertex…
Unlike the case of surfaces of topologically finite type, there are several different Teichm\"uller spaces that are associated to a surface of topological infinite type. These Teichm\"uller spaces first depend (set-theoretically) on whether…
Let S be an infinite-dimensional manifold of all symplectic, or hyperkahler, structures on a compact manifold M, and $Diff_0$ the connected component of its diffeomorphism group. The quotient $S/\Diff_0$ is called the Teichmuller space of…
We introduce a variant of horocompactification which takes "directions" into account. As an application, we construct a compactification of the Teichm\"uller spaces via the renormalized volume of quasi-Fuchsian manifolds. Although we…
The canonical formalism of the (2+2) formulation of general relativity of 4 spacetime dimensions is studied under no symmetry assumptions, where the spacetime is viewed as a local product of a 2 dimensional base manifold of Lorentzian…
J Hempel [Topology, 2001] showed that the set of distances of the Heegaard splittings (S,V, h^n(V)) is unbounded, as long as the stable and unstable laminations of h avoid the closure of V in PML(S). Here h is a pseudo-Anosov homeomorphism…
Let $$1 \to H \to G \to Q \to 1$$ be an exact sequence where $H= \pi_1(S)$ is the fundamental group of a closed surface $S$ of genus greater than one, $G$ is hyperbolic and $Q$ is finitely generated free. The aim of this paper is to provide…
We give a simple combinatorial criterion, in terms of an action on a hyperbolic simplicial complex, for a group to be hierarchically hyperbolic. We apply this to show that quotients of mapping class groups by large powers of Dehn twists are…
In the space $\mathcal{H}^2$ of hyperbolic surfaces decorated with a base unit vector, the topology induced by the Gromov-Hausdorff convergence coincides with the Chabauty topology on the space of discrete torsion-free subgroups of…
We establish the geometric ergodicity of the preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in [Beskos et al., Stochastic Process. Appl., 2011]. This algorithm can be…
We show that nontrivial solutions to higher and fractional order equations with certain nonlinearity are radially symmetric and nonincreasing on geodesic balls in the hyperbolic space $\mathbb{H}^n$ as well as on the entire space…
We provide a full classification of complete maximal $p$-dimensional spacelike submanifolds in the pseudo-hyperbolic space $\mathbf{H}^{p,q}$, and we study its applications to Teichm\"uller theory and to the theory of Anosov representations…
Hyperspaces form a powerful tool in some branches of mathematics: lots of fractal and other geometric objects can be viewed as fixed points of some functions in suitable hyperspaces - as well as interesting classes of formal languages in…
In this survey paper we give a proof of hyperbolicity of the complex of curves for a non-exceptional surface S of finite type combining ideas of Masur/Minsky and Bowditch. We also shortly discuss the relation between the geometry of the…
Let $\psi$ be a conformal map of the unit disk $\mathbb{D}$ onto an unbounded domain and, for $\alpha >0$, let ${F_\alpha }=\left\{ {z \in \mathbb{D}:\left| {\psi \left( z \right)} \right| = \alpha } \right\}$. If ${H^p}\left( \mathbb{D}…
The classical Reifenberg's theorem says that a set which is sufficiently well approximated by planes uniformly at all scales is a topological H\"older manifold. Remarkably, this generalizes to metric spaces, where the approximation by…