University of Chicago
Department of Mathematics

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.

The (stable) homotopy groups of spheres are very complicated, and the computations seem at first glance chaotic. This chaos actually breaks up into periodic patterns, however, and understanding these periodic phenomena has been a focal point in the study of the homotopy groups of spheres. The first layer of periodicity is well understood, and is closely linked with Bott periodicity and K-theory. The second layer of periodicity displays much more complicated patterns. I will discuss an emerging picture of how these patterns are actually dictated by the arithmetic of modular forms. The role that K-theory had in the first layer of periodicity is replaced by the cohomology theory of topological modular forms.

Goerss, Hopkins, Miller, and their collaborators constructed a spectrum of topological modular forms, and Goerss, Henn, Mahowald, and Rezk used this theory to give a resolution of the K(2)-local sphere at the prime 3. Half of this resolution may be recovered at all primes using isogenies of elliptic curves and the building for GL_2(Q_l). In this talk, I will describe how recent work of Lurie may be exploited to associate spectra of topological automorphic forms to certain Shimura varieties associated to a form of the unitary group U(1,n-1). A piece of the resolution of the K(n)-local sphere is recovered by considering the building for GL_n(Q_l).

Self-distributive binary operations have appeared extensively in knot theory in recent years, specifically in algebraic structures called `quandles.' A quandle is a set equipped with two binary operations satisfying axioms that capture the essential properties of the operations of conjugation in a group. The self-distributive axioms of a quandle correspond to the third Reidemeister move in knot theory. Thus, quandles give a solution to the Yang-Baxter equation, which is an algebraic distillation of the third Reidemeister move. We formulate analogues of self-distributivity in the categories of coalgebras and Hopf algebras and use these to construct additional solutions to the Yang-Baxter equation. Please note that the seminar meets at 3:00 this week only.

In this talk, I will briefly outline the emerging program to understand the K-theory of the sphere spectrum via a chromatic filtration in K-theory, due to Rognes (based on an idea of Waldhausen). Part of this program requires the study of the K-theory of certain nonconnective ring spectra, conjecturally via localization sequences. I will discuss the resolution of the first nontrivial case of this conjecture, a particular localization cofiber sequence K(Z) -> K(ku) -> K(KU) which arises from the localization ku -> KU. This is joint work with Mike Mandell.