arXiv research

We produce a sequence of finite dimensional representations of the fundamental group $π_1(S)$ of a closed surface where all simple closed curves act with finite order, but where each non--simple closed curve eventually acts with infinite order. As a consequence, we obtain a representation theoretic algorithm which deci…

A new probabilistic polygonal curve representation using Gaussian Mixture Models.

We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.

In this paper, new representations of a Bertrand curve pair in three dimensional Lie groups with bi-invariant metric are given. Besides, the spherical indicatrices of a Bertrand curve pair are obtain and the relations between the spherical indicatrices and new representations of Bertrand curve pair are shown.

A relation between the Goldstein-Petrich hierarchy for plane curves and the Toda lattice hierarchy is investigated. A representation formula for plane curves is given in terms of a special class of $τ$-functions of the Toda lattice hierarchy. A representation formula for discretized plane curves is also discussed.

The study examines dynamics on SU(2)-representation varieties for surfaces and non-orientable surfaces.

New parametrizations for minimal timelike surfaces discovered.

In this work, we studied the properties of the spherical indicatrices of a Bertrand curve and its mate curve and presented some characteristic properties in the cases that Bertrand curve and its mate curve are slant helices, spherical indicatrices are slant helices and we also researched that whether the spherical indi…

In this paper, we study the problem of finding a hypersurface family from a given spatial geodesic curve in R4. We obtain the parametric representation for a hypersurface family whose members have the same curve as a given geodesic curve. Using the Frenet frame of the given geodesic curve, we present the hypersurface a…

Paper connects surfaces in 4D and 3D spacetime.

In this paper, we consider the intensity surface of a 2D image, we study the evolution of the symmetry sets (and medial axes) of 1-parameter families of iso-intensity curves. This extends the investigation done on 1-parameter families of smooth plane curves (Bruce and Giblin, Giblin and Kimia, etc.) to the general case…

Neural networks are capable of learning rich, nonlinear feature representations shown to be beneficial in many predictive tasks. In this work, we use such models to explore different geographical feature representations in the context of predicting colorectal cancer survival curves for patients in the state of Iowa, sp…

Abstract: Characterizes frontals and wavefronts with formulas.

The Frey--Mazur conjecture states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p\geq 17$. We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multip…

In 1990, Hitchin's proved a component of the space of representations of a surface group in SL(n,R) is homeomorphic to a ball. For n=2,3 this component has been identified with the holonomies of geometric structures (hyperbolic for n=2, or real projective for n=3). In the preprint "Anosov flows, Surface groups and Curv…

Proposes FunNoL for better curve classification and reconstruction in multivariate functional data.

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

Accelerates pulsar light curve inference with learned representations and optimization.

Essential curves in certain graph manifolds imply non-linear groups.

In this paper, we study the monodromy of the Hitchin fibration for rank 2 vector bundles over hyperelliptic curves. We reduce the problem to studying a surface braid group generalization of the classical Burau representation, and give a combinatorial method for computing this representation.

New proofs given for space curves with totally positive torsion.

Inspired by the paper on quantum knots and knot mosaics [23] and grid diagrams (or arc presentations), used extensively in the computations of Heegaard-Floer knot homology [2,3,7,24], we construct the more concise representation of knot mosaics and grid diagrams via mirror-curves. Tame knot theory is equivalent to knot…

In this work, we studied the properties of the spherical indicatrices of involute curve of a space curve and presented some characteristic properties in the cases that involute curve and evolute curve are slant helices and helices, spherical indicatrices are slant helices and helices and we introduced new representatio…

Develops correlation number for specific potentials and Hitchin representations.

Given a pair of planar curves, one can define its generalized area distance, a concept that generalizes the area distance of a single curve. In this paper, we show that the generalized area distance of a pair of planar curves is an improper indefinite affine spheres with singularities, and, reciprocally, every indefini…

We prove that for each sufficiently complicated orientable surface $S$, there exists an infinite image linear representation $ρ$ of $π_1(S)$ such that if $γ\inπ_1(S)$ is freely homotopic to a simple closed curve on $S$, then $ρ(γ)$ has finite order. Furthermore, we prove that given a sufficiently complicated orientable…

We consider the Alexander polynomial of a plane algebraic curve twisted by a linear representation. We show that it divides the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is given by the determinant of its Blanchfield intersection form. Specializing in the…

Introduces Nakajima bundles on algebraic curves, generalizing quiver representations and bundles.

The paper connects hyperbolic spinors to non-null framed curves in Minkowski 3-space.

Extends curve functions to geodesic currents with a simple criterion.

In this paper, we focus on some characterizations for curves in the Galilean and Pseudo-Galilean space.

Extends particle classification to curved space-times using groupoids.

Holomorphic curves found in compact quotients of SL(2,C).

We write the Dirac equation in curved 4-dimensional Lorentzian spacetime using concepts from the analysis of partial differential equations as opposed to geometric concepts.

Transformer model removes noise from light curves efficiently.

We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t=-1 and also the fundament…

We construct discrete and faithful representations into the isometry group of a hyperbolic space of the fundamental groups of acute negatively curved even-sided polygons of finite groups.

We present a novel, log-radius profile representation for convex curves and define a new operation for combining the shape features of curves. Unlike the standard, angle profile-based methods, this operation accurately combines the shape features in a visually intuitive manner. This method have implications in shape an…

We investigate minimal surfaces passing a given curve in $R^{3}$. Using the Frenet frame of a given curve and isothermal parameter, we derive the necessary and sufficient condition for minimal surface. Also we derive the parametric representation of two minimal surface families passing a circle and a helix as examples.

Study shows infinite kernels in topological monodromy for curve families.

Minimal energy local systems on curves are compact components of character varieties.

The reduced Burau representation $V_n$ of the braid group $B_n$ is obtained from the action of $B_n$ on the homology of an infinite cyclic cover of the disc with $n$ punctures. The group homology $H_*(B_n;V_n)$ of braid groups with coefficients in the complexified reduced Burau representation is calculated. Our topolog…

A focal representation of a generic regular curve γ in E^{m+1} consists of the centers of the osculating hyperplanes. A k-slant helix γ in E^{m+1} is a (generic) regular curve whose unit normal vector V_{k} makes a constant angle with a fixed direction U in E^{m+1}. In the present paper we proved that if γ is a k-slant…

New constraints found for algebro-geometric subgroups of mapping class groups.

Approximate 3D elastic curves with exact constraints

The Burau representation is a natural action of the braid group B_n on the free Z[t,t^{-1}]-module of rank n-1. It is a longstanding open problem to determine for which values of n this representation is faithful. It is known to be faithful for n=3. Moody has shown that it is not faithful for n>8 and Long and Paton imp…

We develop a general framework for the description of instabilities on soap films using the Björling representation of minimal surfaces. The construction is naturally geometric and the instability has the interpretation as being specified by its amplitude and transverse gradient along any curve lying in the minimal sur…

In this paper we give a criterion for pairs of isometries of a nonpositively curved metric space to generate a free group. This criterion holds only in singular spaces, for example in Euclidean buildings. The original motivation for our criterion was to prove that the four dimensional Burau representation is faithful. …