Related papers: Discretization of multidimensional submanifolds as…

We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on larger parameter…

A scheme to form a basis and a frame for a Hilbert space of quaternion valued square integrable function from a basis and a frame, respectively, of a Hilbert space of complex valued square integrable functions is introduced. Using the…

Motivated by recent interest in the spectrum of the Laplacian of incomplete surfaces with isolated conical singularities, we consider more general incomplete m-dimensional manifolds with singularities on sets of codimension at least 2. With…

Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine…

We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual, if their…

We consider in this paper discrete improper affine spheres based on asymptotic nets. In this context, we distinguish the discrete edges and vertices that must be considered singular. The singular edges can be considered as discrete cuspidal…

We provide a convincing discretisation of Demoulin's $\Omega$-surfaces along with their specialisations to Guichard and isothermic surfaces with no loss of integrable structure.

Let $\Delta$ denote a non-degenerate $k$-simplex in $\mathbb{R}^k$. The set $\text{Sim}(\Delta)$ of simplices in $\mathbb{R}^k$ similar to $\Delta$ is diffeomorphic to $O(k)\times [0,\infty)\times \mathbb{R}^k$, where the factor in $O(k)$…

In this article, first we give two formulae for the delta invariant of a complex curve singularity that can be embedded as a ${\mathbb Q}$-Cartier divisor in a normal surface singularity with rational homology sphere link. Next, we consider…

The spin network simulator model represents a bridge between (generalized) circuit schemes for standard quantum computation and approaches based on notions from Topological Quantum Field Theories (TQFT). More precisely, when working with…

Based on the well-established theory of discrete conjugate nets in discrete differential geometry, we propose and examine discrete analogues of important objects and notions in the theory of semi-Hamiltonian systems of hydrodynamic type. In…

This paper develops a discrete theory of real Riemann surfaces based on quadrilateral cellular decompositions (quad-graphs) and a linear discretization of the Cauchy-Riemann equations. We construct a discrete analogue of an antiholomorphic…

It is known that the spin structure on a Riemannian manifold can be extended to noncommutative geometry using the notion of a spectral triple. For finite geometries, the corresponding finite spectral triples are completely described in…

Recently, it has been established that the discrete star Laplace and the discrete Dirac operator, i.e. the discrete versions of their continuous counterparts when working on the standard grid, are rotation-invariant. This was done starting…

We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are…

We discuss special quantum group (secret) symmetries of the integrable system associated to the AdS/CFT correspondence. These symmetries have by now been observed in a variety of forms, including the spectral problem, the boundary…

We define the notion of spectral network on manifolds of dimension $\le 3$. For a manifold $X$ equipped with a spectral network, we construct equivalences between Chern-Simons invariants of flat ${\mathrm {SL}}(2,{\mathbb C})$-bundles over…

The Madsen-Tillmann spectra defined by categories of three- and four-dimensional Spin manifolds have a very rich algebraic structure, whose surface is scratched here.

For a positive integer r, an r-spin topological quantum field theory is a 2-dimensional TQFT with tangential structure given by the r-fold cover of SO_2 . In particular, such a TQFT assigns a scalar invariant to every closed r-spin surface…

Given a negatively curved geodesic metric space M, we study the asymptotic penetration behaviour of geodesic lines of M in small neighbourhoods of closed geodesics and of other compact convex subsets of M. We define a spiraling spectrum…