Emmy Murphy

Loose Legendrians and the plastikstufe

We show that the presence of a plastikstufe induces a certain degree of flexibility in contact manifolds of dimension 2nC1> 3. More precisely, we prove that every Legendrian knot whose complement contains a “nice” plastikstufe can be destabilized (and, as a consequence, is loose). As an application, it follows in certain situations that two nonisomorphic contact structures become isomorphic after connect-summing with a manifold containing a plastikstufe. 57R17

Subflexible symplectic manifolds

We introduce a class of Weinstein domains which are sublevel sets of flexible Weinstein manifolds but are not themselves flexible. These manifolds exhibit rather subtle behavior with respect to both holomorphic curve invariants and symplectic flexibility. We construct a large class of examples and prove that every flexible Weinstein manifold can be Weinstein homotoped to have a nonflexible sublevel set.

Amenable groups and smooth topology of 4-manifolds

A smooth five-dimensional s-cobordism becomes a smooth product if stabilized by a finite number n of

Loose Legendrian Embeddings in High Dimensional Contact Manifolds

S^2xS^2x[0,1]

Constructing exact Lagrangian immersions with few double points.

s. We show that for amenable fundamental groups, the minimal n is subextensive in covers, i.e., n(cover)/index(cover) has limit 0. We focus on the notion of sweepout width, which is a bridge between 4-dimensional topology and coarse geometry.