Abstract: The quaternionic Hopf fibration
S7→ S4 lets us view
the round 7-sphere as a family of round 3-spheres. Dilating these
3-spheres by t > 0 yields a family of Riemannian 7-manifolds S7(t)
called "squashed 7-spheres." Each of these spaces admits a natural
(highly symmetric) co-closed G2-structure, making the squashed
7-spheres fundamental examples in G2-geometry.
In this talk, we construct the first non-trivial compact associative
3-folds in S7(t) for every t > 0. Our examples arise from
"twisting" circle bundles over pseudo-holomorphic curves in
CP3 by a meromorphic function. Time permitting, we explain
how our construction generalizes to the larger class of "squashed"
3-Sasakian 7-manifolds. This is joint work with Gavin Ball.