Related papers: Frieze patterns
This basic introduction to tropical geometry is hopefully accessible to a first years student in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's patchworking.…
We investigate special points on the Grassmannian which correspond to friezes with coefficients in the case of rank two. Using representations of arithmetic matroids we obtain a theorem on subpolygons of specializations of the coordinate…
These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM.
These notes are designed for those who either plan to work in differential geometry, or at least want to have a good reason not to do it. We discuss smooth curves and surfaces -- the main gate to differential geometry. We focus on the…
We develop the notion of crystal in the context of derived algebraic geometry, and to connect crystals to more classical objects such as D-modules.
We explore basic properties of number friezes, due to Conway and Coxeter, and their relations to decompositions of rational numbers into continued fractions, Farey sequences, and the modular group acting on the hyperbolic plane. These are…
The array of micro-prisms was described by model of multi-period blazed gratings consisting of triangular apertures. The origins of hexagram-shaped diffraction patterns were interpreted based on multiple-beam interference and diffraction…
In this work we propose a geometric view of the azeotropy problem, using some simplified models. We demonstrate that the occurrence of azeotropes in binary mixtures can be viewed-geometrically-as the intersection of curves in the plane (for…
An expository approach is given on the relationship between algebraic and geometric approaches to properties of isometries in the plane and the 2-sphere.
I give a short review of the theory of twisted symmetries of differential equations, emphasizing geometrical aspects. Some open problems are also mentioned.
This paper constructs a Riemann surface associated to the icosahedron and discusses the geodesics associated to a flat metric on this surface. Because of the icosahedral symmetry, this is a distinguished special case of the example treated…
In this paper the problem of finding a normal form of triangles and plane quadrilaterals up to similarity is considered. Several normal forms for triangles and a normal form for quadrilaterals of special case are described. Normal forms of…
In this work we derive a class of geometric flow equations for metric-scalar systems. Thereafter, we construct them from some general string frame action by performing volume-preserving fields variations and writing down the associated…
In this paper, we study the differential smoothness of diffusion algebras.
This paper aims to introduce high school students to the intriguing world of continued fractions, a mathematical concept that provides a unique representation of numbers. The study focuses on the exploration and development of the…
Inspired by the root systems of Lie algebras of rank 2, we propose a mathematical method to engineer new 2D materials with double periodic structures tessellating the plane. Concretely, we investigate two geometries relaying on squares and…
We present an overview of some recent developments in the theory of generalized formal series, grounded in diffeological geometric framework. These constructions aim to offer new tools for understanding infinite-dimensional phenomena in…
This paper is about Information Geometry, a relatively new subject within mathematical statistics that attempts to study the problem of inference by using tools from modern differential geometry. This paper provides an overview of some of…
In this paper, flows of a viscid fluids on curves are considered. Symmetry algebras and the corresponding fields of differential invariants are found. We study their dependence on thermodynamic states of media, and provide classification of…
Jets frames, that is a generalisation of ordinary frames on a manifold, are described in a language similar to that of gauge theory. This is achieved by constructing the Cartan geometry of a manifold with respect to the diffeomorphism…