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.     

Nonlinear Fourier transforms for the sine-Gordon equation in the quarter plane

Using the Unified Transform, also known as the Fokas method, the solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann–Hilbert problem whose definition involves four spectral functions $a,b,A,B$. The functions $a(k)$ and $b(k)$ are defined via a nonlinear Fourier transform of the initial data, whereas $A(k)$ and $B(k)$ are defined via a nonlinear Fourier transform of the boundary values. In this paper, we provide an extensive study of these nonlinear Fourier transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values. The results can be used to determine the long-time asymptotics of the sine-Gordon quarter-plane solution via nonlinear steepest descent techniques.     

The division problem for tempered distributions of one variable

We give a characterization, in one variable case, of those $C^{\infty }$ multipliers F such that the division problem is solvable in $S^{\prime }(\mathbb{R})$. For these functions $F\in O_M(\mathbb{R})$ we even prove that the multiplication operator $M_F(G)=FG$ has a continuous linear right inverse on $S^{\prime }(\mathbb{R})$, in contrast to what happens in the several variables case, as was shown by Langenbruch.     

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.     

Subproduct systems over N×N

We develop the theory of subproduct systems over the monoid $\mathbb{N}\times \mathbb{N}$, and the non-self-adjoint operator algebras associated with them. These are double sequences of Hilbert spaces ${\{X(m,n)\}}_{m,n=0}^{\infty }$ equipped with a multiplication given by coisometries from $X(i,j)?X(k,l)$ to $X(i+k,j+l)$. We find that the character space of the norm-closed algebra generated by left multiplication operators (the tensor algebra) is homeomorphic to a complex homogeneous affine algebraic variety intersected with a unit ball. Certain conditions are isolated under which subproduct systems whose tensor algebras are isomorphic must be isomorphic themselves. In the absence of these conditions, we show that two numerical invariants must agree on such subproduct systems. Additionally, we classify the subproduct systems over $\mathbb{N}\times \mathbb{N}$ by means of ideals in algebras of non-commutative polynomials.     

The Ramsey property for Banach spaces and Choquet simplices,and applications

We show that the class of finite-dimensional Banach spaces and the class of finite-dimensional Choquet simplices have the Ramsey property. As an application, we show that the group $\mathrm{Aut}(\mathbb{G})$ of surjective linear isometries of the Gurarij space $\mathbb{G}$ is extremely amenable, and that the canonical action $\mathrm{Aut}(\mathbb{P})?\mathbb{P}$ is the universal minimal flow of the group $\mathrm{Aut}(\mathbb{P})$ of affine homeomorphisms of the Poulsen simplex $\mathbb{P}$. This answers questions of Melleray–Tsankov and Conley–Törnquist.     

Lehmer's totient problem over Fq[x]

In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\in {\mathbb{F}}_q[x]$ and $\phi (q,p(x))$ be the Euler's totient function of $p(x)$ over ${\mathbb{F}}_q[x]$, where ${\mathbb{F}}_q$ is a finite field with q elements. We prove that $\phi (q,p(x))|(q^{\mathrm{deg}(p(x))}?1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3$, $p(x)$ is the product of any 2 non-associate irreducibles of degree 1; or (iii) $q=2$, $p(x)$ is the product of all irreducibles of degree 1, all irreducibles of degree 1 and 2, and the product of any 3 irreducibles one each of degree 1, 2 and 3.