Related papers: Exploring the "Rubik's Magic" universe
The structure of topological spaces is analysed here through the lenses of fibrous preorders. Each topological space has an associated fibrous preorder and those fibrous preorders which return a topological space are called spacial. A…
Finite projective (lattice) geometries defined over rings instead of fields have recently been recognized to be of great importance for quantum information theory. We believe that there is much more potential hidden in these geometries to…
Puzzles based on coloured cubes and other coloured geometrical figures have a long history in the recreational mathematical literature. One of the most commercially famous of these puzzles is the Instant Insanity that consists of four…
We show how to construct a cubic partial cube from any simplicial arrangement of lines or pseudolines in the projective plane. As a consequence, we find nine new infinite families of cubic partial cubes as well as many sporadic examples.
In this paper, we study additively indecomposable quadratic forms over real biquadratic and simplest cubic fields. In particular, we show that over these fields, we can always find such a classical form in 2 variables, which differs from…
A new topological invariant of closed connected orientable four-dimensional manifolds is proposed. The invariant, constructed via surgery on a special link, is a four-dimensional counterpart of the celebrated SU(2) three-manifold invariant…
The first part of this work constructs positive-genus real Gromov-Witten invariants of real-orientable symplectic manifolds of odd "complex" dimensions; the present part focuses on their properties that are essential for actually working…
We introduce a version of Farber's topological complexity suitable for investigating mechanical systems whose configuration spaces exhibit symmetries. Our invariant has vastly different properties to the previous approaches of Colman-Grant,…
Magic sets of observables are minimal structures that capture quantum state-independent advantage for systems of $n\ge 2$ qubits and are, therefore, fundamental tools for investigating the interface between classical and quantum physics. A…
This paper presents a type theory in which it is possible to directly manipulate $n$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways…
In this paper we find the most general self-similar, homogeneous and isotropic, Ricci flat cosmologies in 5D. These cosmologies show a number of interesting features: (i) the field equations allow a complete integration in terms of one…
Going beyond the cohomological invariants attached to tiling spaces via inverse limit constructions, Clark and Hunton introduced shape group invariants, and showed these invariants in dimension one give new information. We show for…
It is a classical result that there are $12$ (irreducible) rational cubic curves through $8$ generic points in $\mathbb{P}_{\mathbb{C}}^2$, but little is known about the non-generic cases. The space of $8$-point configurations is…
The Birman-Williams theorem gives a connection between the collection of unstable periodic orbits (UPOs) contained within a chaotic attractor and the topology of that attractor, for three-dimensional systems. In certain cases, the fractal…
We survey two decades of work on the (sequential) topological complexity of configuration spaces of graphs (ordered and unordered), aiming to give an account that is unifying, elementary, and self-contained. We discuss the traditional…
We construct a smooth complex projective rational surface with infinitely many mutually non-isomorphic real forms. This gives the first definite answer to a long standing open question if a smooth complex projective rational surface has…
Toric $t$-designs, or equivalently $t$-designs on the diagonal subgroup of the unitary group, are sets of points on the torus over which sums reproduce integrals of degree $t$ monomials over the full torus. Motivated by the projective…
This is a survey of the geometry of complex cubic fourfolds with a view toward rationality questions. Topics include classical constructions of rational examples, Hodge structures and special cubic fourfolds, associated K3 surfaces and…
Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…
A new construction of naturally reductive spaces is presented. This construction gives a large amount of new families of naturally reductive spaces. First the infinitesimal models of the new naturally reductive spaces are constructed. A…