arXiv research

Research disproves Matsumoto's length conjecture using indicatrix geometry.

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

Relative cup-length defined for non-Morse functions on manifolds.

Study of special elliptic isometries and their lengths in complex hyperbolic plane.

Matsumoto conjectured that for any Finsler manifold $(M, F)$ for which the restriction of the fundamental tensor to the indicatrix of $F$ is positive definite, the absolute length $F(X)$ of any tangent vector $X \in T_xM$ is the global minimum for the relative length $|X|_y$ as $y$ varies along the indicatrix $I_x \sub…

We prove that the length spectrum metric and the arc-length spectrum metric are almost-isometric on the $ε_0$-relative part of Teichmuller spaces of surfaces with boundary.

Research tackles unequal length time series for classification.

Study finds minimum lengths of curves on a one-holed torus.

Given a principal bundle with a connection, we look for an asymptotic expansion of the holonomy of a loop in terms of its length. This length is defined relative to some Riemannian or sub-Riemannian structure. We are able to give an asymptotic formula that is independent of choice of gauge.

Neural machine translation is a relatively new approach to statistical machine translation based purely on neural networks. The neural machine translation models often consist of an encoder and a decoder. The encoder extracts a fixed-length representation from a variable-length input sentence, and the decoder generates…

Transformer model generates long musical compositions with compelling structure.

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

Uniform spectral gap found for stable commutator length in hyperbolic 2-orbifolds.

Every element of PU(2,1) can be decomposed into at most 4 special elliptic isometries.

The paper studies hanging chains and surfaces in degenerate geometries.

Duality result connects bounded cohomology to relative Gromov seminorm.

Study of closed trajectories in hyperbolic plane with specific curvature constraints.

An arbitrary homomorphism between groups is nonincreasing for stable commutator length, and there are infinitely many (injective) homomorphisms between free groups which strictly decrease the stable commutator length of some elements. However, we show in this paper that a random homomorphism between free groups is almo…

Critical trajectories in a sphere are found for a specific bending functional.

Randomized positional encodings boost transformer performance on longer sequences.

The study examines knot probabilities in confined lattice polygons.

For any group, there is a natural (pseudo-)norm on the vector space B1 of real (group) 1-boundaries, called the stable commutator length norm. This norm is closely related to, and can be thought of as a relative version of, the Gromov (pseudo)-norm on (ordinary) homology. We show that for a free group, the unit ball of…

Following \cite{citeSavelyevVirtualMorsetheoryon$Omega$Ham$(Momega)$.}, we develop here a connection between Morse theory for the (positive) Hofer length functional $L: Ω\text {Ham}(M, ω) \to \mathbb{R}$, with Gromov-Witten/Floer theory, for monotone symplectic manifolds $(M, ω)$. This gives some immediate restrictio…

We consider some metrics and weak metrics defined on the Teichmueller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichmüller metric. The comp…

We study properties of generic elements of groups of isometries of hyperbolic spaces. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e. they have full density with respect to counting in balls for the word metric) and translation length grows linearly. We provide applications t…

Relatively extremal knots are the relative minima of the ropelength functional in C^1 topology. On the set curves of fixed length, they are the relative maxima of thickness (normal injectivity radius) functional, including the ideal knots. We prove that a C^{1,1} relatively extremal knot in R^n has thickness equal to h…

Study proves rigidity of marked length spectra in contracting group actions.

We define and study metrics and weak metrics on the Teichmueller space of a surface of topologically finite type with boundary. These metrics and weak metrics are associated to the hyperbolic length spectrum of simple closed curves and of properly embedded arcs in the surface. We give a comparison between the defined m…

Mirzakhani extended curve counting from simple to all curves.

BiPE blends intra-segment and inter-segment encodings for better length extrapolation.

Subsurface projection has become indispensable in studying the geometry of the mapping class group and the curve complex of a surface. When the subsurface is an annulus, this projection is sometimes called relative twisting. We give two alternate versions of relative twisting for the outer automorphism group of a free …

We study a relative trace formula for a compact Riemann surface with respect to a closed geodesic $C$. This can be expressed as a relation between the period spectrum and the ortholength spectrum of $C$. This provides a new proof of asymptotic results for both the periods of Laplacian eigenforms along $C$ as well estim…

The study examines spaces of non-extendable quasimorphisms for group pairs.

The paper explores conditions for certain submanifolds to be cylinders.

OSIRIS reduces variance in off-policy evaluation by omitting irrelevant states.

In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected $2$-complex $X$ with a linear homological isoperimetric inequality, a bound on the length of attaching maps of $2$-cells and finitely many $2$-cells adjacent to any edge must …

Paper defines a new invariant for 4-manifolds with boundary.

Embeddings preserve stable commutator length for surfaces.

We study the asymptotic behaviour of regularized determinants of certain Laplace type operators with respect to singular deformations of the underlying manifold which are obtained by stretching a tubular neighborhood of an embedded separating hypersurface to a cylinder of infinite length. Using the asymptotic expansion…

Here we present a novel approach to statistical analysis of financial time series. The approach is based on $n$-grams frequency dictionaries derived from the quantized market data. Such dictionaries are studied by evaluating their information capacity using relative entropy. A specific quantization of (originally conti…

New method finds Fuchsian representations dominating others in surface group representations.

Any one-cusped hyperbolic manifold M with an unknotting tunnel tau is obtained by Dehn filling a cusp of a two-cusped hyperbolic manifold. In the case where M is obtained by "generic" Dehn filling, we prove that tau is isotopic to a geodesic, and characterize whether tau is isotopic to an edge in the canonical decompos…

The paper solves the isoperimetric problem in Riemannian optical geometry, proving circles minimize lengths with area constraints.

Study of generalized cones in Lorentzian geometry with causality and curvature analysis.

We study the dynamics of the linear and non-linear serial dependencies in financial time series in a rolling window framework. In particular, we focus on the detection of episodes of statistically significant two- and three-point correlations in the returns of several leading currency exchange rates that could offer so…

Study of group actions on CAT(0) cube complexes, focusing on marked length spectra.

Researchers prove twists can make surfaces parabolic even with large cuff lengths.

Geometric derivation of Einstein equations from causal fermion systems.