arXiv research

Surveying methods to create spaces with non-trivial self covers.

The paper proves conditions for self-covering manifolds to be fiber bundles over a circle.

Closed self-covering manifolds with abelian fundamental groups fiber over tori.

Let $M$ be a closed manifold that admits a self-cover $p:M \to M$ of degree >1. We say p is strongly regular if all its iterates are regular covers. In this case, we establish an algebraic structure theorem for the fundamental group of $M$: We prove that $π_1(M)$ surjects onto a nontrivial free abelian group $A$, and t…

The paper studies knots in modular flows using self-covers.

3-manifolds are chiral if not finitely covered by sphere or product.

The paper characterizes isomorphic covers of surfaces and applies it to distinguish representations.

Bianchi proves Mumford conjecture using branched covers.

A closed 4-manifold (or, more generally, a finite $PD_4$-space) has a finitely dominated infinite regular covering space if and only if either its universal covering space is finitely dominated or it is finitely covered by the mapping torus of a self homotopy equivalence of a $PD_3$-complex.

Self-supervised learning improves few-shot classification and segmentation on point clouds.

For a Riemannian covering $p \colon M_{2} \to M_{1}$, we compare the spectrum of an essentially self-adjoint differential operator $D_{1}$ on a bundle $E_{1} \to M_{1}$ with the spectrum of its lift $D_{2}$ on $p^{*}E_{1} \to M_{2}$. We prove that if the covering is infinite sheeted and amenable, then the spectrum of $…

Introduces self-regularization for analyzing learning algorithms.

Study self-intersections of arcs on a pair of pants, proving natural number spectrum.

Study integral simplicial volume of cyclic covers of torus bundles.

The paper proves generalization bounds and stopping rules for self-selected data in reciprocal learning.

A new framework uses directed information to efficiently select context chunks.

By a construction of Berstein and Edmonds every proper branched cover f between manifolds is a factor of a branched covering orbit map from a locally connected and locally compact Hausdorff space called the monodromy space of f to the target manifold. For proper branched covers between 2-manifolds the monodromy space i…

Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar sets have touching structures the problem of Lipschitz equivalence becomes much …

We study contact manifolds that arise as cyclic branched covers of transverse knots in the standard contact 3-sphere. We discuss properties of these contact manifolds and describe them in terms of open books and contact surgeries. In many cases we show that such branched covers are contactomorphic for smoothly isotopic…

Uniform proof of $π_1$-injectivity for certain maps in low dimensions.

Study classifies twist knots with maximal self-linking number in S^3.

Counterexample disproves HK-conjecture for flat manifolds of dimension 9 and higher.

A new approach to continuous-time universal portfolios using pathwise Itô calculus.

New proof shows surfaces can have identical length spectra but not simple ones.

Proves Kähler-Einstein property for certain Einstein 4-manifolds.

We prove algebraic analogues of the facts that a curve on a surface with self-intersection number zero is homotopic to a cover of a simple curve, and that two simple curves on a surface with intersection number zero can be isotoped to be disjoint.

We give a description of asymptotic quadratic growth rates for geodesic segments on covers of Veech surfaces in terms of the modular fiber parameterizing coverings of a fixed Veech surface. To make the paper self contained we derive the necessary asymptotic formulas from the Gutkin-Judge formula. As an application of t…

We show that a self orbit equivalence of a transitive Anosov flow on a $3$-manifold which is homotopic to identity has to either preserve every orbit or the Anosov flow is $\mathbb{R}$-covered and the orbit equivalence has to be of a specific type. This result shows that one can remove a relatively unnatural assumption…

Let $γ$ be an essential closed curve with at most $k$ self-intersections on a surface $\mathcal{S}$ with negative Euler characteristic. In this paper, we construct a hyperbolic metric $ρ$ for which $γ$ has length at most $M \cdot \sqrt{k}$, where $M$ is a constant depending only on the topology of $\mathcal{S}$. Moreov…

We prove that any asymptotically locally Euclidean scalar-flat Kähler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a $k$-dimensional quaternionic vector space by a $(k-1)$-torus. In order to do so, we first prove that any compact anti…

We give examples of closed hyperbolic 3-manifolds with first Betti number 2 and 3 for which no sequence of finite abelian covering spaces increases the first Betti number. For 3-manifolds $M$ with first Betti number 2 we give a characterization in terms of some generalized self-linking numbers of $M$, for there to exis…

Models for self-equivalences and diffeomorphisms of manifolds.

Generalized Thurston's characterization for branched coverings of the 2-sphere.

We consider the relations between different measures of complexity for free homotopy classes of curves on a surface $Σ$, including the minimum number of self-intersections, the minimum length of the words representing them in a geometric presentation of $π_1(Σ)$, and the minimum degree of the coverings of $Σ$ to which …

Generalizes Anosov flows to partially hyperbolic diffeomorphisms.

Mathematicians embed a Klein's quartic cover in hyperbolic space.

Using the twistor correspondence, this article gives a one-to-one correspondence between germs of toric anti-self-dual conformal classes and certain holomorphic data determined by the induced action on twistor space. Recovering the metric from the holomorphic data leads to the classical problem of prescribing the Cech …

We propose a construction of Kähler and non-Kähler Calabi-Yau manifolds by branched double covers of twistor spaces. In this construction we use the twistor spaces of four-manifolds with self-dual conformal structures, with the examples of connected sum of $n$ $\mathbb{P}^{2}$s. We also construct $K3$-fibered Calabi-Ya…

A new method for portfolio allocation in continuous-time markets.

Study $L^2$ Hilbert complexes on complex manifolds.

Mantis improves time series classification using a transformer model trained on synthetic data.

A tutorial on non-asymptotic system identification methods.

The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centerline, whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions. The quasiperiodic curves are paramet…

We study the smooth structure of convex functions by generalizing a powerful concept so-called self-concordance introduced by Nesterov and Nemirovskii in the early 1990s to a broader class of convex functions, which we call generalized self-concordant functions. This notion allows us to develop a unified framework for …

ASTRA uses unlabeled data and weak rules to train deep models effectively.

Simplified proof for approximations of set systems.

We provide new branched covering representations for bounded and/or non-compact 4-manifolds, which extend the known ones for closed 4-manifolds. Assuming $M$ to be a connected oriented PL 4-manifold, our main results are the following: (1) if $M$ is compact with (possibly empty) boundary, there exists a simple branched…

Extends Thurston's combinatorial characterization to all branched coverings of the 2-sphere.