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 Image may be NSFW.
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Herzog and Srinivasan proved that, for monomial ideals,
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and Avramov, Conca and Iyengar conjectured that
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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).
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