arXiv research

Sub-Riemannian geometry connects bike paths to mathematical curves.

Defines Lewy curves in para-CR geometry and characterizes their path geometries.

Derives path-integrals for superstrings on curved backgrounds using string geometry theory.

Origami edge-paths connect coherent curves on surfaces.

This paper extends the Good Covering Theorem and Jordan Curve Theorem for proximal Alexandrov spaces.

The paper shows that almost every path structure is not variational.

We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.

Given a smooth manifold $M$ and a totally nonholonomic distribution $Δ\subset TM$ of rank $d$, we study the effect of singular curves on the topology of the space of horizontal paths joining two points on $M$. Singular curves are critical points of the endpoint map $F:γ\mapstoγ(1)$ defined on the space $Ω$ of horizonta…

Derives path integrals for perturbative strings on various backgrounds.

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

The paper proves a category of dg manifolds with finite positive amplitude.

This paper shows that a construction, which was introduced by Piotr Minc in connection with a problem that came from Helly type theorems and that allows to replace three PL-arcs with a "sheltered middle path", can in the case of general (non-PL) paths result in the topologist's sine curve.

Study proves hyperfiniteness of mapping class group actions on surface graphs.

New proof confirms rolling objects can follow any path.

While the equality of differential signatures (Calabi et al, Int. J. Comput. Vis. 26: 107-135, 1998) is known to be a necessary condition for congruence, it is not sufficient (Musso and Nicolodi, J. Math Imaging Vis. 35: 68-85, 2009). Hickman (J. Math Imaging Vis. 43: 206-213, 2012, Theorem 2) claimed that for non-dege…

We extend the notion of unicorn paths between two arcs introduced by Hensel, Przytycki and Webb to the case where we replace one arc with a geodesic asymptotic to a lamination. Using these paths, we give new proofs of the results of Klarreich and Schleimer identifying the Gromov boundaries of the curve graph and the ar…

Space of hyperbolic surfaces is path-connected.

We discuss some aspects of the differential geometry of curves in Minkowski space. We establish the Serret-Frenet equations in Minkowski space and use them to give a very simple proof of the fundamental theorem of curves in Minkowski space. We also state and prove two other theorems which represent Minkowskian versions…

A new method uses string method to explore diffusion models.

Study efficient geodesics in curve complex using dot graphs.

New algorithms sample from complex path measures using neural networks.

This note presents a formula for the enumerative invariants of arbitrary genus in toric surfaces. The formula computes the number of curves of a given genus through a collection of generic points in the surface. The answer is given in terms of certain lattice paths in the relevant Newton polygon. If the toric surface i…

Spirals are not shortest paths in certain sub-Riemannian geometries.

We provide techniques for studying the nonnegatively curved left-invariant metrics on a compact Lie group. For "straight" paths of left-invariant metrics starting at bi-invariant metrics and ending at nonnegatively curved metrics, we deduce a nonnegativity property of the initial derivative of curvature. We apply this …

We discuss how the shape of a special Cosserat rod can be represented as a path in the special Euclidean algebra. By shape we mean all those geometric features that are invariant under isometries of the three-dimensional ambient space. The representation of the shape as a path in the special Euclidean algebra is intrin…

We derive a curvature-variation formula for a path of left-invariant metrics on a compact Lie group, beginning at a bi-invariant metric. We prove rigidity theorems for paths which remain nonnegatively curved, and we make progress towards a classification of the left-invariant metrics with nonnegative curvature on SO(4)…

This paper presents a new methodology to compute first-order Greeks for barrier options under the framework of path-dependent payoff functions with European, Lookback, or Asian type and with time-dependent trigger levels. In particular, we develop chain rules for Wiener path integrals between two curves that arise in t…

Path integral method calculates PDBS option prices with time-dependent parameters.

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. We compute the Wilson Loop observable using a path integral with an Einstein-Hilbert action. Using axial-gauge fixing, we can write this path integral as the limit of a sequence of Chern-Simons integrals, studied e…

Geodesics connect model modes in neural network loss landscapes.

The snake charmer algorithm permits us to deform a piecewise smooth curve starting from the origin in R^d, so that its end follows a given path. When this path is a loop, a holonomy phenomenon occurs. We prove that the holonomy orbits are closed manifolds diffeomorphic to real Stiefel manifolds. A survey of the snake c…

Develops derived differential geometry theory.

We introduce a new deep-learning based algorithm to evaluate options in affine rough stochastic volatility models. Viewing the pricing function as the solution to a curve-dependent PDE (CPDE), depending on forward curves rather than the whole path of the process, for which we develop a numerical scheme based on deep le…

In this work we generalize the classical notion of a (compact) twistor line in the period domain of compact complex tori. We introduce two new types of lines, which are non-compact analytic curves in the period domain of complex tori. We study the analytic properties of the compactifications of the curves, the preserva…

The paper conjectures and proves fixed points for certain group actions on nonpositively curved spaces.

We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves and it is related w…

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…

We study the existence of closed trajectories of a particle model on null curves in anti-de Sitter 3-space defined by a functional which is linear in the curvature of the particle path. Explicit expressions for the trajectories are found and the existence of infinitely many closed trajectories is proved.

In this note we prove that the level-set flow of the topologist's sine curve is a smooth closed curve. In previous work it was shown by the second author that under level-set flow, a locally-connected set in the plane evolves to be smooth, either as a curve or as a positive area region bounded by smooth curves. Here we…

In genus two and higher, the fundamental group of a closed surface acts naturally on the curve complex of the surface with one puncture. Combining ideas from previous work of Kent--Leininger--Schleimer and Mitra, we construct a universal Cannon--Thurston map from a subset of the circle at infinity for the closed surfac…

Contact path geometries are curved geometric structures on a contact manifold comprising smooth families of paths modeled on the family of all isotropic lines in the projectivization of a symplectic vector space. Locally such a structure is equivalent to the graphs in the space of independent and depedent variables of …

The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in http://arxiv.org/abs/math.AG/0209253. The result is established with the help of the …

Study shows almost complex structures with certain tensor properties are prevalent.

Shapes can roll downhill following any curve, but often return to initial orientation after crossing multiple copies.

Paper computes optimal matching between curves on manifolds.

We prove two theorems about homotopies of curves on 2-dimensional Riemannian manifolds. We show that, for any epsilon > 0, if two simple closed curves are homotopic through curves of bounded length L, then they are also isotopic through curves of length bounded by L + epsilon. If the manifold is orientable, then for an…

Given a positive and unitarily invariant Lagrangian L defined in the algebra of Hermitian matrices, and a fixed interval $[a,b]\subset\mathbb R$, we study the action defined in the Lie group of $n\times n$ unitary matrices $\mathcal{U}(n)$ by $S(α)=\int_a^b L(\dotα(t))\,dt\,,$ where $α:[a,b]\to\mathcal{U}(n)$ is a …

The symmetries of paths in a manifold $M$ are classified with respect to a given pointwise proper action of a Lie group $G$ on $M$. Here, paths are embeddings of a compact interval into $M$. There are at least two types of symmetries: Firstly, paths that are parts of an integral curve of a fundamental vector field on $…