arXiv research

Paper introduces length measures for curves and convex shapes, proving isoperimetric and distance properties.

Filling length measures the length of the contracting closed loops in a null-homotopy. The filling length function of Gromov for a finitely presented group measures the filling length as a function of length of edge-loops in the Cayley 2-complex. We give a bound on the filling length function in terms of the log of an …

The paper connects Riemann surface length spectra to Brownian loop measures.

Introduces fractional k-dimensional measure bridging fractional length and area.

The radar experiment connects the geometry of spacetime with an observers measurement of spatial length. We investigate the radar experiment on Finsler spacetimes which leads to a general definition of radar orthogonality and radar length. The directions radar orthogonal to an observer form the spatial equal time surfa…

In the paper, we introduce a new measure of correlation between possibly non-stationary series. As the measure is based on the detrending moving-average cross-correlation analysis (DMCA), we label it as the DMCA coefficient $ρ_{DMCA}(λ)$ with a moving average window length $λ$. We analytically show that the coefficient…

Defines magnitude for length spaces with measures, agreeing with finite spaces' magnitude.

Extremal length is an important conformal invariant on Riemann surface. It is closely related to the geometry of Teichmuller metric on Teichmuller space. By identifying extremal length functions with energy of harmonic maps from Riemann surfaces to $\mathbb{R}$-trees, we study the second variation of extremal length fu…

The paper introduces new volume measures and volume comparison inequalities for Lorentzian spaces.

The aim of this paper is to associate a measure for certain sets of paths in the Euclidean plane $\mathbb{R}^2$ with fixed starting and ending points. Then, working on parameterized surfaces with a specific Riemannian metric, we define and calculate the integral of the length over the set of paths obtained as the image…

We define a Hamilton-Jacobi semigroup acting on continuous functions on a compact length space. Following a strategy of Bobkov, Gentil and Ledoux, we use some basic properties of the semigroup to study geometric inequalities related to concentration of measure. Our main results are that (1) a Talagrand inequality on a …

Non-convex extremal length found in surface metrics.

We combine concepts from random matrix theory and free probability together with ideas from the theory of commutator length in groups and maps from surfaces, and establish new connections between the two. More particularly, we study measures induced by free words on the unitary groups $U(n)$. Every word $w$ in the free…

New complexity measure ADL connects to classical complexity measures.

The hitting measure is singular and has dimension less than 1 for cocompact Fuchsian groups.

Fix a translation surface $X$, and consider the measures on $X$ coming from averaging the uniform measures on all the saddle connections of length at most $R$. Then as $R\to\infty$, the weak limit of these measures exists and is equal to the Lebesgue measure on $X$. We also show that any weak limit of a subsequence of …

ReLU networks don't exponentially distort curve lengths as previously thought.

We define a new spectrum for compact length spaces and Riemannian manifolds called the "covering spectrum" which roughly measures the size of the one dimensional holes in the space. More specifically, the covering spectrum is a set of real numbers $δ>0$ which identify the distinct $δ$ covers of the space. We investigat…

Synthetic approach to conformal transformations in metric and Lorentzian spaces.

Estimates dimensions of maximal simplices for rational and irrational trees in Outer space.

We give sufficient conditions for a measured length space (X,d,m) to admit local and global Poincare inequalities. We first introduce a condition DM on (X,d,m), defined in terms of transport of measures. We show that DM, along with a doubling condition on m, implies a scale-invariant local Poincare inequality. We show …

We show that the length function of a measured geodesic lamination is convex in Thurston's shear coordinates over Teichmüller space and strictly convex for generic laminations. We give some consequences of this result in the context of Thurston's asymmetric metric on Teichmüller space.

Critical graphs of quadratic differentials equidistribute in moduli space.

The study of random surfaces reveals asymptotic lengths of separating geodesics.

Study compactifies representations space of hyperbolic surfaces.

The set of osculating circles of a given curve in $\SS^3$ forms a curve in the set of oriented circles in $\SS^3$. We show that its "${\frac12}$-dimensional measure" with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a confor…

We study metric and analytic properties of generalized lemniscates E_t(f)={z:ln|f(z)|=t}, where f is an analytic function. Our main result states that the length function |E_t(f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|E_t(f)| is convex on any interval free of cri…

Paper proves method for calculating NML code length works for continuous models.

We consider the expected value for the total curvature of a random closed polygon. Numerical experiments have suggested that as the number of edges becomes large, the difference between the expected total curvature of a random closed polygon and a random open polygon with the same number of turning angles approaches a …

Study shows non-spectrality of certain curves and line segments.

Develops optimal transport in Lorentzian spaces with synthetic curvature bounds.

We investigate the terms arising in an identity for hyperbolic surfaces proved by Luo and Tan, namely showing that they vary monotonically in terms of lengths and that they verify certain convexity properties. Using these properties, we deduce two results. As a first application, we show how to deduce a theorem of Thur…

Let $X$ be an infinite geodesically complete hyperbolic surface which can be decomposed into geodesic pairs of pants. We introduce Thurston's boundary to the Teichmüller space $T(X)$ of the surface $X$ using the length spectrum analogous to Thurston's construction for finite surfaces. Thurston's boundary using the leng…

We give effective proofs of residual finiteness and conjugacy separability for finitely generated nilpotent groups. In particular, we give precise asymptotic bounds for a function introduced by Bou-Rabee that measures how large the quotients that are need to separate non-identity elements of bounded length from the ide…

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…

Length spectral rigidity is the question of under what circumstances the geometry of a surface can be determined, up to isotopy, by knowing only the lengths of its closed geodesics. It is known that this can be done for negatively curved Riemannian surfaces, as well as for negatively-curved cone surfaces. Steps are tak…

Uniform systole bounds for arithmetic orbifolds and number fields.

Metrics on Lie groupoids and differentiable stacks have been introduced recently, extending the Riemannian geometry of manifolds and orbifolds to more general singular spaces. Here we continue that theory, studying stacky curves on Riemannian stacks, measuring their length using stacky metrics, and introducing stacky g…

A random Heegaard splitting is a 3-manifold obtained by using a random walk of length n on the mapping class group as the gluing map between two handlebodies. We show that the joint distribution of random walks of length n and their inverses is asymptotically independent, and converges to the product of the harmonic an…

Analyzes geodesic lengths in sparse networks, deriving a distribution.

Study shows LLC correlates with neural network compressibility.

Let $S$ be a closed oriented surface of genus at least $2$, and denote by $\mathcal{T}(S)$ its Teichm{ü}ller space. For any isotopy class of closed curves $γ$, we compute the first three derivatives of the length function $\ell\_γ:\mathcal{T}(S)\rightarrow\mathbf{R}\_+$ in the shearing coordinates associated to a maxim…

For a path in a compact finite dimensional Alexandrov space $X$ with curv $\ge κ$, the two basic geometric invariants are the length and the turning angle (which measures the closeness from being a geodesic). We show that the sum of the two invariants of any loop is bounded from below in terms of $κ$, the dimension, di…

Random walks on hyperbolic spaces show linear growth in translation lengths.

In this paper we raise the question whether every closed Riemannian manifold has a spine of minimal area, and we answer it affirmatively in the surface case. On constant curvature surfaces we introduce the spine systole, a continuous real function on moduli space that measures the minimal length of a spine in each surf…

We introduce new definitions of universal and superuniversal computable codes, which are based on a code's ability to approximate Kolmogorov complexity within the prescribed margin for all individual sequences from a given set. Such sets of sequences may be singled out almost surely with respect to certain probability …

In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have been successfully computed using the periodic orbit sum method, which are compar…

We define a notion of a measured length space X having nonnegative N-Ricci curvature, for N finite, or having infinity-Ricci curvature bounded below by K, for K a real number. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein space P_2(X) of probability measur…