David Bachman

Title: Topological index theory for surfaces in 3-manifolds

Abstract: We utilize the curve complex to define an isotopy invariant
index of a surface in a 3-manifold, and show that it mimics the index
of a (unstable) minimal surface. This allows us to find purely
topological analogues to familiar techniques from geometry, such as
barrier arguments.  Deep results follow concerning Heegaard
splittings, bridge positions of knots, and normal surfaces. In
addition, this new viewpoint opens up a host of exciting new questions
for the field of 3-manifold topology.