arXiv research

Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.

Generic Hitchin representations generate dense subgroups.

Neural networks are dense among Lipschitz functions with fixed Lipschitz constant.

In each Menger manifold $M$ we construct: (i) a closed nowhere dense subset $M_0$ which is homeomorphic to $M$ and is universal nowhere dense in the sense that for each nowhere dense set $A\subset M$ there is a homeomorphism $h$ of $M$ such that $h(A)\subset M_0$; (ii) a meager $F_σ$-set $Σ_0\subset M$ which is univers…

Random 3-manifolds have no totally geodesic submanifolds.

Non-compact manifolds prevent $C^0$-stable mappings from being dense.

We prove that if an analytic subset $A$ of a linear metric space $X$ is not contained in a $σZ_ω$-subset of $X$ then for every Polish convex set $K$ with dense affine hull in $X$ the sum $A+K$ is non-meager in $X$ and the sets $A+A+K$ and $A-A+K$ have non-empty interior in the completion $\bar X$ of $X$. This implies t…

Study generalizes Lebesgue curves to new space-filling and fractal sets.

In each manifold $M$ modeled on a finite or infinite dimensional cube $[0,1]^n$ we construct a closed nowhere dense subset $S\subset M$ (called a spongy set) which is a universal nowhere dense set in $M$ in the sense that for each nowhere dense subset $A\subset M$ there is a homeomorphism $h:M\to M$ such that $h(A)\sub…

We show that any grafting ray in Teichmüller space determined by an arational lamination or a multi-curve is (strongly) asymptotic to a Teichmüller geodesic ray. As a consequence the projection of a generic grafting ray to moduli space is dense. We also show that the set of points in Teichmüller space obtained by integ…

In high dimensions, find paths connecting points with intermediate steps in a dense set.

Dense neural networks can't approximate all functions.

New framework for dense weighted networks with community-specific patterns.

In an earlier paper of the authors it was shown that the sheaf theoretically based recently developed abstract differential geometry of the first author can in an easy and natural manner incorporate singularities on arbitrary closed nowhere dense sets in Euclidean spaces, singularities which therefore can have arbitrar…

Currents on cusped hyperbolic surfaces have a denseness property similar to compact surfaces.

Busemann points are sparse in Teichmüller spaces.

We show that for every sequence $(n_i)$, where each $n_i$ is either an integer greater than 1 or is $\infty$, there exists a simply connected open 3-manifold $M$ with a countable dense set of ends $\{e_i\}$ so that, for every $i$, the genus of end $e_i$ is equal to $n_i$. In addition, the genus of the ends not in the d…

The paper finds bounds on shortest dense curves on surfaces.

In this paper, we study dense subsets of boundaries of CAT(0) groups. Suppose that a group $G$ acts geometrically on a CAT(0) space $X$ and suppose that there exists an element $g_0\in G$ such that (1) $Z_{g_0}$ is finite, (2) $X\setminus F_{g_0}$ is not connnected, and (3) each component of $X\setminus F_{g_0}$ is con…

Generic metrics make geodesic nets dense.

We show that when $r \geq 5$ is prime, the SO(3) Witten-Reshetikhin-Turaev quantum invariants for three-manifolds at the level $r$ form a dense set in the complex plane. This confirms a conjecture of Larsen and Wang.

Method generates dense fields from sparse measurements without needing spatial statistics or examples.

A dense amalgam connects boundaries of groups split by finite subgroups.

In this paper we show that certain generalizations of the $C^r$-Whitney topology, which include the Hölder-Whitney and Sobolev-Whitney topologies on smooth manifolds, satisfy the Baire property, to wit, the countable intersection of open and dense sets is dense.

In this article we verify an orbifold version of a conjecture of Nimershiem from 1998. Namely, for every flat $n$-manifold $M$, we show that the set of similarity classes of flat metrics on $M$ which occur as a cusp cross-section of a hyperbolic $(n+1)$-orbifold is dense in the space of similarity classes of flat metri…

We consider dense 2-generator multiplicative subgroups in $\mathbb C$ and show that for each point $z\in \mathbb C$ the set of limit values for the arguments of the powers of each generator at the point $z$ is either finite or is $[-π,π]$

We show that the topological groups $Diff_{+}^{1}(I)$ and $Diff_{+}^{1}(\mathbb{S}^1)$ of orientation-preserving $C^1$-diffeomorphisms of the interval and the circle, respectively, admit finitely generated dense subgroups. We also investigate the question of genericity (in the sense of Baire category) of such finite to…

We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dens…

Mendes Lopes and Pardini showed that minimal general type surfaces of Albanese dimension one have slopes $K^2/χ$ dense in the interval $[2,8]$. This result was completed to cover the admissible interval $[2,9]$ by Roulleau and Urzua, who proved that surfaces with fundamental group equal to that of any curve of genus $g…

Fully augmented links have dense volume densities but discrete in certain ranges.

The paper shows dense and residual sets of continuous maps with positive metric mean dimension.

We give examples of rank one compact surfaces on which there exist recurrent geodesics that cannot be shadowed by periodic geodesics. We build rank one compact surfaces such that ergodic measures on the unit tangent bundle of the surface are not dense in the set of probability measures invariant by the geodesic flow. F…

We study the regularity properties of the value function associated with an affine optimal control problem with quadratic cost plus a potential, for a fixed final time and initial point. Without assuming any condition on singular minimizers, we prove that the value function is continuous on an open and dense subset of …

We prove sharp criteria on the behavior of radial curvature for the existence of asymptotically flat or hyperbolic Riemannian manifolds with prescribed sets of eigenvalues embedded in the spectrum of the Laplacian. In particular, we construct such manifolds with dense embedded point spectrum and sharp curvature bounds.

Complex projective manifolds without rational curves are quotients of Abelian varieties.

We introduce and study the operation, called dense amalgam, which to any tuple X_1,...,X_k of non-empty compact metric spaces associates some disconnected perfect compact metric space, denoted $\widetilde\sqcup(X_1,...,X_k)$, in which there are many appropriately distributed copies of the spaces X_1,...,X_k. We then sh…

We prove that the set of closed finite gap curves in hyperbolic 3-space $\mathbb{H}^3$ is $W^{2,2}$-dense in the Sobolev space of all closed $W^{2,2}$-curves in $\mathbb{H}^3$. We also show that the set of closed finite gap curves in any 2-dimensional space form $\mathbb{E}^2$ is $W^{2,2}$-dense in the Sobolev space of…

The purpose of this paper is to present, for all $n\ge 3$, very simple examples of continuous maps $f:M^{n-1} \to M^{n}$ from closed $(n-1)$-manifolds $M^{n-1}$ into closed $n$-manifold $M^n$ such that even though the singular set $S(f)$ of $f$ is countable and dense, the map $f$ can nevertheless be approximated by an …

ViCE uses superpixels to enhance self-supervised learning for better dense visual embeddings.

In this paper, we give a new generalization error bound of Multiple Kernel Learning (MKL) for a general class of regularizations, and discuss what kind of regularization gives a favorable predictive accuracy. Our main target in this paper is dense type regularizations including \ellp-MKL. According to the recent numeri…

This note introduces and studies an open set of PSL(2,C) characters of a nonabelian free group, on which the action of the outer automorphism group is properly discontinuous, and which is strictly larger than the set of discrete, faithful convex-cocompact (i.e. Schottky) characters. This implies, in particular, that th…

Cataclysm deformations study Anosov representations, leading to new formulas and non-open sets.

Estimates for Schrödinger operators on manifolds with bounded Ricci curvature.

A pair of points (x,y) in a Riemannian manifold (M,g) is said to have the finite blocking property if there is a finite set P contained in M\{x,y} such that every geodesic segment from x to y passes through a point of P. We show that for every closed C-infinity manifold M of dimension at least two and every pair (x,y) …

New lattices in higher dimensions have dense surface subgroups.

We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dens…

Cataclysm deformations study Anosov representations and their convergence.

Generic geodesic nets are dense in high-dimensional manifolds.