Cosmetic surgeries and Heegaard Floer homology

For about 5 decades, it has been known that every 3-dimensional
space (aka 3-manifold) can be obtained by Dehn surgery on a "framed link".
Dehn surgery is an operation for building 3-manifolds that starts with a
collection of disjoint circles (a link) in the 3-sphere, "drills" out
tubular neighborhoods of these circles, and replaces them with solid tori
in a novel way - the precise re-insertion of the tori being specified by
the "framing". After explaining these notions in detail, and going over
some examples, we will focus on those 3-dimensional spaces for which the
collection of circles consists of just a single circle, i.e. a knot. In
this context, one can ask both about the uniqueness of the knot as well as
the uniqueness of the framing (with the knot fixed) that yield the same
3-dimensional space via Dehn surgery. The "Cosmetic Surgery Conjecture"
asserts that no two distinct Dehn surgeries on the same nontrivial knot
can ever yield the same 3-dimensional space.

While this conjecture, whose dynamic name is due to Peter Bleiler, is
still largely open, there has been some recent progress driven by Heegaard
Floer theory. The second half of the talk will focus on describing the
Heegaard Floer techniques used and detailing the results thus obtained.

Most of the talk will be accessible to a general mathematical audience.