I will give a little introduction to the classical Adams spectral sequence. Prerequisites will be minimal: if you know what Ext groups over a ring are, and maybe what a triangulated category is, you should be able to follow. I should be able to state what the Hopf Invariant 1 (Adams' theorem) and the Kervaire Invariant 1 (now a theorem of Hopkins, Ravenel, and our own Mike Hill) problems are from this point of view.

If $A$ is a square of ring spectra (satisfying hypotheses essentlially saying that it is opposite to a gluing of closed embeddings of schemes), then the cube induced by Goodwillie's integral cyclotomic trace $K(A)\to TC(A)$ is homotopy cartesian. In other words, the homotopy fiber of the cyclotomic trace satisfies "excision" for closed embeddings.
So, if one, for instance, is able to calculate $TC$, this means that its K-theory is accessible through assembling the K-theories of simpler closed subspaces. The theorem works equally well for the non-commutative case.

Heegaard Floer homology is an attempt to compute the Seiberg- Witten invariant of smooth 4-manifolds by cutting those manifolds up along 3-manifolds. Unfortunately, the resulting 3-manifold invariants remain mysterious, as well as hard to compute. We will discuss joint work with Peter Ozsvath and Dylan Thurston to understand the 3- manifold invariants by further decomposing along surfaces. The talk will focus on the formal structure of the theory, rather than the (somewhat technical) definitions of the invariants.

I will report on a talk by Jacob Lurie that I heard last weekend. He is playing with the favorite toyes of ffolks (Goodwillie, Weiss, Greg A and collaborators) who study "Embedding calculus".

We will consider the space of compactly supported smooth embeddings (modulo immersions) of R^m into R^n. For m=1 this is sometimes called the space of long knots in R^n. When n>2m+1, both rational homology and the rational homotopy groups are isomorphic to the homology of a direct sum of rather explicit finite chain complexes. The proof uses calculus of functors, the theory of operads, and some homological algebra a la Pirashvili. I will try to explain the splitting and its proof. This is a report on joint work with V. Turchin.

We trace an idea similar to the existence of a basis for R^j in linear algebra; that, in some sense, closed n-1 dimensional manifolds seem not to admit "true" Z^n PL actions. We outline the proof that Z \wr Z^2 fails to embed in PL(I), and our separate (easy) proof that Z \wr Z^n embeds in PL(I^n) with no difficulty. We then explain why, if Z^n is acting on an (n-1) cube in PL fashion so as to fix the boundary, the local orbit of interior points in the cube will appear as a Z^j orbit, for some j < n. We also mention the non-embedding of Z \wr Z^2 into PL(S^1) as evidence that a non-embedding result for Z \wr Z^n into PL(I^{n-1}) suggests that Z \wr Z^n will also fail to embed into PL(M) for M any closed PL manifold of dimension n-1. The work is joint with Benjamin Krause.