Dima Sinapova of Rutgers University presents a set theory seminar talk at the Institute for Advanced Study on the powerset function at singular cardinals. She frames the problem through the singular cardinal hypothesis, the analogue of the continuum hypothesis for cardinals that can be written as a union of fewer, smaller sets. Building on Cohen's forcing technique and its role in proving CH's independence, Sinapova explains why forcing constructions that violate SCH are notoriously difficult and intricate compared to CH-violating forcing. The talk centers on recent research results linking failures of SCH to combinatorial properties of the singular cardinal in question. Delivered to the IAS Set Theory Group, this is a technical research talk aimed at set theorists already familiar with forcing and cardinal arithmetic, covering current work rather than introductory material.