Related papers: A Very Intuitive Geometric Picture of the 24-cell,…
We investigate the use of programmable optical lattices for quantum simulation of Hubbard models, determining analytic expressions for the hopping and Hubbard U, finding that they are suitable for emulating strongly correlated systems with…
We use the automorphism group $Aut(H)$, of holes in the lattice $L_8=A_2\oplus A_2\oplus D_4$, as the starting point in the construction of sphere packings in 10 and 12 dimensions. A second lattice, $L_4=A_2\oplus A_2$, enters the…
This work presents a computational method for the design of architected truss lattice materials where each strut can be made of one of a set of available materials. We design the lattices to extremize effective properties. As customary in…
Algebraic lattices are those obtained from modules in the ring of integers of algebraic number fields through the canonical or twisted embeddings. In turn, well-rounded lattices are those with maximal cardinality of linearly independent…
Haptic sensation is an important modality for interacting with the real world. This paper proposes a general framework of inferring haptic forces on the surface of a 3D structure from internal deformations using a small number of physical…
We present a general way to define a topology on orthomodular lattices. We show that in the case of a Hilbert lattice, this topology is equivalent to that induced by the metrics of the corresponding Hilbert space. Moreover, we show that in…
We propose a construction of lattices from codes corresponding to lattices of type $A_n$, $D_n$ and $E_n$. This construction is a generalization of construction A of lattices from $p$-ary codes corresponding to a lattice of type $A_{p-1}$.…
We show that under certain symmetry, the images of complete harmonic embeddings from the complex plane into the hyperbolic plane is completely determined by the geometric information of the vertical measured foliation and is independent of…
We here define a cell structure for real, complex and quaternionic flag manifolds in a unified way. Our method is geometric in nature and is inspired from a method due to Milnor and Stasheff, which they used to define a cell structure for…
Hyperbolic lattices are a revolutionary platform for tabletop simulations of holography and quantum physics in curved space and facilitate efficient quantum error correcting codes. Their underlying geometry is non-Euclidean, and the absence…
We prove that there are precisely two embedded exact Lagrangian fillings of the standard Legendrian Hopf link, up to compactly supported Hamiltonian isotopy. It was known that the standard Legendrian Hopf link admitted at least two such…
In this paper, using compute-and-forward as an example, we provide an overview of constructions of lattices from codes that possess the right algebraic structures for harnessing interference. This includes Construction A, Construction D,…
We study the vortex formation in optical lattices submitted to artificial gauge potentials. We compute the superfluid density for Abelian and non-Abelian gauge potentials with a mean-field approach of the Bose-Hubbard model and we determine…
Let $L/F$ be a Galois extension of fields with Galois group isomorphic to the quaternion group of order $ 8 $. We describe all of the Hopf-Galois structures admitted by $ L/F $, and determine which of the Hopf algebras that appear are…
Motivated by recent realizations of two-dimensional (2D) superconducting-qubit lattices, we propose a protocol to simulate Hofstadter butterfly with synthetic gauge fields in superconducting circuits. Based on the existing 2D…
We consider the reductions of $2p$-dimensional particle system ($p=2,4,8$), associated with the Hopf map. For the third Hopf map we explicitly construct the functions associated to the symmetry related to the rotations in the fiber.
We construct a new 2-parameter family E_mn of self-dual 2-simple and 2-simplicial 4-polytopes, with flexible geometric realisations. E_44 is the 24-cell. For large m,n the f-vectors have ``fatness'' close to 6. The E_t-construction of…
Latent fibrations are an adaptation, appropriate for categories of partial maps (as presented by restriction categories), of the usual notion of fibration. The paper initiates the development of the basic theory of latent fibrations and…
We introduce new partial order structures on the underlying sets of free nonsymmetric operads. These posets involve decorated ordered rooted trees, and their terminal intervals are lattices. These lattices are not graded, not self-dual, and…
We use the Hopf fibration to explicitly compute generators of the second homotopy group of the flag manifolds of a compact Lie group. We show that these $2$-spheres have nice geometrical properties such as being totally geodesic surfaces…