Related papers: Some Applications of a Multiplicative Structure on…

Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this paper, we study counting…

We introduce a numerical method for reconstructing a multidimensional surface using the gradient of the surface measured at some values of the coordinates. The method consists of defining a multidimensional spline function and minimizing…

We classify the singular loci of real surfaces in three-space that contain two circles through each point. We characterize how a circle in such a surface meets this loci as it moves in its pencil and as such provide insight into the…

We present an elementary introduction to the construction of integrable models on hyper-elliptic surfaces for non specialists; also, we present some of the details of the paper `solv-int/9504002' for the more interested readers. (Based on a…

We characterise which simplicial surfaces can be folded onto a triangle. We define a notion of folding that incorporates the non-intersection-properties of real materials. All of the surfaces foldable onto a triangle admit a…

Several results from differential geometry of hypersurfaces in R^n are derived to form a tool box for the direct mapping method. The latter technique has been widely employed to solve problems with moving interfaces, and to study the…

This survey paper addresses uniqueness questions for symplectic forms on closed manifolds, explains what is known in several examples, and reviews some open problems.

We construct new monomorphisms between mapping class groups of surfaces. The first family of examples injects the mapping class group of a closed surface into that of a different closed surface. The second family of examples are defined on…

We study the family of rational curves on arbitrary smooth hypersurfaces of low degree using tools from analytic number theory.

We consider a rational elliptic surface with a relatively minimal fibration. We compute the number of integral sections in the above rational elliptic surface. As an application, we obtain an estimate of polynomial solutions of some…

We consider surfaces embedded in a Riemannian manifold of arbitrary dimension and prove that many aspects of their differential geometry can be expressed in terms of a Poisson algebraic structure on the space of smooth functions of the…

A simplicial complex is a set equipped with a down-closed family of distinguished finite subsets. This structure, usually viewed as codifying a triangulated space, is used here directly, to describe "spaces" whose geometric realisation can…

The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…

This paper describes, in the case of the unit circle, several applications of a geometrically intrinsic treatment of counterparts of classical electrostatics, previously developed in [4] and [5].

In the spirit of topological entropy we introduce new complexity functions for general dynamical systems (namely groups and semigroups acting on closed manifolds) but with an emphasis on the dynamics induced on simplicial complexes. For…

We present four counterexamples in surface homology. The first example shows that even if the loops inducing a homology basis intersect each other at most once, they still may separate the surface into two parts. The other three examples…

By considering appropriate finite covering spaces of closed non-orientable surfaces, we construct linear representations of their mapping class group which have finite index image in certain big arithmetic groups.

We relate the author's Lie cobracket in the module additively generated by loops on a surface with the Connes-Kreimer Lie bracket in the module additively generated by trees. To this end we introduce a pre-Lie coalgebra and a (commutative)…

We introduce explicit families of good interpolation points for interpolation on a triangle in $\mathbb{R}^2$ that may be used for either polynomial interpolation or a certain rational interpolation for which we give explicit formulas.

This note deals with two topics of linear algebra. We give a simple and short proof of the multiplicative property of the determinant and provide a constructive formula for rotations. The derivation of the rotation matrix relies on simple…