This is a transcription of my talk on the work of Doug Ravenel. While it was intended as a one-time performance, the organizers felt it would be a good idea to have some representation appear as part of the conference proceedings. Don Davis went to the great trouble of transcribing the lecture from a video, and he sent it to me. After reading it I agreed to have it included in these proceedings. Even though the text is ungrammatical, repetitious and meandering (apparently that's how I talk), it feels to me as if it conveys something about how the world of algebraic topology has felt all these years with Doug in it. So with heartfelt thanks to Don Davis, here it is.

This paper is a survey of Steve's work on BP and periodic cohomology theories. It was presented as a talk given in March 2007 at a conference celebrating Steve's 60th birthday.

Let $E_n$ be the $n$-th Lubin-Tate spectrum at a prime $p$ . There is a commutative $S$-algebra $E^{\rm{nr}}_n$ whose coefficients a.re built from the coefficients of $E_n$ and contain all roots of unity whose order is not divisible by $p$. For odd primes $p$ we show that $E^{\rm{nr}}_n$ does not have any non-trivial connected finite Galois extensions and is thus separably closed in the sense of Rognes. At the prime 2 we prove that there a.re no non-trivial connected Galois extensions of $E^{\rm{nr}}_n$ with Galois group a finite group $G$ with cyclic quotient. Our results carry over to the $K(n)$-local context.

Let $M(1)$ be the mod 2 Moore spectrum. J.F. Adams proved that $M(1)$ admits a minimal $v_1$-self map $v^4_1 : \Sigma^8 M (1) \to M (1)$. Let $M(1, 4)$ be the cofiber of this self-map. The purpose of this paper is to prove that $M(1, 4)$ admits a minimal $v_2$-self map of the form $v^{32}_2 : \Sigma^{192} M (1,4) \to M (1,4)$. The existence of this map implies the existence of many 192-periodic families of elements in the stable homotopy groups of spheres..

Steenrod operations were defined by Voedvodsky in motivic cohomology in order to prove the Milnor and Bloch-Kato conjectures. These operations have also been constructed by Brosnan for Chow rings. The purpose of this paper is to provide a setting for the construction of the Steenrod operations in algebraic geometry, for generalized cohomology theories whose formal group law has order two. We adapt the methods used by Bisson-Joyal in studying Steenrod and Dyer-Lashof operations in unoriented cobordism and mod 2 cohomology.

This paper is a companion to the Boardman-Wilson paper on the ring spectrum $P(n)$. When the prime is 2, this spectrum is not commutative, which introduces several complications. Here, we supply the necessary details of the relevant Hopf algebroids and Hopf ring for this case.

If $C$ is the model category of simplicial presheaves on a site with enough points, with fibrations equal to the global fibrations, then it is well-known that the fibrant objects are, in general, mysterious. Thus, it is not surprising that, when $G$ is a profinite group, the fibrant objects in the model category of discrete $G$-spectra are also difficult to get a handle on. However, with simplicial presheaves, it is possible to construct an explicit fibrant model for an object in $C$, under certain finiteness conditions. Similarly, in this paper, we show that if $G$ has finite virtual cohomological dimension and $X$ is a discrete $G$-spectrum, then there is an explicit fibrant model for $X$. Also, we give several applications of this concrete model related to closed subgroups of $G$.

In a 2002 paper, the authors and Bruner used the new spectrum tmf to obtain some new nonimmersions of real projective spaces. In this note, we complete/correct two oversights in that paper. The first is to note that in that paper a general nonimmersion result was stated which yielded new nonimmersions for $RP^n$ with $n$ as small as 48, and yet it was stated there that the first new result occurred when $n = 1536$. Here we give a simple proof of those overlooked results. Secondly, we fill in a gap in the proof of the 2002 paper. There it was claimed that an axial map $f$ must satisfy $f^*(X) = X_1 + X_2$. We realized recently that this is not clear. However, here we show that it is true up multiplication by a unit in the appropriate ring, and so we retrieve all the nonimmersion results claimed in the 2002 paper. Finally, we completely determine $tmf^{8*}(RP^\infty × RP^infty)$ and $tmf^*(CP^\infty × CP^\infty)$ in positive dimensions.

We study the Brown-Peterson (co)homology of a product of two real projective spaces via the Landweber short exact sequence. The image of the tensor product is well understood. Our contribution is to understand those elements not in the tensor product and to show how they behave under maps. The results are partially extended to the case where one of the factors is replaced by a $2^e$-torsion lens space.

Let $I$ be a small indexing category, $G: I^{op} \to \mathcal{C}at$ be a functor and $BG \in \mathcal{C}$ at denote the Grothendieck construction on $G$. We define and study Quillen pairs between the category of diagrams of simplicial sets (resp. categories) indexed on $BG$ and the category of $I$-diagrams over $N(G)$ (resp. $G$). As an application we obtain a Quillen equivalence between the categories of presheaves of simplicial sets (resp. groupoids) on a stack $\mathcal{M}$ and presheaves of simplicial sets (resp. groupoids) over $\mathcal{M}$.

Hu and Kriz construct the real Johnson-Wilson spectrum, $ER(n)$, which is $2^{n+2}(2^n - 1)$-periodic, from the $2(2^n - 1)$- periodic spectrum $E(n)$. $ER(1)$ is just $KO_{(2)}$ and $E(1)$ is just $KU_{(2)}$. We compute $ER(n) * (RP^\infty)$ and set up a Bockstein spectral sequence to compute $ER(n)*(-)$ from $E(n) * (-)$. We combine these to compute $ER(2) * (RP^{2n})$ and use this to get new nonimmersions for real projective spaces. Our lowest dimensional new example is an improvement of 2 for $RP^{48}$.

This paper is a continuation of the study begun in the previous paper with the same title. We analyze $ER(2)^{16*+8}(RP^{2n})$ and compute $ER(2)*(RP^{16K+1})$, and use these to prove more nonimmersion theorems for $RP^n$, including many in fairly low dimensions. In particular, we get 12 new nonimmersion results for $RP^n$ where $n < 192$, the range included in the tables Don Davis keeps. These complement the 10 already found in the first paper.

In the mid 1980s, Pete Bousfield and I constructed certain $p$-local `telescopic' functors $Φ_n$ from spaces to spectra, for each prime $p$, and each $n ≥ 1$. They are constructed using the full strength of the Nilpotence and Periodicity Theorems of Devanitz-Hopkins-Smith, and have some striking properties that relate the chromatic approach to homotopy theory to infinite loopspace theory.

Recently there have been a variety of new uses of these functors, suggesting that they have a central role to play in calculations of periodic phenomena. Here I offer a guide to their construction, characterization, application, and computation.

As is well known, the image of the $J$-homomorphism in the stable homotopy groups of spheres is described in terms of the first line of the Adams-Novikov $E_2$-term. In this paper we consider an algebraic analogue of the image of $J$ using the spectrum $T(m)_(j)$ defined by Ravenel and determine the Adams-Novikov first line for small values of $j$.

Let $p$ be a prime and $f$ a positive integer, greater than 1 if $p = 2$. We construct liftings of the Artin-Schreier curve $C(p, f)$ in characteristic $p$ defined by the equation $y^e = x - x^p$ (where $e = p^f - 1)$ to a curve $\tilde{C}(p, f)$ over a certain polynomial ring $R^\prime$ in characteristic 0 which shares the following property with $C(p, f)$. Over a certain quotient of $R^\prime$, the formal completion of the Jacobian $J( \tilde{C}(p, f))$ has a 1-dimensional formal summand of height $(p - 1)f$. Along the way we show how Honda’s theory of commutative formal group laws can be extended to more general rings and prove a conjecture of his about the Fermat curve.