University of Chicago
Department of Mathematics

(Joint work with Kasper Andersen.) A p-compact groups is a homotopical version of a Lie group, but with all the structure concentrated at a single prime p. I'll state and outline a proof of the classification of 2-compact groups, hence completing the classification at all primes (the case of odd primes is proven in earlier work by Andersen-Grodal-Moller-Viruel). The theorem says that there is a 1-1-correspondence between connected 2-compact groups and root data over the 2-adic integers.

A Hopf algebra developed by Andre Henriques and the speaker and used for computing the $tmf$ homology of a space or spectrum is presented. This is done using a variant of the Adams spectral sequence in the world of $tmf$-module spectra. Following a presentation of this, it will be used to show that the $tmf$ homology of the cofiber of the transfer map $B\Sigma_3->S^0$ is a torsion free indecomposable module over $tmf_*$.

Recent calculations of surgery obstruction groups makes it possible to classify completely all topological manifolds doubly covered by by S^1xS^n, n>2. I will begin by describing this classification and then go on to study closer an infinite family of such manifolds in dimension 4, all homotopy equivalent to RP^4#RP^4.

I will give a new construction of commutative algebras over a commutative ring spectrum, generalizing the classical construction of Thom spectra. Examples include iterated Thom Thom Thom ... spectra. The ideas are elementary, conceptual, and entertaining. Thinking in terms of parametrized spaces (don't be afraid: there is NO use of parametrized spectra), the construction just abstracts and codifies classical classifying spaces and universal bundles as we actually see them ``in nature''. *What I did in my Christmas vacation.

A classical result of Stasheff says that one can define the bar construction on an A-infinity H-space using a certain family of polyhedra called Stasheff associahedra. I will show how one can define the cyclic bar and cobar construction in a similar way, using another family of polyhedra called cyclohedra. I will then use this to define THH of an A-infinity ring spectrum, and explain how the A-infinity structure is responsible for extensions in the canonical spectral sequences calculating the homotopy groups of THH.

Working with $tmf$ Goerss, Henn, Mahowald and Rezk gave a small topological resolution of $L_&ob;K(2)&cb;S^0$ at the prime 3. This resolution has, as its component, spectrum $tmf(N)$ arising from elliptic curves with N-level structures. For M. Behrens studied $tmf(N)$ for $N$ prime to 3 [at the prime 3]. In general $N$ is prime to a fixed prime $p$. The question we can ask is whether there are spectra similar to $tmf(N)$ for $p^n$-level structures on elliptic curves at the prime $p$? It turns out that there is such a spectrum. Throughout the talk I will give some background information and will explain how can we produce the spectrum in question.