arXiv research

Determines the crossing number of polynomial curve systems on surfaces.

Constructs coordinate systems from spectral curve sheaves.

Paper finds minimal number of curves in surface systems.

33 curves on a 3-genus surface, all intersecting at most once.

Unique maximal curve systems found for up to 5 punctures.

Sparse curves on surfaces grow at a specific intermediate rate.

Minimal crossing number found in arithmetic curve systems.

We study the limiting case of the Krichever construction of orthogonal curvilinear coordinate systems when the spectral curve becomes singular. We show that the case when the curve is reducible and all its irreducible components are rational curves the construction procedure reduces to solving systems of linear equatio…

A typical solution of an integrable system is described in terms of a holomorphic curve and a line bundle over it. The curve provides the action variables while the time evolution is a linear flow on the curve's Jacobian. Even though the system of Nahm equations is closely related to the Hitchin system, the curves appe…

Study minimizes crossing points of up to 12 curves on a genus 2 surface.

Contact Lie systems analyze integral curves of Hamiltonian vector fields.

Let $D_n$ denote the $n$-punctured disk in the complex plane, where the punctures are on the real axis. An $n$-braid $α$ is said to be \emph{reducible} if there exists an essential curve system $\C$ in $D_n$, called a \emph{reduction system} of $α$, such that $α*\C=\C$ where $α*\C$ denotes the action of the braid $α$ o…

Study geodesic curves on Heisenberg group, classify them, and compute first step of quadrature.

A collection $Δ$ of simple closed curves on an orientable surface is an algebraic $k$-system if the algebraic intersection number $\langle α,β\rangle$ is equal to $k$ in absolute value for every $α, β\in Δ$ distinct. Generalizing a theorem of [MRT14] we compute that the maximum size of an algebraic $k$-system of c…

Superintegrable systems on curved manifolds found to have Hessian structures.

This paper describes how to define and work with differential equations in the abstract setting of tangent categories. The key notion is that of a curve object which is, for differential geometry, the structural analogue of a natural number object. A curve object is a preinitial object for dynamical systems; dynamical …

Higher KdV flows on spaces of closed equicentroaffine plane curves are studied and it is shown that the flows are described as certain multi-Hamiltonian systems on the spaces. Multi-Hamiltonian systems describing higher mKdV flows are also given on spaces of closed Euclidean plane curves via the geometric Miura transfo…

Let $σ$ be an involution of a real semi-simple Lie group $U$, $U_0$ the subgroup fixed by $σ$, and $U/U_0$ the corresponding symmetric space. Ferus and Pedit called a submanifold $M$ of a rank $r$ symmetric space $U/U_0$ a {\it curved flat} if $T_pM$ is tangent to an $r$-dimensional flat of $U/U_0$ at $p$ for each $p\i…

New connections share geodesics with superintegrable systems.

Optimizes curves on Riemannian manifolds to minimize curvature.

We investigate the arithmetic of algebraic curves on coarse moduli spaces for special linear rank two local systems on surfaces with fixed boundary traces. We prove a structure theorem for morphisms from the affine line into the moduli space. We show that the set of integral points on any nondegenerate algebraic curve …

In this paper we introduce a new dynamical system which we call Angular billiard. It acts on the exterior points of a convex curve in Euclidean plane. In a neighborhood of the boundary curve this system turns out to be dual to the Birkhoff billiard. Using this system we get new results on algebraic Birkhoff conjecture …

The paper studies curve evolution using the PLR equation and its solutions.

Method for generating new curves from plane curves on cylinders.

The systems of complex analytic second order ordinary differential equations whose solutions close up to become rational curves (after analytic continuation) are characterized by the vanishing of an explicit differential invariant, and turn out to provide an infinite dimensional family of integrable systems.

This paper establishes three relations between the Toda field theory associated to a simple Lie algebra and the integral curves of the standard differential system on the corresponding complete flag variety. The motivation comes from the viewpoint on the Toda field theories as Darboux integrable differential systems as…

Quadratic growth of intersecting curves on surfaces resolved.

Constructs universal local deformations for curves and differential forms.

Generalizes Riemann-Hilbert correspondence for curved local systems.

Geometry of holomorphic curves from point of view of open Toda systems is discussed. Parametrization of curves related this way to non-exceptional simple Lie algebras is given. This gives rise to explicit formulas for minimal surfaces in real, complex and quaternionic projective spaces or complex quadrics. The paper ge…

Curves in Lagrange Grassmannians appear naturally in the intrinsic study of geometric structures on manifolds. By a smooth geometric structure on a manifold we mean any submanifold of its tangent bundle, transversal to the fibers. One can consider the time-optimal problem naturally associate with a geometric structure.…

An example of mechanical system whose configuration space is direct product of a curved space and the local group of rotations, is presented. The system is considered as a model of spinning particle moving in the space. The Hamiltonian formalism for this system and possible method for its quantization are discussed. It…

The paper establishes a correspondence between Higgs torsors and connections on curves.

New method calculates geodesic distances in Gaussian random field manifolds.

Conditions for curves on a torus with specific pairwise intersections.

A correspondence between 1) rank 2 completely integrable systems of Jacobians of algebraic curves and 2) (holomorphically) symplectic surfaces was established in a previous paper by the first author. A more general abelian variety that occurs as a Liouville torus of integrable systems is a prym variety associated to a …

Curves in Lagrange Grassmannians naturally appear when one studies intrinsically "the Jacobi equations for extremals", associated with control systems and geometric structures. In this way one reduces the problem of construction of the curvature-type invariants for these objects to the much more concrete problem of fin…

Unified framework identifies nonlinear systems using characteristic curves and neural networks.

New solutions to SU(n+1) Toda system found on compact Riemann surfaces with cone singularities.

We describe a polynomial-time algorithm to compute a (tight) geodesic between two curves in the curve graph. As well as enabling us to compute the distance between a pair of curves, this has several applications to mapping classes. For example, we can use these geodesics to compute the asymptotic translation length, Ni…

Study on dynamic curves with elastic energy and spontaneous curvature.

The paper explores how complex models can improve system identification beyond traditional limits.

Curved Frobenius manifolds link to Hessian metrics in geometry.

Study shows invariant curves in tubular origami dynamics, revealing geometric barriers to folding transitions.

Study proves uniqueness of corrugated negatively curved immersions in differential geometry.

Optimizes the crossing number for curve systems on surfaces.

Derives new orthogonal coordinates for evolving surfaces and curves.

Method constructs orthogonal curvilinear coordinates in constant curvature spaces.