Abstract: With motivation from condensed matter physics and materials science in collaboration with physicists, I’ll discuss explicit examples of eigenbundles associated to both gapped and gapless quantum mechanical systems, that is, to families of matrices with prescribed symmetry, as well as their homotopical classifications.  These include Hopf bundles and, more generally, certain rank-2 and rank-3 Higgs bundles.  The geometry and topology of such mathematical objects and their moduli spaces shed light on questions in physics, namely, hyperbolic band theory (after A. J. Kollár et al.) and bulk–edge correspondence, which I will at least indicate.

Abstract: The ring spectrum of spheres plays a similar role in spectral algebraic geometry that the ring of integers does for classical algebraic geometry.   It recovers the latter in a natural way while encoding additional higher-homotopical information.   With motivation from structure and computation centered around the homotopy groups of spheres, I’ll explain joint work with Xuecai Ma of Westlake University.   We define relative effective Cartier divisors for a spectral Deligne–Mumford stack.   We then apply the representability of their moduli to questions at the crossroad of algebraic topology and arithmetic geometry.