Related papers: Tiling Spaces have aspherical shape
We present a new construction of tubular neighborhoods in (possibly infinite dimensional) Riemannian manifolds M, which allows us to show that if G is an arbitrary group acting isometrically on M, then every G-invariant submanifold with…
This is a survey of topological properties of open, complete nonpositively curved manifolds which may have infinite volume. Topics include topology of ends, restrictions on the fundamental group, as well as a review of known examples.
We give a complete classification of edge-to-edge tilings of the sphere by regular polygons under a unified framework. Without assuming convexity of the tiles or polyhedrality of the underlying graph, our proof is independent of the…
It is shown that there are primitive substitution tilings with dense tile orientations invariant under n-fold rotation for n=2,3,4,5,6,8. The proof for dense tile orientations uses a general result about irrationality of angles in certain…
We classify edge-to-edge tilings of the sphere by congruent almost equilateral pentagons, in which four edges have the same length. Together with our earlier classifications of edge-to-edge tilings of the sphere by congruent equilateral…
In 2023, two striking, nearly simultaneous, mathematical discoveries have excited their respective communities, one by Greenfeld and Tao, the other (the Hat tile) by Smith, Myers, Kaplan and Goodman-Strauss, which can both be summed up as…
We introduce a formalism for handling general spaces of hierarchical tilings, a category that includes substitution tilings, Bratteli-Vershik systems, S-adic transformations, and multi-dimensional cut-and-stack transformations. We explore…
The space of topological decompositions into triangulations of a surface has a natural graph structure where two triangulations share an edge if they are related by a so-called flip. This space is a sort of combinatorial Teichm\"uller space…
We define the notion of a holomorphic bundle on the noncommutative toric orbifold $T_{\theta}/G$ associated with an action of a finite cyclic group $G$ on an irrational rotation algebra. We prove that the category of such holomorphic…
We investigate the dynamics of tiling dynamical systems and their deformations. If two tiling systems have identical combinatorics, then the tiling spaces are homeomorphic, but their dynamical properties may differ. There is a natural map…
We find universal spaces for Alexandroff and finite spaces and explore some of its topological properties as well as their description as inverse limits of finite spaces and Alexandroff extensions. They can be used as a natural environment…
This short note is some obvious mathematical addendum to our papers on Wilson loops on polygon-like contours with circular edges \cite{Dorn:2020meb,Dorn:2020vzj}. Using the technique of osculating spheres and circles we identify the…
We consider closed and orientable immersed hypersurfaces of translational manifolds. Given a vector field on such a hypersurface, we define a perturbation of its Gauss map, which allows us to obtain topological invariants for the immersion…
We show how the Turaev--Viro invariant can be understood within the framework of Chern--Simons theory with gauge group SU(2). We also describe a new invariant for certain class of graphs by interpreting the triangulation of a manifold as a…
We present a detailed study of the curvature and symplectic asphericity properties of symmetric products of surfaces. We show that these spaces can be used to answer nuanced questions arising in the study of closed Riemannian manifolds with…
For a finite planar graph, it associates with some metric spaces, called (regular) spherical polyhedral surfaces, by replacing faces with regular spherical polygons in the unit sphere and gluing them edge-to-edge. We consider the class of…
The fundamental representations of the special linear group ${\rm SL}_n$ over the complex numbers are the exterior powers of $\mathbb{C}^n$. We consider the invariant rings of sums of arbitrary many copies of these ${\rm SL}_n$-modules. The…
We show that $U(k)$-invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in $\mathfrak{gl}(k,{\mathbb C})$ correspond to algebraic curves $C$ of genus $(k-1)^2$, equipped with a flat projection…
In this paper, generalizing the techniques of Bour's theorem, we prove that every generic cuspidal edge, more generally, generic $n$-type edge, which is invariant under a helicoidal motion in Euclidean $3$-space admits non-trivial isometric…
Recent studies showed that topologically trivial insulators may have fractionally quantized corner charges due to the topological invariant called a filling anomaly. Such crystal shapes in three dimensions are restricted to…