arXiv research

Study finds a limiting distribution for free path lengths on flat surfaces with circular obstacles.

Formula derived for $G_2$-manifolds, showing moduli spaces are incomplete.

We construct an action of the free group $F_n$ on the homotopy category of projective modules over a finite dimensional zigzag algebra. The main theorem in the paper is that this action is faithful. We describe the relationship between homotopy classes of paths in the punctured disc and complexes of projective zigzag m…

The paper defines and studies discrete p-density and compression-radius profiles of lattice knots.

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…

In this paper we consider the length minimizing properties of Hamiltonian paths generated by quasi-autonomous Hamiltonians on symplectically aspherical manifolds. Motivated by the work of L. Polterovich and M. Schwarz, we study the role of the fixed global extrema in the Floer complex of the generating Hamiltonian. Our…

Characterizes paths minimizing anisotropic lengths in Euclidean space.

We study adaptive regret bounds in terms of the variation of the losses (the so-called path-length bounds) for both multi-armed bandit and more generally linear bandit. We first show that the seemingly suboptimal path-length bound of (Wei and Luo, 2018) is in fact not improvable for adaptive adversary. Despite this neg…

Study geodesic paths on flat surfaces, comparing length and singularity counts.

When two free factors A and B of a free group F_n are in "general position" we define the projection of B to the splitting complex (alternatively, the complex of free factors) of A. We show that the projections satisfy properties analogous to subsurface projections introduced by Masur and Minsky. We use the subfactor p…

We use the criteria of Lalonde and McDuff to determine a new class of examples of length minimizing paths in the group $Ham(M)$. For a compact symplectic manifold $M$ of dimension two or four, we show that a path in $Ham(M)$, generated by an autonomous Hamiltonian and starting at the identity, which induces no non-cons…

We start by studying the distribution of (cyclically reduced) elements of the free groups Fn with respect to their abelianization (or equivalently, their integer homology class. We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev pol…

In this paper we first show that the necessary condition introduced in our previous paper is also a sufficient condition for a path to be a geodesic in the group $\Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms. This applies with no restriction on $M$. We then discuss conditions which guarantee that su…

In this paper, we use Floer theory to study the Hofer length functional for paths of Hamiltonian diffeomorphisms which are sufficiently short. In particular, the length minimizing properties of a short Hamiltonian path are related to the properties and number of its periodic orbits.

Link between Teichmüller and anti de Sitter geometry via length functions.

Choose two points in the tangent bundle of the Euclidean plane $(x,X),(y,Y)\in T{ \mathbb R}^2$. In this work we characterise the immersed length minimising paths with a prescribed bound on the curvature starting at $x$, tangent to $X$; finishing at $y$, tangent to $Y$, in each connected component of the space of paths…

This paper, the second of a series, deals with the function space of all smooth Kähler metrics in any given closed complex manifold $M$ in a fixed cohomology class. The previous result of the second author \cite{chen991} showed that the space is a path length space and it is geodesically convex in the sense that any tw…

GIST adapts HMC by tuning parameters based on position and momentum.

We study side-lengths of triangles in path metric spaces. We prove that unless such a space X is bounded, or quasi-isometric to line or half-line, every triple of real numbers satisfying the strict triangle inequalities, is realized by the side-lengths of a triangle in X. We construct an example of a complete path metr…

In his PhD thesis, Abrams proved that, for a natural number n and a graph G with at least n vertices, the n-strand configuration space of G deformation retracts to a compact subspace, the discretized n-strand configuration space, provided G satisfies two conditions: each path between distinct essential vertices (vertic…

We present the first treatment of the arc length of the Gaussian Process (GP) with more than a single output dimension. GPs are commonly used for tasks such as trajectory modelling, where path length is a crucial quantity of interest. Previously, only paths in one dimension have been considered, with no theoretical con…

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…

In this paper we use a time-evolving graph which consists of a sequence of graph snapshots over time to model many real-world networks. We study the path classification problem in a time-evolving graph, which has many applications in real-world scenarios, for example, predicting path failure in a telecommunication netw…

Connectivity proven in large rank Gromov boundary of free factor complex.

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

We show that stable commutator length is rational on free products of free Abelian groups amalgamated over $\mathbb{Z}^k$, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parameterize all surfaces with specified boundary mapping to th…

Study curves evolving on hypersurfaces with free boundaries, preserving length.

We introduce here a natural functional associated to any $b \in QH_* (M, ω)$: \emph{spectral length functional}, on the space of "generalized paths" in $\text {Ham}(M, ω)$, closely related to both the Hofer length functional and spectral invariants and establish some of its properties. This functional is smooth on its…

Improved dynamic regret analysis for strongly convex and smooth functions.

Given two points on a soup can or conical cup with lid, we find and classify all paths of minimal length connecting them. When the number of minimal paths is finite, there are at most four on a can and three on a cup. At worst, minimal paths are piece-wise smooth with three components, each of which is a classical geod…

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…

Study on stable commutator length in RAAGs and Coxeter groups, proving spectral gaps and hardness results.

Handel and Mosher have proved that the free splitting complex FS for the free group is Gromov hyperbolic. This is a deep and much sought-after result, since it establishes FS as a good analogue of the curve complex for surfaces. We give a shorter alternative proof of this theorem, using surgery paths in Hatcher's spher…

In this paper we study the convexity properties of geodesics and balls in Outer space equipped with the Lipschitz metric. We introduce a class of geodesics called balanced folding paths and show that, for every loop $α$, the length of $α$ along a balanced folding path is not larger than the maximum of its lengths at th…

Paper explores rough path theory for frictionless markets, linking NCFL to unbiased rough integrators.

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 …

Unfolding paths in Outer space accumulate on a simplex, not converge.

Bicycle paths form geodesics in 3D subspaces, related to Kirchhoff rods.

Improved sampling efficiency for molecular systems using path gradients after Flow Matching.

We give an algorithm to compute stable commutator length in free products of cyclic groups which is polynomial time in the length of the input, the number of factors, and the orders of the finite factors. We also describe some experimental and theoretical applications of this algorithm.

The paper extends Lipschitz metric isometries between Outer Spaces of virtually free groups.

Develops a new model-free approach to portfolio theory using rough paths.

We develop a novel and generic algorithm for the adversarial multi-armed bandit problem (or more generally the combinatorial semi-bandit problem). When instantiated differently, our algorithm achieves various new data-dependent regret bounds improving previous work. Examples include: 1) a regret bound depending on the …

Compactness proven for CMC surfaces with bounded topology and boundary length.

Improved bounds on geodesic lengths in Riemannian surfaces.

In this paper, we address the challenging problem of selecting tuning parameters for high-dimensional sparse regression. We propose a simple and computationally efficient method, called path thresholding (PaTh), that transforms any tuning parameter-dependent sparse regression algorithm into an asymptotically tuning-fre…

New method uses path signatures for causal discovery in time series data.

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…