(Joint work in progress with Curtis Pro)
I will survey a problem that has a beautiful history that goes back to
Cheeger's 1967 thesis. It is connected to a large body of the Riemannian geometry
that was developed in the later part of the 20th century.
The talk will survey this history and briefly outline a strategy to prove
the following conjecture.
Conjecture 1. For any k in R, v > 0, and D > 0 there are only finitely many diffeomorphism types of Riemannian 4-manifolds with sectional curvature ≥ k; volume ≥ v, and diameter ≤ D.