Zoom link: https://princeton.zoom.us/j/96282936122

Passcode: 998749

I will review the classic problems that arise when attempting to classify smooth, closed (2n)-manifolds with three non-zero homology groups, going back to 1960s work of CTC Wall. Joint work with Burklund and Senger completes the classification away from finitely many dimensions, and work of Burklund and Senger analyzes the remaining dimensions using topological modular forms. Our proofs combine modern higher algebra with a classical vanishing region in pictures of the stable homotopy groups of spheres. Understanding this vanishing region, and its boundary, has general implications for m-manifolds with reduced homology concentrated above dimension m/3.