arXiv research

This paper proves a conjecture about trisections with a specific length.

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 …

Constructs non-isometric iso-length-spectral surfaces.

Study on stable torsion length in groups, showing it vanishes in crystallographic groups and providing algorithms for computation.

Study on stable translation lengths of surface homeomorphisms and their approximations.

We study the length, weak length and complex length spectrum of closed geodesics of a compact flat Riemannian manifold, comparing length-isospectrality with isospectrality of the Laplacian acting on p-forms. Using integral roots of the Krawtchouk polynomials, we give many pairs of p-isospectral flat manifolds having di…

J.-B. Meilhan and the second author showed that any Milnor $\barμ$-invariant of length between 3 and $2k+1$ can be represented as a combination of HOMFLYPT polynomial of knots obtained by certain band sum of the link components, if all $\barμ$-invariants of length $\leq k$ vanish. They also showed that their formula do…

New proof shows surfaces can have identical length spectra but not simple ones.

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

I show that Matsumoto conjectured inequality between relative length and Finsler length is false. The incorrectness of the claim is easily inferred from the geometry of the indicatrix.

This paper establishes the existence of a gap for the stable length spectrum on a hyperbolic manifold. If M is a hyperbolic n-manifold, for every positive e there is a positive d depending only on n and on e such that an element of pi_1(M) with stable commutator length less than d is represented by a geodesic with leng…

New results on the convexity of geodesic-length functions on Teichmüller space are presented. A formula for the Hessian of geodesic-length is presented. New bounds for the gradient and Hessian of geodesic-length are described. A relationship of geodesic-length functions to Weil-Petersson distance is described. Applicat…

Study gluing of Lorentzian length spaces and their causal ladder properties.

New theorem shows metrics of certain groups are close if their lengths are identical.

The paper proves rigidity of length identities for simple closed curves on hyperbolic surfaces.

Each free homotopy class of directed closed curves on a surface with boundary can be described by a cyclic reduced word in the generators of the fundamental group and their inverses. The word length is the number of letters of the cyclic word. If the surface has a hyperbolic metric with geodesic boundary, the geometric…

The paper provides conditions for realizing graphs and polytopes with specified edge lengths.

We study decompositions of complex hyperbolic isometries as products of involutions. We show that PU(2,1) has involution length 4 and commutator length 1, and that for all $n \geqslant 3$ PU($n$,1) has involution length at most 8.

The study examines arithmetic orbifolds and their length spectra, proving uniform discreteness and linear dependence of geodesic lengths.

Sharp proof of sub-Riemannian length-minimizing curves being at least $C^2$

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…

Extremal length systole is maximized at the Bolza surface.

Study extends null distance concept to Lorentzian length spaces for spacetime analysis.

Study approximate marked length spectrum rigidity in non-positively curved groups.

Study finds numerical moduli in special 2-flags of length 5.

Curvature bounds preserved in length-minimizing disks.

Minimum Description Length prevents overfitting in noisy data.

Timelike curves in Lorentzian length spaces have a total curvature notion that agrees with smooth curves.

We introduce the stable presentation length of a finitely presented group. The stable presentation length of the fundamental group of a 3-manifold can be considered as an analogue of the simplicial volume. We show that the stable presentation length have some additive properties like the simplicial volume, and the simp…

Two free homotopy classes of closed curves in an orientable surface with negative Euler characteristic are said to be length equivalent if for any hyperbolic structure on the surface, the length of the geodesic in one class is equal to the length of the geodesic in the other class. We show that there are elements in th…

The paper proves uniqueness of closed timelike geodesics on de-Sitter tori with one singularity.

This paper discovers new identities linking geodesic and orthogeodesic lengths on hyperbolic surfaces.

The simple length spectrum of a Riemannian manifold is the set of lengths of its simple closed geodesics. We prove a theorem claimed by Lusternik: in any Riemannian 2-sphere whose simple length spectrum consists of only one element L, any geodesic is simple closed with length L.

Each element of the commutator subgroup of a group can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of the element. The commutator length of a group is defined as the supremum of commutator lengths of elements of its commutator subgroup. We …

We show that certain families of iso-length spectral hyperbolic surfaces obtained via the Sunada construction are not generally simple iso-length spectral.

New bounds on specific torsion lengths for periodic mapping classes.

New theorem shows certain curved surfaces are uniquely identified by their geodesic lengths.

Algorithms compute length spectra of torus graphs efficiently.

Shorter adversarial prompts help protect LLMs from jailbreak attacks.

New proof shows Fuchsian groups have irrational length spectra.

In this article, we prove that every arithmetic locally symmetric orbifold of classical type without Euclidean or compact factors has arbitrarily long arithmetic progressions in its primitive length spectrum. Moreover, we show the stronger property that every primitive length occurs in arbitrarily long arithmetic progr…

Two Riemannian manifolds are called eigenvalue equivalent when their sets of eigenvalues of the Laplace-Beltrami operator are equal (ignoring multiplicities). They are (primitive) length equivalent when the sets of lengths of their (primitive) closed geodesics are equal. We give a general construction of eigenvalue equ…

Task hinting improves transformer performance on longer tasks.

We prove a strong multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces i.e. if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then all closed geodesics have the same length.

Recurrent neural networks and sequence to sequence models require a predetermined length for prediction output length. Our model addresses this by allowing the network to predict a variable length output in inference. A new loss function with a tailored gradient computation is developed that trades off prediction accur…

Paper proves flat metrics from holomorphic quadratic differentials can be identified by length spectrum.

Study shows surfaces with similar length spectra are smoothly deformable.

Study on length spectrum of random hyperbolic 3-manifolds.