Discrete Tomography of Mathematcal Quasicrystals: A Primer

This text is a report on work progress. We introduce the class of cyclotomic model sets (mathematical quasicrystals) $\Lambda \subset \mathbb{Z}[{\xi }_n]$, where $\mathbb{Z}[{\xi }_n]$ is the ring of integers in the nth cyclotomic field $\mathbb{Q}({\xi }_n)$, and discuss the corresponding decomposition, consistency and reconstruction problems of the discrete tomography of these sets. Our solution of the so-called decomposition problem also applies to the case of the square lattice ${\mathbb{Z}}^2=\mathbb{Z}[{\xi }_4]$, which corresponds to the classical setting of discrete tomography.     

Deformation of central charges,vertex operator algebras whose Griess algebras are Jordan algebras

If a vertex operator algebra $V={\oplus }_{n=0}^{\infty }{V}_n$ satisfies $\dim V_0=1$, $V_1=0$, then $V_2$ has a commutative (nonassociative) algebra structure called Griess algebra. One of the typical examples of commutative (nonassociative) algebras is a Jordan algebra. For example, the set ${\mathrm{Sym}}_d(\mathbb{C})$ of symmetric matrices of degree d becomes a Jordan algebra. On the other hand, in the theory of vertex operator algebras, central charges influence the properties of vertex operator algebras. In this paper, we construct vertex operator algebras with central charge c and its Griess algebra is isomorphic to ${\mathrm{Sym}}_d(\mathbb{C})$ for any complex number c and a positive integer d.     

A topological nullstellensatz for tensor-triangulated categories

Let $\mathrm{Spec}(\mathbb{T})$ be the spectrum of a tensor-triangulated category $(\mathbb{T},?,\mathbf{1})$. We show that there is a homeomorphism between the spectral space of radical thick tensor ideals in $(\mathbb{T},?,\mathbf{1})$ and the collection of open subsets of $\mathrm{Spec}(\mathbb{T})$ in inverse topology. In fact, we prove a more general result in terms of supports on $(\mathbb{T},?,\mathbf{1})$ and work by combining methods from commutative algebra, topology and tensor triangular geometry.     

G-Ham Sandwich Theorems: Balancing measures by finite subgroups of spheres

Equivariant Ham Sandwich Theorems are obtained for the classical algebras $\mathbb{F}=\mathbb{R},\mathbb{C}\text{, and}\mathbb{H}$ and the finite subgroups G of their unit spheres. Given any n $\mathbb{F}$-valued Borel measures on ${\mathbb{F}}^n$ and any n-dimensional free $\mathbb{F}$-unitary representation of G, it is shown that there exists a Voronoi partition of ${\mathbb{F}}^n$ naturally determined by G which “G-balances” each measure, as realized by the simultaneous vanishing of each “G-average” of the measures of the partition?s isometric fundamental domains. Applications for real measures follow, among them that any n signed mass distributions on ${\mathbb{C}}^{(p?1)n/2}$ can be equipartitioned by a single complex regular p-fan if p is an odd prime.     

On the irreducible action of PSL(2,R) on the 3-dimensional Einstein universe

In this paper, we study the irreducible representation of $\text{PSL}(2,\mathbb{R})$ in $\text{PSL}(5,\mathbb{R})$. This action preserves a quadratic form with signature $(2,3)$. Thus, it acts conformally on the 3-dimensional Einstein universe $\mathbb{E}{\mathrm{in}}^{1,2}$. We describe the orbits induced in $\mathbb{E}{\mathrm{in}}^{1,2}$ and its complement in $\mathbb{R}{\mathbb{P}}^4$. This work completes the study in [2], and is one element of the classification of cohomogeneity one actions on $\mathbb{E}{\mathrm{in}}^{1,2}$[5].