Georgia Institute of Technology College of Sciences

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.

The Dehn Nielsen Baer Theorem states that the extended mapping class group is isomorphic to the outer automorphisms of π1(Sg). The theorem highlights the connection between the topological invariant of distinct symmetries of a space and its fundamental group. This talk will incorporate ideas from algebra, topology, and hyperbolic geometry!

This is the sixth (and last) of several talks discussing embeddings of manifolds. I will discuss some general results for smooth manifolds, but focus on embeddings of contact manifolds into other contact manifolds. Particular attention will be paid to embeddings of contact 3-manifolds in contact 5-manifolds. I will discuss two approaches to this last problem that are being developed jointly with Yanki Lekili.

The algebraic K-theory of the sphere spectrum, K(S), encodes significant information in both homotopy theory and differential topology. In order to understand K(S), one can apply the techniques of chromatic homotopy theory in an attempt to approximate K(S) by certain localizations K(L_n S). The L_n S are in turn approximated by the Johnson-Wilson spectra E(n) = BP[v_n^{-1}], and it is not unreasonable to expect to be able to compute K(BP). This would lead inductively to information about K(E(n)) via the conjectural fiber sequence K(BP) --> K(BP) --> K(E(n)).