New year; same me; new math. During the fall I said that I wanted to post on this blog monthly, but that did not happen for some reason. I thought I’d try again this semester, but maybe I am setting this not that high bar still too high for my self. For some reason there seems to be fewer hours in the day than it used to. The cover page for the post is generated by Dalle·2 using the prompt “A man finding the universal piece, old painting” – a true work of art....
In a recent lunch conversation with Nils Baas we, among a plethora of other things, discussed the chromatic redshift phenomenon in stable homotopy theory. Nils was explaining some things about the results he had published together with Bjørn Dundas and John Rognes on using $2$-vector bundles to explain the algebraic K-theory of topological K-theory, i.e. $K(ku)$. This spectrum has height $2$, and since $ku$ has height $1$ this exhibits a phenomenon called redshift....
Hi, long time no see. I used to be very persistent about writing at least one blog-post each month, but as one can see, I have taken a break for a couple months due to vacation and increased teaching and lecturing duties at NTNU. Incidentally it coincided exactly with the 2 year mark of having posted at least once a month, often more. I have been writing stuff, but not anything worth posting....
It has been some time since we studied at the correlation between formal group laws, which were certain power series that looked like Taylor expansion of multiplication on a Lie group, and complex oriented cohomology theories. In particular, we learned that these two completely separate notions had a common universal object. The universal formal group law over the Lazard ring was the same as the formal group law determined by the universal complex oriented cohomology theory — complex cobordism cohomology....
This semester I am taking part in a seminar on $\infty$-categories, administered by Rune Haugseng. So far we have covered roughly: the basic definitions, fibrations, limits, colimits, Joyal’s lifting theorem, equivalences, straightening, Yoneda lemma, adjunctions and Kan extensions. This week it is my turn to give a talk on stable $\infty$-categories, and this blog post will hopefully be some sort of lecture notes for this talk. The intersection of things in this post and the contents of the talk should at least be non-empty....
Recently my friend Elias started his own math blog adventure, and his first post gave a nice introduction to spectral sequences. Reading it I remembered that I should really understand some of the parts better myself, because a lot of the arguments one makes in chromatic homotopy theory are based on spectral sequences. There is a framework for constructing spectral sequences that are not covered in my old post on them, as well as Elias’ post, and that is creating spectral sequences from exact couples....
Over the holidays sadly Edgar H. Brown passed away. He was one of the influential men behind many of the concepts this blog has featured and will feature in the future. This post is in particular focused on one of these concepts, namely Brown-Peterson cohomology and the Brown-Peterson spectrum. In the last post we developed the category of $p$-local spectra, and in the post before that we explored complex cobordism cohomology....
Topology, particularly homotopy theory, is hard. The scenes where these kind of mathematics happen are immensely complicated; the category of topological spaces; the category of spectra. The problem is that there is simply too much information to try to capture by using simple tools that we can actually understand properly. Trying to classify topological spaces or spectra is a feat that many deem impossible, it is simply too difficult. So, how can we try to fix this?...
In the next couple years I will need to understand the ins and outs of different cohomology theories and the spectra that represents them. Some of the most important of these (for my research) can be described using $MU$ — the complex cobordism spectrum. We briefly met this spectrum — or at least its cohomology theory — when we discussed formal group laws. There we explained briefly a theorem of Quillen, stating that the universal formal group law over the Lazard ring corresponds to complex cobordism cohomology....
In the previous post we studied some “easy” cases of homotopy groups of spheres. We focused most on the group $\pi_3(S^2)$ and its computation from the Hopf fibration. All groups calculated last time were part of the so-called unstable range, meaning that they are not invariant under suspension. Due to the Freudenthal suspension theorem we know precicely the stable range for homotopy groups of spheres, and these are given by the stable homotopy groups....