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.     

Sieving and expanders

Let V be an orbit in ${\mathbb{Z}}^n$ of a finitely generated subgroup Λ of ${\mathrm{GL}}_n(\mathbb{Z})$ whose Zariski closure $\mathrm{Zcl}(\Lambda )$ is suitably large (e.g. isomorphic to ${\mathrm{SL}}_2$). We develop a Brun combinatorial sieve for estimating the number of points on V for which a fixed set of integral polynomials take prime or almost prime values. A crucial role is played by the expansion property of the ‘congruence graphs’ that we associate with V. This expansion property is established when $\mathrm{Zcl}(\Lambda )={\mathrm{SL}}_2$. To cite this article: J. Bourgain et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

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.     

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.