Maya Sankar of the Institute for Advanced Study gives this Computer Science/Discrete Mathematics seminar on extremal combinatorics. The Turán problem asks, for a fixed forbidden hypergraph F, how many edges an F-free hypergraph can have as a function of its vertex count. Sankar opens with a survey of foundational tools in the field, including Lagrangian methods and supersaturation arguments, before turning to her own recent result on the Turán numbers of long tight cycles, a hypergraph generalization of ordinary cycles. The core technical move is a hypergraph analogue of the classical fact that a graph is bipartite exactly when it has no odd closed walks, recast here as a coloring condition on tuples of vertices. She works through the proof in full and shows how it unifies earlier results in the three-uniform case by Kamčev-Letzter-Pokrovskiy and Balogh-Luo, extending the approach to a wider family of cycle-like hypergraphs. The talk runs just over two hours, delivered at IAS's Simonyi Hall.