Power operations have been constructed and successfully utilized in the Adams and Homological Homotopy Fixed Point Spectral Sequences by Bruner and Bruner-Rognes. It was thought that such results were not specific to the spectral sequence, but rather that they arose because highly structured ring spectra are involved. In this talk, we show that while the Kunneth Spectral Sequence enjoys some nice multiplicative properties, there are no non-zero operations on the $E_2$ page of the spectral sequence. Despite the negative results we are able to use old computations of Steinbergers with our current work to compute operations in the homotopy of some relative smash products.

The noncommutative two torus $A_theta$ equipped with a general complex structure and Weyl conformal factor, is a noncommutative Riemannian manifold where the metric information is encoded in the Dirac operator $D$ of a spectral triple over this C*-algebra. In a recent joint work with M. Khalkhali, we computed a local expression for the scalar curvature of $A_theta$. This was achieved by finding an explicit formula for the value at the origin of the analytic continuation of the spectra zeta function $\Zeta_a(s) := Trace (a|D|^{-s}) (Real(s) >> 0)$ as a linear functional in $a \in A_theta$ . This local expression was also computed by Connes and Moscovici independently. In this talk, I will explain how they have used this local formula and variational methods to compute the determinant of the Laplacian D2 on $A_theta$.

We review the study of the generating hypothesis (GH) in derived categories and stable module categories. For $p$-groups, only $C_2$ and $C_3$ satisfy the GH. We present results on the ghost numbers of $p$-groups, which test the failure of the generating hypothesis in stable module categories. We also introduce a strong version of GH for stable module categories.

Let $G \times X \rightarrow X$ be an action of the algebraic group $G$ on the affine, algebraic variety $X$. There are two quite different notions of quotient associated with this situation. If we embrace the conventional approach, we just accept the object $Y =$ Spec$(k[X]^G)$, along with the natural map $\pi : X \rightarrow Y$, as the inevitable thing to study. If $G$ is reductive then this "quotient" is a variety and it has the anticipated universal property, even if $Y$ is not an orbit space. Similar results hold if $X$ is a projective variety. Many important moduli spaces have been constructed using this approach. But maybe there is another approach, where the emphasis is on orbits of maximal dimension rather than on closed orbits. In this scenario we consider sufficiently small open, $G$-subsets $U$ of $X$ such that each $G \times U \rightarrow U$ has as many desirable properties as the situation will tolerate. If we define $U/G$ by the equation $k[U/G] = k[U]^G$, then we can ask for the following. (1) $k[U/G]$ is finitely generated. (2) $k[U/G]$ is a regular ring. (3) $\pi : U \rightarrow U/G$ is surjective. (4) $\pi : U \rightarrow U/G$ separates orbits of maximal dimension. (5) $\pi : U \rightarrow U/G$ has no exceptional divisors. (6) $\pi : U \rightarrow U/G$ is flat. We do this so as to discard only a small portion of $X$. We then try to glue all these $U/G$'s together to get a separated quotient variety, without the help of semi-invariants.