Michal Szachniewicz of the Institute for Advanced Study presents joint work with Antoine Sedillot on sup-norms of sections of metrized line bundles for families of arithmetic varieties, delivered as part of the Workshop on Recent Developments in Hodge Theory and O-minimality held in Simonyi Hall. He builds the talk around Riemann-Zariski spaces, the inverse limits of all blow-ups of a variety, and shows how formulas derived from this construction yield new definability results in the setting of globally valued fields, a framework bridging arithmetic geometry and model theory. Szachniewicz lays out the technical setup carefully before turning to motivation, situating the sup-norm formulas within broader questions about tameness and definability in arithmetic contexts. The talk assumes graduate-level familiarity with algebraic geometry and model theory, and runs just over an hour as a focused research seminar rather than an introductory survey, aimed at specialists working at the intersection of Hodge theory, o-minimality, and arithmetic geometry.