Related papers: Iterating evolutes and involutes
We investigate a system of geometric evolution equations describing a curvature and torsion driven motion of a family of 3D curves in the normal and binormal directions. We explore the direct Lagrangian approach for treating the geometric…
We classify conjugacy classes of involutions in the isometry groups of nondegenerate, symmetric bilinear forms over the field of two elements. The new component of this work focuses on the case of an orthogonal form on an even dimensional…
We review the concept of a graded bundle as a natural generalisation of a vector bundle. Such geometries are particularly nice examples of more general graded manifolds. With hindsight there are many examples of graded bundles that appear…
We classify involutions acting on spherical 3-manifolds up to conjugacy. Our geometric approach provides insights into numerous topological properties of these involutions.
We investigate the regularity of invariant curves of rotation number $1/2$ for a special class of symplectic twist maps of the annulus, billiard maps. We construct strictly convex smooth tables close to the circle having singular (i.e. not…
Modifications of bundles over complex curves is an operation that allows one to construct a new bundle from a given one. Modifications can change a topological type of bundle. We describe the topological type in terms of the characteristic…
We describe limits of line bundles on nodal curves in terms of toric arrangements associated to Voronoi tilings of Euclidean spaces. These tilings encode information on the relationship between the possibly infinitely many limits, and…
We construct moduli stacks of stable sheaves for surfaces fibered over marked nodal curves by using expanded degenerations. These moduli stacks carry a virtual class and therefore give rise to enumerative invariants. In the case of a…
We introduce a novel class of rotation invariants of two dimensional curves based on iterated integrals. The invariants we present are in some sense complete and we describe an algorithm to calculate them, giving explicit computations up to…
Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such…
We introduce pseudo-spherical non-null framed curves in the three-dimensional anti-de Sitter spacetime and establish the existence and uniqueness of these curves. We then give moving frames along pseudo-spherical framed curves, which are…
We provide a complete structure theorem for involutory matrices. This yields a new approach to principal angles between subspaces and provide a series of nice formulae for these angles.
We formalize the intuitive idea of a labelled discrete surface which evolves in time, subject to two natural constraints: the evolution does not propagate information too fast; and it acts everywhere the same.
One of the general problems in algebraic geometry is to determine algorithmically whether or not a given geometric object, defined by explicit polynomial equations (e.g. a curve or a surface), satisfies a given property (e.g. has…
In this paper we define a Poisson structure on some moduli spaces related to principal G-bundles on elliptic curves, the simplest example being the moduli space of stable pairs: a vector bundle and its global section. We also study…
We show how quiver representations and their invariant theory natu- rally arise in the study of some moduli spaces parametrizing bundles dened on an algebraic curve, and how they lead to ne results regarding the geometry of these spaces.
This article deals with the study of the birational transformations of the projective complex plane which leave invariant an irreducible algebraic curve. We try to describe the state of art and provide some new results on this subject.
In this paper we deal with curves with degeneration degree two in pseudo-Euclidean spaces of index two. We characterize Bertrand curves. We show a correspondence between the evolute of a null curve and the involute of a certain spacelike…
An involution is usually defined as a mapping that is its own inverse. In this paper, we study quaternion involutions that have the additional properties of distribution over addition and multiplication. We review formal axioms for such…
We construct the infinite sequence of invariants for curves in surfaces by using word theory that V. Turaev introduced. For plane closed curves, we add some extra terms, e.g. the rotation number. From these modified invariants, we get the…