Related papers: Geometric realization for substitution tilings

Let $H$ be a complex Hilbert space and let ${\mathcal P}(H)$ be the associated projective space (the set of rank-one projections). Suppose that $\dim H\ge 3$. We prove the following Wigner-type theorem: if $H$ is finite-dimensional, then…

We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant related to a subgroup of rotations and…

Some positive answers to the problem of endowing a dynamical system with a Hamiltonian formulation are presented within the class of Poisson structures in a geometric framework. We address this problem on orientable manifolds and by using…

Semi-topological Galois theory associates a canonical finite splitting covering to a monic Weierstrass polynomial. The inverse limit of the corresponding deck groups defines the absolute semi-topological Galois group, $\PiST(X,x)$. This…

The continuity, in a suitable topology, of algebraic and geometric operations on real analytic manifolds and vector bundles is proved. This is carried out using recently arrived at seminorms for the real analytic topology. A new…

We study compatible families of four-dimensional Galois representations constructed in the \'{e}tale cohomology of a smooth projective variety. We prove a theorem asserting that the images will be generically large if certain conditions are…

A combinatorial substitution is a map over tilings which allows to define sets of tilings with a strong hierarchical structure. In this paper, we show that such sets of tilings are sofic, that is, can be enforced by finitely many local…

The aim of this paper is to give the geometric realization of regular path complexes via (co)homology groups with coefficients in a ring $R$. Concretely, for each regular path complex $P$, we associate it with a singular $\Delta$-complex…

This paper aims to give some examples of diffeomorphic (or homeomorphic) low-dimensional complete intersections, which can be considered as a geometrical realization of classification theorems about complete intersections. A conjecture of…

This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include…

We study the middle convolution of local systems on the punctured affine line in the setting of singular cohomology and in the setting of \'etale cohomology. We derive a formula to compute the topological monodromy of the middle convolution…

We prove that for the uniquely ergodic ${\bf R}^d$ action associated with a primitive substitution tiling of finite local complexity, every measurable eigenfunction coincides with a continuous function almost everywhere. Thus, topological…

For any primitive substitution whose Perron eigenvalue is Pisot unit, we construct a domain exchange measurably conjugated to the subshift. And we give a condition for the subshift to be a finite extension of a torus translation. For the…

We study the topology and dynamics of subshifts and tiling spaces associated to non-primitive substitutions in one dimension. We identify a property of a substitution, which we call tameness, in the presence of which most of the possible…

On a compact connected Lie group $G$, we study the global solvability and the cohomology spaces of the differential complex associated with an essentially real involutive structure that is invariant under left translations. We prove that…

For every half-translation surface with marked points $(M,\Sigma)$, we construct an associated tessellation $\Pi(M,\Sigma)$ of the Poincar\'e upper half plane whose tiles have finitely many sides and area at most $\pi$. The tessellation…

Recent work has shown that two-dimensional non-linear $\sigma$-models on group manifolds with Poisson-Lie symmetry can be understood within generalised geometry as exemplars of generalised parallelisable spaces. Here we extend this idea to…

Briefly: Using a novel $(1,1)$ superspace formulation of semichiral sigma models with $4D$ target space, we investigate if an extended supersymmetry in terms of semichirals is compatible with having a $4D$ target space with torsion. In more…

It is known that any Poisson manifold can be embedded into a bigger space which admites a description in terms of the canonical Poisson structure, i.e., Darboux coordinates. Such a procedure is known as a symplectic realization and has a…

The realizations of the basic representation of $\widehat\mathfrak{gl}_r$ are known to be parametrized by partitions of r and have an explicit description in terms of vertex operators on the bosonic/fermionic Fock space. In this paper, we…