arXiv research

Embeds skein algebras into quantum tori using Dehn-Thurston coordinates.

The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.

Given a compact orientable surface of negative Euler characteristic, there exists a natural pairing between the Teichmueuller space of the surface and the set of homotopy classes of simple loops and arcs. The length pairing sends a hyperbolic metric and a homotopy class of a simple loop or arc to the length of geodesic…

Let $Σ$ be a compact, orientable surface of negative Euler characteristic, and let $h$ be a complete hyperbolic metric on $Σ$. A geodesic curve $γ$ in $Σ$ is filling, if it cuts the surface into topological disks and annuli. We propose an efficient algorithm for deciding whether a geodesic curve, represented as a word …

Let $Σ$ be a surface of negative Euler characteristic together with a pants decomposition $¶$. Kra's plumbing construction endows $Σ$ with a projective structure as follows. Replace each pair of pants by a triply punctured sphere and glue, or `plumb', adjacent pants by gluing punctured disk neighbourhoods of the punctu…

This research connects combinatorial Teichmüller space geometry to Weil-Petersson geometry.

Introduces LRY skein algebras generalizing Kauffman bracket and Roger-Yang skein algebras.

The Maskit embedding M of a surface Σis the space of geometrically finite groups on the boundary of quasifuchsian space for which the `top' end is homeomorphic to Σ, while the `bottom' end consists of two triply punctured spheres, the remains of Σwhen two fixed disjoint curves have been pinched. As such representations…

We provide analogues for non-orientable surfaces with or without boundary or punctures of several basic theorems in the setting of the Thurston theory of surfaces which were developed so far only in the case of orientable surfaces. Namely, we provide natural analogues for non-orientable surfaces of the Fenchel-Nielsen …

This paper is a sequel of arxiv:1709.09045 and deals with privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold equipped with a filtration by subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. In this paper, we single…

Coordinate descent methods usually minimize a cost function by updating a random decision variable (corresponding to one coordinate) at a time. Ideally, we would update the decision variable that yields the largest decrease in the cost function. However, finding this coordinate would require checking all of them, which…

New findings on Kähler manifolds restrict orthogonal coordinates existence.

Defines Fenchel-Nielsen coordinates for SL(3,C) representations.

Submanifolds of coordinate finite-type were introduced in HV1. A submanifold of a Euclidean space is called a coordinate finite-type submanifold if its coordinate functions are eigenfunctions of Δ. In the present study we consider coordinate finite-type surfaces in E^4. We give necessary and sufficient conditions for g…

We consider complex Fenchel-Nielsen coordinates on the quasi-Fuchsian space of punctured tori. These coordinates arise from a generalisation of Kra's plumbing construction and are related to earthquakes on Teichmueller space. They also allow us to interpolate between two coordinate systems on Teichmueller space, namely…

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.

Method constructs orthogonal curvilinear coordinates in constant curvature spaces.

In a previous paper, we parametrized boundary-unipotent representations of a 3-manifold group into SL(n,C) using Ptolemy coordinates, which were inspired by A-coordinates on higher Teichmüller space due to Fock and Goncharov. In this paper, we parametrize representations into PGL(n,C) using shape coordinates which are …

Study on conformal harmonic coordinates on manifolds, proving existence and properties.

Accelerates coordinate descent methods for machine learning problems.

We construct a tangent bundle exponential map and locally autoparallel coordinates for geometries based on a general connection on the tangent bundle of a manifold. As concrete application we use these new coordinates for Finslerian geometries and obtain Finslerian geodesic coordinates. They generalise normal coordinat…

Invariants of braids found using shear coordinates in hyperbolic geometry.

Extends Dynnikov coordinates to punctured torus.

We study the limiting case of the Krichever construction of orthogonal curvilinear coordinate systems when the spectral curve becomes singular. We show that the case when the curve is reducible and all its irreducible components are rational curves the construction procedure reduces to solving systems of linear equatio…

Normalizes pseudo-Einstein contact forms for easier analysis.

DP-SGD can update fewer coordinates while maintaining privacy.

Novel deep learning method predicts reaction coordinates and future MD trajectories.

Despite being studied for over a century, the use of quadrupoles have been limited to Cartesian coordinates in flat spacetime due to the incorrect transformation rules used to define them. Here the correct transformation rules are derived, which are particularly unusual as they involve second derivatives of the coordin…

The study examines the regularity of branched immersions using special coordinate systems.

This monograph presents a class of algorithms called coordinate descent algorithms for mathematicians, statisticians, and engineers outside the field of optimization. This particular class of algorithms has recently gained popularity due to their effectiveness in solving large-scale optimization problems in machine lea…

Flat coordinates found for algebraic Frobenius manifolds in low dimensions.

This paper constructs a family of coordinate systems about a point on a quaternionic contact manifold, called quaternionic contact pseudohermitian normal coordinates. Once defined, conformal variations of the quaternionic contact structure induce changes on the coordinates which are studied in an effort to simplify the…

Given a finite collection of $C^1$ vector fields on a $C^2$ manifold which span the tangent space at every point, we consider the question of when there is locally a coordinate system in which these vector fields are real analytic. We give necessary and sufficient, coordinate-free conditions for the existence of such a…

We find necessary and sufficient conditions under which the complex coordinates on a flag manifold of a classical group described in [2] are Bochner coordinates.

If one could assume that local coordinates in a Riemannian manifold were orthogonal, then local expressions for differential operators, and curvature computations, would be simplified. It is always possible on 2-manifolds, using geometric normal coordinates or isothermal coordinates. In 1984, Dennis DeTurck and Dean Ya…

Study uses Dynnikov coordinates to analyze actions of Dehn twists on a thrice-punctured disc.

Constructs coordinate systems from spectral curve sheaves.

Study curvature and torsion in Gaussian distribution's dual coordinate system.

Develops DP-SCD for stochastic coordinate descent, making it differentially private.

Carnot groups can be polarized if they have specific coordinate systems.

It is shown that, in four dimensions, it is possible to introduce coordinates so that an analytic metric locally takes block diagonal form. i.e. one can find coordinates such that $g_{αβ} = 0$ for $(α, β) \in S$ where $S = {(1, 3), (1, 4), (2, 3), (2, 4)}$. We call a coordinate system in which the metric takes this for…

Robustly computes intrinsic coordinates on point clouds using resampling and averaging.

Paper proposes a method to improve circular coordinate representation for detecting changes in high-dimensional datasets.

This paper focuses on coordinate update methods, which are useful for solving problems involving large or high-dimensional datasets. They decompose a problem into simple subproblems, where each updates one, or a small block of, variables while fixing others. These methods can deal with linear and nonlinear mappings, sm…

Enhanced Teichmüller space for surfaces with decorations and enhancements.

Generalizes Riemann's results on flat coordinates for non-symmetric bilinear forms.

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

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 …