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.
I will talk about the following things for Loday algebras: (i) the existence of a Gerstenhaber algebra structure on cohomology and (ii) an analogue of Deligne's conjecture.
From an operad C of topological spaces with an action of a group G, we construct new operads in spectra from the homotopy fixed point and orbit spectra. These new operads are shown to be equivalent when the generalized G-Tate cohomology of C is trivial. Applying this theory to the little disk operads C_k (which are SO(k)-operads) we obtain an operad governing the Chas-Sullivan string bracket and conjecturally higher dimensional versions. If time permits, we will comment on how these constructions fit into Ginzburg-Kapranov's notion of Kozul duality for operads.
Understanding fixed point spectra of topological Hochschild homology can aid in algebraic K-theory computations. Taking homotopy groups of such spectra we arrive at TR groups, an integer-graded theory that fits into a rigid algebraic structure, namely a Witt complex. In this talk we will review this classical TR theory, and introduce RO(S^1)-graded TR groups. We will then address the question of how to define the RO(S^1)-graded analog of a Witt complex. We conclude by describing explicit computations of the RO(S^1)-graded TR theory of F_p.
I will give a introduction to the theory of Chevalley p-local finite groups. These p-local finite groups arise from p-compact groups in a way similar to the way Chevalley finite groups arise from reductive algebraic groups.
The talk will report on joint work with Arthur Bartels and Wolfgang Lueck. The algebraic K-theory of a group ring RG can be studied via the so called assembly map. The assembly map is conjectured to be an isomorphism if R is the ring of integers and G is torsionfree. We prove this conjecture for torsionfree hyperbolic groups. This is a step towards conjectured rigidity results about aspherical manifolds in geometric topology. In order to achieve the result we will in fact study generalized assembly maps that were introduced and very successfully studied by Farrell and Jones.
The Becker-Gottlieb transfer of a fibration $p:E \to B$ is a stable map in the other direction, $\tau(p):Q(B_+) \to Q(E_+)$. Associated to the same fibration one also has the algebraic K-theory transfer $p^*:Q(B_+) \to A(E)$, whose target is Waldhausen's algebraic K-theory of the total space. Finally, one always has the Trace map $tr:A(E) \to Q(E_+)$ and the claim is that $tr \circ p^* \simeq \tau(p)$ for compact ANR fibrations. The proof is surprisingly clean, thanks to the axiomatic description of $\tau(p)$ for this type of fibration given by Becker and Schultz. We simply verify these axioms hold for our composite $tr \circ p^*$, although this requires us to work with relative versions of all of the above constructions.