The colorful Helly theorem and colorful resolutions of ideals

We demonstrate that the topological Helly theorem and the algebraic Auslander-Buchsbaum may be viewed as different versions of the same phenomenon. Using this correspondence we show how the colorful Helly theorem of I.Barany and its generalizations by G.Kalai and R.Meshulam translates to the algebraic side. Our main results are algebraic generalizations of these translations, which in particular gives a syzygetic version of Hellys theorem.Comment: 13 pages, minor change

Asymptotic Lattices, Good Labellings, and the Rotation Number for Quantum Integrable Systems

This article introduces the notion of good labellings for asymptotic lattices in order to study joint spectra of quantum integrable systems from the point of view of inverse spectral theory. As an application, we consider a new spectral quantity for a quantum integrable system, the quantum rotation number. In the case of two degrees of freedom, we obtain a constructive algorithm for the detection of appropriate labellings for joint eigenvalues, which we use to prove that, in the semiclassical limit, the quantum rotation number can be calculated on a joint spectrum in a robust way, and converges to the well-known classical rotation number. The general results are applied to the semitoric case where formulas become particularly natural