arXiv research

In this paper we deduce a local deformation lemma for uniform embeddings in a metric covering space over a compact manifold from the deformation lemma for embeddings of a compact subspace in a manifold. This implies the local contractibility of the group of uniform homeomorphisms of such a metric covering space under t…

Whenever a finitely generated group $G$ acts properly discontinuously by isometries on a metric space $X$, there is an induced uniform embedding (a Lipschitz and uniformly proper map) $ρ: G \rightarrow X$ given by mapping $G$ to an orbit. We study when there is a difference between a finitely generated group $G$ acting…

We prove a uniform extension result for contracting maps defined on subsets of Hadamard manifolds subject to curvature bounds.

The paper proves a conjecture about manifold limits and characterizes their structure.

This paper examines limits of Riemannian 2-manifolds with bounded curvature.

We will show that for a polynomially contractible manifold of bounded geometry and of polynomial volume growth every coarse and rough cohomology class pairs continuously with the K-theory of the uniform Roe algebra. As an application we will discuss non-vanishing of rough index classes of Dirac operators over such mani…

Let H denote the standard one-point completion of a real Hilbert space. Given any non-trivial proper sub-set U of H one may define the so-called `Apollonian' metric d_U on U. When U \subset V \subset H are nested proper subsets we show that their associated Apollonian metrics satisfy the following uniform contraction p…

In the present paper, we consider the family of all compact Alexandrov spaces with curvature bound below having a definite upper diameter bound of a fixed dimension. We introduce the notion of essential coverings by contractible metric balls, and provide a uniform bound on the numbers of contractible metric balls formi…

Let $X$ be a proper CAT(0) space and let $G$ be a cocompact group of isometries of $X$ which acts properly discontinuously. Charney and Sultan constructed a quasi-isometry invariant boundary for proper CAT(0) spaces which they called the contracting boundary. The contracting boundary imitates the Gromov boundary for $δ…

New bounds for SGLD show error decreases with more data.

Uniform TD(0) bound derived for function approximation with Markov noise.

A local deformation property for uniform embeddings in metric manifolds (LD) is formulated and its behaviour is studied in a formal view point. It is shown that any metric manifold with a geometric group action, typical metric spaces (Euclidean space, hyperbolic space and cylinders) and for κ\leq 0 the κ-cone ends over…

Study uses machine learning to estimate effective policies in settings with hidden individual actions.

Study proves rigidity of marked length spectra in contracting group actions.

Uniform systole bounds for arithmetic orbifolds and number fields.

Bayesian framework for sphere regression using Gaussian fields.

We address the problem of estimating the mixing time of a Markov chain from a single trajectory of observations. Unlike most previous works which employed Hilbert space methods to estimate spectral gaps, we opt for an approach based on contraction with respect to total variation. Specifically, we estimate the contracti…

Three themes of general topology: quotient spaces; absolute retracts; and inverse limits - are reapproached here in the setting of metrizable uniform spaces, with an eye to applications in geometric and algebraic topology. The results include: 1) If f: A -> Y is a uniformly continuous map, where X and Y are metric spac…

Develops a new framework for temporal anchoring in deep embedding spaces.

Let $X$ be a proper geodesic metric space and let $G$ be a group of isometries of $X$ which acts geometrically. Cordes constructed the Morse boundary of $X$ which generalizes the contracting boundary for CAT(0) spaces and the visual boundary for hyperbolic spaces. We characterize Morse elements in $G$ by their fixed po…

No 5D aspherical manifolds can have uniformly positive scalar curvature.

This paper proves properties of uniformly hyperbolic sets and constructs Markov partitions.

Let $N$ be a complete manifold with bounded geometry, such that $\sec_N\le -σ< 0$ for some positive constant $σ$. We investigate the mean curvature flow of the graphs of smooth length-decreasing maps $f:\mathbb{R}^m\to N$. In this case, the solution exists for all times and the evolving submanifold stays the graph of a…

We prove the existence of positive lower bounds on the Cheeger constants of manifolds of the form $X/Γ$ where $X$ is a contractible Riemannian manifold and $Γ<\Isom(X)$ is a discrete subgroup, typically with infinite co-volume. The existence depends on the $L^2$-Betti numbers of $Γ$, its subgroups and of a uniform latt…

Tensor CANDECOMP/PARAFAC (CP) decomposition has wide applications in statistical learning of latent variable models and in data mining. In this paper, we propose fast and randomized tensor CP decomposition algorithms based on sketching. We build on the idea of count sketches, but introduce many novel ideas which are un…

New methods for estimating and inferring nonparametric structural functions and elasticities.

New bound relaxes uniform gradient norm assumptions for PAC-Bayesian bounds.

Proves transitivity of real Anosov diffeomorphisms with specific properties.

New algorithm uniformly samples high-dimensional convex bodies efficiently.

We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, non-elementary group $G$ acting on a $G$--space $\mathcal{X}$, we prove that if $G$ contains a strongly contracting eleme…

Geodesics found in spacetime satisfy curvature conditions.

A triangulation of a $3$-manifold can be shown to be homeomorphic to the $3$-sphere by describing a discrete Morse function on it with only two critical faces, that is, a sequence of elementary collapses from the triangulation with one tetrahedron removed down to a single vertex. Unfortunately, deciding whether such a …

Let $M^n$ be a closed immersed hypersurface lying in a contractible ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reilly's inequality via sectional curvature upper bound of $B(p,R)$, 1st eigenvalue and mean curvature of $M$, not only $M$ is Hausdorff close to a geodesic sph…

In this paper, we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold $(X,ξ)$. By using such a chart, we show that every holomorphic Legendrian immersion $R\to X$ from an open Riemann surface can be approximated on relatively compact subsets by holo…

The paper explores coalescent contractions in contractible spaces, providing criteria and examples.

This is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in a metric space $(X,\mathsf{d})$ that can be characterized by Evolution Variational Inequalities. We present new results concerning the structural properties of solutions to the $\mathrm{EVI}$ formulation, such as co…

Holomorphic automorphisms on hyperkähler manifolds with high entropy are Kummer examples.

Randomly initialized transformers show extreme token preferences.

In the first part Busemann concavity as non-negative curvature is introduced and a bi-Lipschitz splitting theorem is shown. Furthermore, if the Hausdorff measure of a Busemann concave space is non-trivial then the space is doubling and satisfies a Poincaré condition and the measure contraction property. Using a compari…

Computable contracts simplify financial transactions and reduce legal costs.

In an online contract selection problem there is a seller which offers a set of contracts to sequentially arriving buyers whose types are drawn from an unknown distribution. If there exists a profitable contract for the buyer in the offered set, i.e., a contract with payoff higher than the payoff of not accepting any c…

We study proper, isometric actions of nonsolvable discrete groups Gamma on the 3-dimensional Minkowski space R^{2,1} as limits of actions on the 3-dimensional anti-de Sitter space AdS^3. To each such action is associated a deformation of a hyperbolic surface group Gamma_0 inside O(2,1). When Gamma_0 is convex cocompact…

Optimal execution strategy for merger & acquisition contracts with price impact.

The paper proves bounds on curvature and injectivity radius for convex sums of Riemannian metrics.

This paper develops a method to select a reference contract for multi-contract quoting to minimize execution risk.

We prove that if a family of metrics, $g_i$, on a compact Riemannian manifold, $M^n$, have a uniform lower Ricci curvature bound and converge to $g_\infty$ smoothly away from a singular set, $S$, with Hausdorff measure, $H^{n-1}(S) = 0$, and if there exists connected precompact exhaustion, $W_j$, of $M^n \setminus S$ s…

Proposes a probabilistic framework for smart contract risk quantification.

We consider a general framework of optimal mechanism design under adverse selection and ambiguity about the type distribution of agents. We prove the existence of optimal mechanisms under minimal assumptions on the contract space and prove that centralized contracting implemented via mechanisms is equivalent to delegat…