Let me start with some overdue news, finally released after passive aggressive messages and open bullying. And England’s prime minister got usurped by lettuce. I promise I will start making more sense now.

Erdős and Szekeres famously showed that every set of general position points in the plane contains a large subset of points that is in convex position. And quite recently Andrew Suk improved this so much, that by now we have an essentially tight bound on how many points we can find in convex position, at least in the plane. This was improved a little furtherby Holmsen, Mojarrad, Pach and Tardos here.

But some found that all too plain, and pointed out that in higher dimensions, convex sets could have an even harder time hiding, and that there could be substantially larger convex sets in higher dimensional pointclouds. Alas, noone could prove this until recently, when my friend Cosmin provided, in joint work with Dmitrii Zakharov the first substantial improvement in dimension 3. Congratulations Dima and Cosmin (and see you soon, looking forward to a wintery northeast).

Secondly, our (announcement) preprint on the unimodality of the h*-polynomial is finally online. One curious discovery in this research is that there is a Parseval-Raleigh type identity (Lemma 5.2) for the degree map of the Chow ring associated to the monoid.

Seen as a rational function.

In characteristic two.

In dependence on the torus action/the linear system of parameters.

This really remains to be understood, and we have no clue where it comes from. No phenomenon like that seems to be explored, and we suspect some deep connections to residue theory.