Georgia Institute of Technology College of Sciences

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)).

I will sketch how to detect nontrivial higher homotopy groups of the space of complete nonnegatively curved metrics on an open manifold.

This is the fifth 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.

We build a family of spectral triples for a discrete aperiodic tiling space, and derive the associated Connes distances. (These are non commutative geometry generalisations of Riemannian structures, and associated geodesic distances.) We show how their metric properties lead to a characterisation of high aperiodic order of the tiling. This is based on joint works with J. Kellendonk and D. Lenz.

A surface with negative Euler characteristic has a hyperbolic metric. However, this metric is not unique. We will consider the Teichmüller space of a surface, which is the space of hyperbolic structures up to an equivalence relation. We will discuss the topology of and how to put coordinates on this space. If there is time, we will see that the lengths of 9g-9 curves determine the hyperbolic structure.