Georgia Institute of Technology College of Sciences

TBA

Under the operation of connected sum, the set of knots in the 3-sphere forms a monoid. Modulo an equivalence relation called concordance, this monoid becomes a group called the knot concordance group. We will consider various algebraic methods -- both classical and modern -- for better understanding the structure of this group.

Groups, rings, modules, and compact Hausdorff spaces have underlying sets ("forgetting" structure) and admit "free" constructions. Moreover, each type of object is completely characterized by the shadow of this free-forgetful duality cast on the category of sets, and this syntactic encoding provides formulas for direct and inverse limits.

I will present a result of Klarreich on the boundary at infinity of the complex of curves of a compact orientable surface. The complex of curves is a delta-hyperbolic space so it has a boundary which is the set of equivalence classes of quasi-geodesic rays. Klarreich shows that the resulting space is homeomorphic to the space of minimal foliations of the surface.

The Lickorish-Wallace theorem states that every closed, connected, orientable three-manifold can be expressed as surgery on a link in the three-sphere (i.e., remove a neighborhood of a disjoint union of embedded $S^1$'s from $S^3$ and re-glue). It is natural to ask which three-manifolds can be obtained by surgery on a single knot in the three-sphere. We discuss a new way to obstruct integer homology spheres from being surgery on a knot and give some examples. This is joint work with Jennifer Hom and Cagri Karakurt.