Title: Classification of highly-connected manifolds
Abstract: The problem of classifying smooth, closed, highly-connected
n-manifolds (for n \geq 5) up to diffeomorphism has a long history in
the study of geometric topology—in the 1950s, Milnor's study of this
problem led to his discovery of exotic spheres. In this talk, I will
survey this classical problem and discuss its solution, which builds
on the work of many people, including Wall, Stolz, and Mahowald. This solution,
obtained in joint works with Burklund, Hahn and Zhang, relies on
modern techniques in the higher algebra of E_\infty-rings. Time
permitting, I will also explain how to generalize this classification
from highly-connected manifolds to the wider class of
metastably-connected manifolds.