Activities This Week

A Grassmannian complex is a family of linear subspaces of a given linear space, closed under inclusion. In the talk we will explore the properties of Grassmannian complexes over a finite field and define boundary maps that give rise to notions of connectivity and high dimensional expansion. In contrast to the simplex, where all the homology groups are trivial, the complete Grassmannian (consisting of all subspaces of a given linear space) may have a non-trivial homology, and other exciting phenomena.

We will show analogues to the theorems of Linial, Meshulam and Wallach on the expansion of the complete Grassmannian, and to the phase transition of the connectivity of a random complex.

If time permits, we will discuss related extremal problems and topological overlap.

Based on joint work with Ran Tessler.

The noncommutative (nc) disc algebra $\mathcal{A}_d$ was studied extensively first by Popescu. It is the norm closed operator algebra generated by the left creation operators on the full Fock space. This algebra is semi-Dirichlet. Namely, $\mathcal{A}_d^* \mathcal{A}_d \subset \overline{\mathcal{A}_d + \mathcal{A}_d^*} = \mathcal{S}_d$. Therefore, one can perform a GNS type construction to obtain representations of $\mathcal{A}_d$ from states on $\mathcal{S}_d.$ This observation is one of the ingredients in the nc Clark theory developed by Jury and Martin.

In this talk, I will focus on nc rational functions and, in particular, inner ones. I will show how one obtains from an nc inner rational a finitely-correlated state (Bratteli and Jorgensen) on the Cuntz algebra. Connect the finitely-correlated states to minimal isometric dilations of finite-dimensional row coisometries and the work of Davidson, Kribbs, and Shpigel. Lastly, I will show that many finitely-correlated states are peak states in the sense of Clouatre and Thomson.

This talk is based on joint works with Mike Jury and Rob Martin.