arXiv research

Uniform bounds on ends for non-branching CD spaces with nonnegative curvature outside a compact set.

In "Rips complexes and covers in the uniform category" \cite{Rips} the authors define, following James \cite{J}, covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given…

We prove that if $Y$ is the Gromov-Hausdorff limit of a sequence of compact manifolds, $M^n_i$, with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, then $Y$ has a universal cover. We then show that, for $i$ sufficiently large, the fundamental group of $M_i$ has a surjective homeomorphis…

Berestovskii and Plaut introduced the concept of a coverable uniform space when developing their theory of generalized universal covering maps for uniform spaces. Brodskiy, Dydak, LaBuz, and Mitra introduced the concept of a locally uniformly joinable uniform space when developing their theory of generalized uniform co…

Adding noise controls capacity of function compositions.

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…

Uniform waist inequalities proven for manifolds with Kazhdan groups in codimension two.

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…

The paper studies random covers of torus knot complements and their statistical properties.

The study explores maps of 2- and 3-uniform tilings on the torus.

We give conceptual proofs of some well known results concerning compact non-positively curved locally symmetric spaces. We discuss vanishing and non-vanishing of Pontrjagin numbers and Euler characteristics for these locally symmetric spaces. We also establish vanishing results for Stiefel-Whitney numbers of (finite co…

The paper proves rigidity results for Einstein manifolds with specific geometric constraints.

With the help of hyper-ideal circle pattern theory, we have developed a discrete version of the classical uniformization theorems for surfaces represented as finite branched covers over the Riemann sphere as well as compact polyhedral surfaces with non-positive curvature. We show that in the case of such surfaces discr…

Harmonic maps from surfaces to CAT(k) spheres are branched coverings.

In this paper, we study extremal subsets in Alexandrov spaces with dimension $n$, curvature $\geκ$, and diameter $\le D$. We show that the following three quantities are uniformly bounded above in terms of $n$, $κ$, and $D$: (1) the number of extremal subsets in an Alexandrov space; (2) the Betti numbers of an extremal…

New SGD covering technique yields dimension-independent generalization bounds.

We study Betti numbers of sequences of Riemannian manifolds which Benjamini-Schramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, under a negative Ricci curvature assumption and no assumption on sign of the sectional curvature…

Let $Γ$ be a Zariski dense convex cocompact subgroup contained in an arithmetic lattice of $\operatorname{SO}(n, 1)^{\circ}$. We prove uniform exponential mixing of the geodesic flow for congruence covers of the hyperbolic manifold $Γ\backslash \mathbb{H}^n$ avoiding finitely many prime ideals. This extends the work of…

Graph manifolds are manifolds that decompose along tori into pieces with a tame $S^1$-structure. In this paper, we prove that the simplicial volume of graph manifolds (which is known to be zero) can be approximated by integral simplicial volumes of their finite coverings. This gives a uniform proof of the vanishing of …

Minimal action on universal circle for foliations on 3-manifolds.

Uniform foliations with Reeb components on 3-manifolds.

We prove that if $Y$ is the Gromov-Hausdorff limit of a sequence of complete manifolds, $M^n_i$, with a uniform lower bound on Ricci curvature then $Y$ has a universal cover.

We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows' techniques, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed …

Uniform spectral gap for convex cocompact hyperbolic surfaces and expanders.

We show relationships between uniform K-stability and plt blowups of log Fano pairs. We see that it is enough to evaluate certain invariants defined by volume functions for all plt blowups in order to test uniform K-stability of log Fano pairs. We also discuss the uniform K-stability of two log Fano pairs under crepant…

Uniform scaling limits in AdamW-trained transformers converge to ODEs.

Equivalence proven between divisorial stability and quotient log divisorial stability.

It is well known that the collection of uniformizations of a closed Riemann surface $S$ is partially ordered; the lowest ones are the Schottky unformizations, that is, tuples $(Ω,Γ,P:Ω\to S)$, where $Γ$ is a Schottky group with region of discontinuity $Ω$ and $P:Ω\to S$ is a regular holomorphic cover map with $Γ$ as it…

Uniform comparison of hyperbolic ball volumes on universal cover.

New method calculates asymptotic expectation of fixed points in covering spaces.

We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, $n$-dimensional Riemannian manifold with pinched negative sectional curvature follows …

Uniform covers with a finite-dimensional nerve are rare (i.e., do not form a cofinal family) in many separable metric spaces of interest. To get hold on uniform homotopy properties of these spaces, a reasonably behaved notion of an infinite-dimensional metric polyhedron is needed; a specific list of desired properties …

Study uniform convergence of random walk Laplacians to diffusion Laplacian on smooth manifolds.

We announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the Lück Approximation Theorem \cite{luck}, which is much stronger than the linear upper bounds on Betti numbers given by Gr…

New property ensures neural networks generalize well with limited data.

This paper improves bounds on DNN generalization to adversarial examples.

We establish conditions for a continuous map of nonzero degree between a smooth closed manifold and a negatively curved manifold of dimension greater than four to be homotopic to a smooth cover, and in particular a diffeomorphism when the degree is one. The conditions hold when the volumes or entropy-volumes of the two…

Equality in Miyaoka-Yau inequality implies uniformization of Klt pairs.

We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems. A basic idea is to adapt the notion o…

New framework improves restless bandit policies for large numbers of arms.

We produce examples of taut foliations of hyperbolic 3-manifolds which are R-covered but not uniform --- ie the leaf space of the universal cover is R, but pairs of leaves are not contained in bounded neighborhoods of each other. This answers in the negative a conjecture of Thurston `Three-manifolds, foliations and cir…

Study ergodic properties of geodesic flows on specific manifolds without conjugate points.

New algorithm learns multiclass concepts with finite Littlestone dimension.

In this article we obtain a simple topological and dynamical systems condition which is necessary and sufficient for an arbitrary pseudo-Anosov flow in a closed, hyperbolic three manifold to be quasigeodesic. Quasigeodesic means that orbits are efficient in measuring length up to a bounded multiplicative distortion whe…

Uniform Poincaré inequalities established for various metric spaces.

Associated to each material body $\mathcal{B}$ there exists a groupoid $Ω\left( \mathcal{B} \right)$ consisting of all the material isomorphisms connecting the points of $\mathcal{B}$. The uniformity character of $\mathcal{B}$ is reflected in the properties of $Ω\left( \mathcal{B} \right)$: $\mathcal{B}$ is uniform if,…

Researchers prove a spectral gap for Hecke covers of Schottky surfaces.

We give uniform, explicit, and simple face-pairing descriptions of all the branched cyclic covers of the 3-sphere, branched over two-bridge knots. Our method is to use the bi-twisted face-pairing constructions of Cannon, Floyd, and Parry; these examples show that the bi-twist construction is often efficient and natural…