Activities This Week

A cut-and-project set is constructed by restricting a lattice $L$ in $(d+m)$-space to a domain bounded in the last m coordinates, and projecting these points to the the space spanned by its d-dimensional orthogonal complement. These point sets constitute an important example of so-called quasicrystals.

During the talk, we shall present and give some classification results of the moduli spaces of cut-and-project sets, which were introduced by Marklof-Strömbergsson. These are obtained by considering the orbit closure of the special linear group in $d$-space acting on the lattice $L$ inside the space of unimodular lattices of rank $d+m$. Theorems of Ratner imply that these are meaningful objects.

We then describe quantitative counting result for patches in generic cut-and-project sets. Patches are local configuration of point sets whose multitude reflects aperiodicity.

The count follows some old argument of Schmidt using moment bounds. These bounds are obtained by integrability properties of the Siegel transform, which in turn follow from reduction theory and a symmetrisation argument of Rogers. This argument is of independent interest, giving an alternative account to recent work of Kelmer-Yu (which is based on the theory of Eisenstein series) on counting points in generic symplectic lattices.

This is a joint endeavour with Yotam Smilansky and Barak Weiss.

Attached

Following work of Evert, Helton, Klep and McCullough on free linear matrix inequality domains, we ask when a matrix convex set is the closed convex hull of its (absolute) extreme points. This is a finite-dimensional version of Arveson’s non-commutative Krein-Milman theorem, which may generally fail completely since some matrix convex sets have no (absolute) extreme points. In this talk we will explain why the Arveson-Krein-Milman property for a given matrix convex set is difficult to determine. More precisely, we show that this property for certain commuting tensor products of matrix convex sets is equivalent to a weak version of Tsirelson’s problem from quantum information. This weak variant of Tsirelson’s problem was shown, by a combination of results of Kirchberg, Junge et. al., Fritz and Ozawa, to be equivalent to Connes’ embedding conjecture; considered to be one of the most important open problems in operator algebras. We do more than just provide another equivalent formulation of Connes’ embedding conjecture. Our approach provides new matrix-geometric variants of weak Tsirelson type problems for pairs of convex polytopes, which may be easier to rule out than the original weak Tsirelson problem.

Based on joint work with Roy Araiza and Thomas Sinclair

For a quasi-split group $G$ over a local field $F$, with Borel subgroup $B=TU$ and Weyl group $W$, there is a natural geometric action of $G\times T$ on $L^2(X),$ where $X=G/U$ is the basic affine space of $G$. For split groups, Gelfand and Graev have extended this action to an action of $G\times (T\rtimes W)$ by generalized Fourier transforms $\Phi_w$. We shall extend this result for quasi-split groups, using a new interpretation of Fourier transforms for quasi-split groups of rank one.

This is joint work with David Kazhdan.