University of Chicago
Department of Mathematics

Abstract: Given a functor F from S-modules (spectra) to S-modules, let P_dF be its universal degree d (or d-excisive) approximation, in the sense of Goodwille.
Theorem P_dF --> P_d-1F splits after applying Bousfield localization with respect to any Morava K-theory. Thus a polynomial endo-functor of spectra splits as the wedge of its homogeneous components after applying L_K(n).
The proof is an application of the Greenlees-Hovey-Sadofksy Tate homology vanishing theorem, which itself is dependent on the Nilpotence Theorems. Also used is work of Randy McCarthy relating Goodwillie towers to Tate homology.

I will discuss the primary v₂ family in the stable stems and it's relation to the root invariant at the prime 3. I will begin by saying a bit about some v₂-periodic elements in the context of some joint work with Satya Pemmaraju concerning the existence of the map v₂⁹ on the Smith-Toda complex V(1). Then some root invariants of the primary v₁ family will be computed using a collection of elements of the Adams-Novikov spectral sequence that relate root invariants to Adams-Novikov differentials. These finite computations will then extend to computations of root invariants of infinite families through a propagation theorem related to the map v₂⁹ alluded to earlier.

Let $\sigma$ denote the area form on $S^2$ and consider the symplectic form $\omega=\sigma \oplus (1+\lambda)\sigma$ on $S^2 \times S^2$. Let $G_\lambda$ denote the group of symplectomorphisms of $S^2 \times S^2$ with respect to this symplectic structure. Using previous work of Gromov, Abreu, Abreu-McDuff and Anjos we will show that a certain canonical map \[ SO(3) \times SO(3) \coprod_{SO(3)} S^1 \times SO(3) \to G_\lambda\] is a weak equivalence for $1<\labda \leq 2$. This expresses $G_\lambda$ as an amalgamated sum of certain interesting compact subgroups. This is joint work with Silvia Anjos.

The study of unimodular rows, and their orbit spaces, over a commutative ring with $1$, lies in the fertile cross-section of ideas from Algebra, Algebraic Topology, Number Theory, and Algebraic Geometry. Witt group structures, Cohomotopy groups, Mennicke symbols, Reciprocity Laws, etc. make their appearance very naturally. We shall discuss the connection of the study of orbit spaces, via a symbiosis of constructions of L.N. Vaserstein, A. Suslin, and its relation to problems in classical K-theory, and to the program of J.-P. Serre, which interconnected the study of projective $R$-modules with problems of efficient generation of ideals of $R$.

Ando, Hopkins, and Strickland have shown that there is a canonical orientation $\sigma:MU<6>\to E$ for an elliptic spectrum $E$. Here $MU<6>$ is the Thom spectrum associated to the 5-connected cover $BU<6>$ of $BU$. So, given a lift $M\to BU<6>$ of the classifying map for an $SU$-bundle $V$, the $\sigma$-orientation provides an $E$-Thom class for $V$. Such lifts, or $BU<6>$-structures, exist when $c_2(V)=0$, but there may be many. In fact, the set of lifts is a quotient of $H^3(M)$. To gain insight into how variations of lifts affect the $\sigma$-orientation, we consider a manifold $M$ with a finite group action, and we define an orbifold $\sigma$-orientation for $G$-bundles on $M$. Using classes in $H^3(BG)$, we can vary choices of $BU<6>$-structure for a $G$-bundle over $M$. We will study the effect of such variations on the orbifold $\sigma$-orientation. We will also remark on how our formulas are reflected in some recent discoveries in the physics of conformal field theories.

We describe how the theory of power operations gives rise to an action of ``Hecke operators'' on the cohomology theory $E=E_n$; this is the cohomology theory associated to Lubin-Tate's deformation space of formal groups of height $n$. We then interpret the formula for a certain logarithmic cohomology operation $E^0(X)^\times \ra E^0(X)$ in terms of such operators.

Every space can be built out of spheres in the sense that every space is weakly equivalent to a CW-complex. I'll describe an analogous situation for algebraic varieties in the context of A^1-homotopy theory. In this case, not every object can be built from spheres. So, which varieties can be built from spheres? And what does that tell us about these varieties?

In Joint work with Carles Broto, Laia Saumell, Nitu Kitchloo and Albert Ruiz, we have been able to extend some of the methods and results of the homotopy theory of compact Lie groups to Kac-Moody groups. In this particular talk, I will focus on our results on the cohomology of classifying spaces of central quotients of rank two Kac-Moody groups. The computations of these cohomology algebras display interesting relationships between homotopy theory, representation theory and invariant theory.