arXiv research

Invariants of braids found using shear coordinates in hyperbolic geometry.

New definition of twisted 1-loop invariant using Ptolemy coordinates.

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

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

Most neural networks are trained using first-order optimization methods, which are sensitive to the parameterization of the model. Natural gradient descent is invariant to smooth reparameterizations because it is defined in a coordinate-free way, but tractable approximations are typically defined in terms of coordinate…

In integrable hydrodynamic systems, coordinates exist where generators and symmetries are simple.

We define Dynnikov coordinates on virtual braid groups. We prove that they are faithful invariants of virtual 2-braids, and present evidence that they are also very powerful invariants for general virtual braids.

Paper classifies symmetries of cross caps using invariants.

Two popular examples of first-order optimization methods over linear spaces are coordinate descent and matching pursuit algorithms, with their randomized variants. While the former targets the optimization by moving along coordinates, the latter considers a generalized notion of directions. Exploiting the connection be…

Study on special coordinates for Dubrovin-Frobenius manifolds in low dimensions.

We define Ptolemy coordinates for representations that are not necessarily boundary-unipotent. This gives rise to a new algorithm for computing the SL(2,C) A-polynomial, and more generally the SL(n,C) A-varieties. We also give a formula for the Dehn invariant of an SL(n,C)-representation.

We consider a variant of online convex optimization in which both the instances (input vectors) and the comparator (weight vector) are unconstrained. We exploit a natural scale invariance symmetry in our unconstrained setting: the predictions of the optimal comparator are invariant under any linear transformation of th…

We construct a geometric, real analytic parametrization of the Hitchin component Hit_n(S) of the PSL_n(R)-character variety R_{PSL_n(R)}(S) of a closed surface S. The approach is explicit and constructive. In essence, our parametrization is an extension of Thurston's shear coordinates for the Teichmueller space of a cl…

DNNs improve SIMP method but not spatially invariant, study shows.

We study coordinate-invariance of some asymptotic invariants such as the ADM mass or the Chruściel-Herzlich momentum, given by an integral over a "boundary at infinity". When changing the coordinates at infinity, some terms in the change of integrand do not decay fast enough to have a vanishing integral at infinity; bu…

Flat coordinates for Frobenius manifolds defined on the orbit space of a Coxeter group W are specified through a certain system of generators of W-invariant polynomials. In this note, starting from basic invariants proposed by M.Mehta, we calculate flat coordinates for the exceptional groups of type E_7 and E_8, leadin…

Flat coordinates found for algebraic Frobenius manifolds in low dimensions.

In \cite{rigidity}, Luo introduced a $ψ_λ$ edge invariant which turns out to be a coordinate of the Teichmüller space of a surface with boundary. And he proved that for $λ\geq 0$, the image of the Teichmüller space under $ψ_λ$ edge invariant coordinate is an open cell. In this paper we verify his conjecture that for $λ…

The study examines surfaces in isotropic space with specific Gauss map properties.

The paper connects isomonodromic and isospectral deformations for $\mathfrak{sl}_2(\mathbb{C})$ connections.

The area of a convex projective surface of genus $g\ge 2$ is at least $(g-1)π^2/2+\|τ\|^2/8$ where $τ=(\log t_i)$ is the vector of triangle invariants of Bonahon-Dreyer and $t_i$ are the Fock-Goncharov triangle coordinates.

We point out a new view on slow invariant manifolds (SIM) in dynamical systems which departs from a purely geometric covariant characterization implying coordinate independency. The fundamental idea is to treat the SIM as a well-defined geometric object in phase space and elucidate characterizing geometric properties t…

A family of coordinates $ψ_h$ for the Teichmüller space of a compact surface with boundary was introduced in \cite{l2}. In the work \cite{m1}, Mondello showed that the coordinate $ψ_0$ can be used to produce a natural cell decomposition of the Teichmüller space invariant under the action of the mapping class group. In …

A general class of Newton algorithms on Graßmann and Lagrange-Graßmann manifolds is introduced, that depends on an arbitrary pair of local coordinates. Local quadratic convergence of the algorithm is shown under a suitable condition on the choice of coordinate systems. Our result extends and unifies previous convergenc…

In these notes, after an introduction to toric Kahler geometry, we present Calabi's family of U(n)-invariant extremal Kahler metrics in symplectic action-angle coordinates and show that it actually contains, as particular cases, many interesting cohomogeneity one examples of constant scalar curvature.

Researchers geometrically define asymptotic coordinates in General Relativity.

New Frobenius manifold structures found on Dicyclic group orbits.

Computes knot types using HOMFLY-PT polynomial.

Study algebraic invariants from lightning self-attention models.

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).

We determine the explicit transformation under duality of generic configurations of four flags in $\PGL(3,\bC)$ in cross-ratio coordinates. As an application we prove invariance under duality of an invariant in the Bloch group obtained from decorated triangulations of 3-manifolds.

In this paper we give a complete description of the set of discrete faithful representations SH(M) uniformizing a compact, orientable, hyperbolizable 3-manifold M with incompressible boundary, equipped with the strong topology, with the description given in term of the end invariants of the quotient manifolds. As part …

We introduce canonical coordinates on minimal time-like surfaces in the n-dimensional Minkowski space and prove the existence and the uniqueness of these parameters. With respect to these coordinates the coefficients of the first fundamental form are expressed by the invariants of the surface. On any time-like surface …

We develop Fenchel-Nielsen coordinates for representations of surface groups into Sp(2n,R) with maximal Toledo invariant. Analogous to classical Fenchel-Nielsen coordinates on the Teichmüller space they consist of a parametrization of representations of the fundamental group of a pair of pants and a careful investigati…

The paper derives explicit geodesic equations for a specific type of group structure.

The problem of the invariant classification of the orthogonal coordinate webs defined in Euclidean space is solved within the framework of Felix Klein's Erlangen Program. The results are applied to the problem of integrability of the Calogero-Moser model.

Proves harmonic coordinates for weak immersions in even dimensions.

Conformal Autoencoders infer intrinsic dimensionality and impose invariance.

We introduce a new method for computing the heat invariants of a 2-dimensional Riemannian manifold based on a result by S.Agmon and Y.Kannai. Two explicit expressions for the heat invariants are presented. The first one depends on the choice of a certain coordinate system; the second involves only invariant terms but h…

Researchers find optimal paths on a specific geometric group.

The authors found extremals of arbitrary left-invariant sub-Finsler metric on the Engel group defined by a distribution of rank two. They use for this the Pontryagin Maximum Principle for the corresponding time-optimal problem in coordinates of the first kind. The obtained results are applied to the case of left-invari…

Study hyperplanes in abelian groups and their signatures for manifold identification.

We discuss bases of the space of holomorphic quadratic differentials that are dual to the differentials of Fenchel-Nielsen coordinates and hence appear naturally when considering functions on the set of hyperbolic metrics which are invariant under pull-back by diffeomorphisms, such as eigenvalues of the Laplacian. The …

New form of $D_4^-$-singularities for fronts in 3D space.

We describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented punctured surface $S$ with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of a triple of flags in the complex hyperpolic plane. We establish a bijection between …

Let $M$ be a smooth manifold and $\mathcal{S}$ a semi-spray defined on a sub-bundle $\mathcal{C}$ of the tangent bundle $TM$. In this work it is proved that the only non-trivial $k$-jet approximation to the exact geodesic deviation equation of $\mathcal{S}$, linear on the deviation functions and invariant under an spec…

Neural network learns atomic coordinates from Patterson maps in a simplified case.

Study $D_4^-$-front singularities, compute invariants, and derive a Gauss-Bonnet theorem.