David Nadler of UC Berkeley delivers this symplectic geometry seminar talk, part of the joint IAS/Princeton/Montreal/Paris/Tel Aviv Symplectic Geometry Zoominar series. He introduces microlocal sheaves on Weinstein manifolds, a framework for studying symplectic spaces through sheaf-theoretic invariants, and surveys recent results drawn from papers he co-authored with Laurent Cote, Chris Kuo, Wenyuan Li, and Vivek Shende. The talk sits at the research frontier of symplectic topology, addressing the tension in the title between local or singular descriptions of these sheaves and more global structural statements. Expect blackboard-style mathematical argument aimed at specialists already familiar with Weinstein manifolds, Floer theory, and microlocal analysis rather than an introductory overview. The seminar format means definitions move quickly and the payoff is in the survey of how several recent papers fit together into a coherent picture of singular support and globalization phenomena in this setting.