I had lunch with László Babai yesterday, and he asked me the following problem. I changed it up a little to make it less findable online (though I did not find the problem in this variant).

Say you have shakshuka with a friend every Monday at the same restaurant. The meal is always the same, but the price can vary every week; the restaurant adjusts it to market prizes for the many vegetables that enter the dish (shakshuka literally means something like mixed stuff from the market). There is no way of knowing what the prizes next week will be, and it can even be that the restaurant pays you a little for taking overdue vegetables from their hand.

Now your friend is quite gentlemanly, and every second week, he allows you to look at the check, and decide whether you want to pick up the bill, or let him do it, with the understanding that you alternate in the week after.

You are not such a polite person, and try to save money in a sneaky way. What should be your strategy?

Here, the strategy works even if your friend knows the strategy. And even if he has set the prices in advance for eternity.

It is one of those cases where your gutfeeling is correct.

The summer and spring was quite eventful and interesting, aside from lazy rhymes. Apart from exciting new work that I will talk about later, I just submitted a paper that proves subadditivity of shifts in minimal resolutions:

Given a homogeneous ideal I in a polynomial ring S, the quotient S/I admits a minimal resolution; a way of writing S/I at the end of an exact sequences involving shifted copies of S; the maximal shift in place i is denoted by .

Herzog and Srinivasan proved that, for monomial ideals,

and Avramov, Conca and Iyengar conjectured that

This is false already for binomial ideals, though, as Ein and Lazarsfeld showed. But for monomial ideals I showed it is true. In the end, it is a quite beautiful reduction to a vanishing theorem for lattices (as in, poset lattices) and a bit of of Eilenberg-Zilber shuffle products. You can grab the paper here.

Currently working on a rather cool construction due to Danzer. Incidentally, Ludwig Danzer is one of the first professors I had in a class (he taught a course in tiling theory that I attended) and I fell in love recently with a construction of his: given a simplicial complex X, he constructs a cubical complex such that the neighborhood of every vertex is isomorphic to X. If X is the clique complex of a graph, this cubical complex has the fundamental group that is the commutator of the associated right-angled Coxeter group. And the cohomology ring is quite interesting indeed. More later.

Aside from that, travels to Greece (marvelous new projects with Vasso Petrotou and Stavros Papadakis), Beijing (amazing conference organized by Yau) and Los Angeles (working with Igor).

The past few weeks/months/years have been interesting to say the least; and all surrounds the recent activities surrounding basic laws; my colleague David Enoch summarizes the situation here quite well.

I will leave it to him to explain it, and just add a sideobservation: Even though the struggle is against an eroding of constitutional principles in Israel primarily, I have the entirely subjective feeling that also discussions of the occupation and relations to the Palestinians have become more commonplace; perhaps it is the prospect of losing civil liberties, or it is awakening to the slowly increasing temperature in the cookpot, but I feel many discussions have been more open; and though there is more fighting among colleagues and friends, I also appreciate how alive and democracy are, not in the Knesset, perhaps, but on the street protesting.