arXiv research

Example shows no global coordinates on 2-torus's cover.

Some results on existence of global Chebyshev coordinates on a Riemannian manifold or, more generally, on Aleksandrov surface are proved. For instance, if the positive and the negative parts of integral curvature of a Riemannian manifold M are less than 2πeach, then there exist global Chebyshev coordinates on M. These …

Global geometric expressions derived for manifold embeddings.

In this paper we combine our recent work on regular globally hyperbolic maximal anti-de Sitter structures with the classical theory of globally hyperbolic maximal Cauchy-compact anti-de Sitter manifolds in order to define an augmented moduli space. Moreover, we introduce a coordinate system in this space that resembles…

The Dynnikov coordinate system puts global coordinates on the boundary of Teichmüller space of an $n$--punctured disk. We survey the Dynnikov coordinate system, and investigate how we use this coordinate system to study pseudo--Anosov braids making use of results from Thurston's theory on surface homeomorphisms.

The study uses isothermic coordinates to analyze space-like surfaces with constant curvature.

We construct explicit global symplectic coordinates for the Calabi's inhomogeneous Kaehler-Einstein metric on tubular domains.

The abstract introduces golden Finsler structures and explores their local and global properties.

BCD algorithm finds global minima in neural networks.

The paper finds global Darboux coordinates for a new family of symplectic forms on the deformation space of $\mathbb{R}\mathbb{P}^2$-structures.

In this article we define new flows on the Hitchin components for PGL(V). Special examples of these flows are associated to simple closed curves on the surface and give generalized twist flows. Other examples, so called eruption flows, are associated to pair of pants in S and capture new phenomena which are not present…

New action-angle coordinates found for singular symplectic manifolds.

Stability proved for open Milne spacetime, showing gravity's long-term behavior.

We develop and analyze efficient "coordinate-wise" methods for finding the leading eigenvector, where each step involves only a vector-vector product. We establish global convergence with overall runtime guarantees that are at least as good as Lanczos's method and dominate it for slowly decaying spectrum. Our methods a…

Proposes local coordinate frames for improving model performance in complex dynamical systems.

Study counts orbits of mapping class group in shearing coordinates.

While convergence of the Alternating Direction Method of Multipliers (ADMM) on convex problems is well studied, convergence on nonconvex problems is only partially understood. In this paper, we consider the Gaussian phase retrieval problem, formulated as a linear constrained optimization problem with a biconvex objecti…

Let P(S) be the space of convex projective structures on a surface S with negative Euler characteristic. Goldman and Bonahon-Dreyer constructed two different sets of global coordinates for P(S), both associated to a pair of pants decomposition of the surface S. The article explicitly describes the coordinate change bet…

Goldman parametrizes the $\mathrm{PSL}_3(\mathbb{R})$-Hitchin component of a closed oriented hyperbolic surface of genus $g$ by $16g-16$ parameters. Among them, $10g-10$ coordinates are canonical. We prove that the $\mathrm{PSL}_3(\mathbb{R})$-Hitchin component equipped with the Atiyah-Bott-Goldman symplectic form admi…

A classical result of D. McDuff asserts that a simply-connected complete Kaehler manifold $(M,g,ω)$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $Ψ: M\rightarrow R^{2n}$ (where $n$ is the complex dimension of $M$), satisfying the following property (proved by E.…

We show global existence theorems for Gowdy symmetric spacetimes with type IIB stringy matter. The areal and constant mean curvature time coordinates are used. Before coming to that, it is shown that a wave map describes the evolution of this system.

Recently several methods were proposed for sparse optimization which make careful use of second-order information [10, 28, 16, 3] to improve local convergence rates. These methods construct a composite quadratic approximation using Hessian information, optimize this approximation using a first-order method, such as coo…

Deroin and Tholozan's representations are mapped to complex projective space via action-angle coordinates.

We describe a minimal global coordinate system of order 30 on the SL(4,C)-character variety of a rank 2 free group. Using symmetry within this system, we obtain a smaller collection of 22 coordinates subject to 5 further real relations that determine conjugation classes of generic pairs of matrices in SU(3,1).

Shared workspace improves neural module coordination in deep learning.

Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge fields needed for the con…

This article is the second of a pair of articles about the Goldman symplectic form on the PGL(V)-Hitchin component of a closed, connected, oriented, hyperbolic surface S. We show that any ideal triangulation on S and any compatible bridge system determine a symplectic trivialization of the tangent bundle to the PGL(V)-…

Global calculus for manifolds with boundary, solving evolution problems.

For every abelian compact Lie group A, we prove that the homotopical A-equivariant complex bordism ring, introduced by tom Dieck (1970), is isomorphic to the A-equivariant Lazard ring, introduced by Cole-Greenlees-Kriz (2000). This settles a conjecture of Greenlees. We also show an analog for homotopical real bordism r…

Let $M\subset{\complex}^n$ be a complex domain of ${\complex}^n$ endowed with a rotation invariant \K form $ω_Φ= \frac{i}{2} \partial\bar\partialΦ$. In this paper we describe sufficient conditions on the \K potential $Φ$ for $(M, ω_Φ)$ to admit a symplectic embedding (explicitely described in terms of $Φ$) into a compl…

Explicit computation of symplectic form for $\mathrm{PGL}_n(\mathbb{R})$-Hitchin component.

There are two important statements regarding the Trautman-Bondi mass at null infinity: one is the positivity, and the other is the Bondi mass loss formula, which are both global in nature. In this note, we compute the limit of the Wang-Yau quasi-local mass on unit spheres at null infinity of an asymptotically flat spac…

We underline some differences between the geometric aspect of Berezin's approach to quantization on homogeneous Kähler manifolds and Bergman's construction for bounded domains in $\mathbb{C}^n$. We construct explicitly the Bergman representative coordinates for the Siegel-Jacobi disk $\mathcal{D}^J_1$, which is a parti…

Deep learning has aroused extensive attention due to its great empirical success. The efficiency of the block coordinate descent (BCD) methods has been recently demonstrated in deep neural network (DNN) training. However, theoretical studies on their convergence properties are limited due to the highly nonconvex nature…

Proves harmonic coordinates for weak immersions in even dimensions.

We investigate the existence of coordinate transformations which bring a given vector field on a manifold equipped with an involutive distribution into the form of a second-order differential equation field with parameters. We define associated connections and we give a coordinate-independent criterion for determining …

We develop a deep autoencoder architecture that can be used to find a coordinate transformation which turns a nonlinear PDE into a linear PDE. Our architecture is motivated by the linearizing transformations provided by the Cole-Hopf transform for Burgers equation and the inverse scattering transform for completely int…

A multi-neck spacetime wormhole is constructed with a simple metric tensor.

An analytic extension of the Reissner-Nordstrom solution at and beyond the singularity is presented. The extension is obtained by using new coordinates in which the metric becomes degenerate at $r=0$. The metric is still singular in the new coordinates, but its components become finite and smooth. Using this extension …

The paper explores the geometric structure of cost functions in multiple dimensions.

Symplectic coordinates found on projective structures on orbifolds.

New method deflates manifolds to visualize high-dimensional data.

We construct a C-space associated with every closed 3-form on a spacetime $M$ and show that it depends on the class of the form in $H^3(M, Z)$. We also demonstrate that C-spaces have a relation to generalized geometry and to gerbes. C-spaces are constructed after introducing additional coordinates at the open sets and …

This paper introduces a new method for semi-supervised learning on high dimensional nonlinear manifolds, which includes a phase of unsupervised basis learning and a phase of supervised function learning. The learned bases provide a set of anchor points to form a local coordinate system, such that each data point $x$ on…

We develop randomized (block) coordinate descent (CD) methods for linearly constrained convex optimization. Unlike most CD methods, we do not assume the constraints to be separable, but let them be coupled linearly. To our knowledge, ours is the first CD method that allows linear coupling constraints, without making th…

Timelike Thomsen surfaces are timelike minimal surfaces that are also affine minimal. In this paper, we make use of both the Lorentz conformal coordinates and the null coordinates, and their respective representation theorems of timelike minimal surfaces, to obtain a complete global classification of these surfaces and…

This paper develops a theory of graded manifolds in differential geometry.

The paper extends Newlander-Nirenberg theorem to domains with $C^2$ boundary.