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Welcome to my mathematics blog. On here I write my way through my mathematics education, mostly focusing on algebraic topology, but also other things that interests me.

Johnson-Wilson theory

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....

April 29, 2022 · 10 min · Torgeir Aambø

Stable infinity-categories

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....

March 30, 2022 · 17 min · Torgeir Aambø

The Adams spectral sequence

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....

February 17, 2022 · 13 min · Torgeir Aambø

Brown-Peterson cohomology

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....

January 20, 2022 · 14 min · Torgeir Aambø

Bousfield localization

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?...

December 2, 2021 · 19 min · Torgeir Aambø

Complex cobordism cohomology

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....

November 18, 2021 · 16 min · Torgeir Aambø

The homotopy groups of the spheres. Part 2

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....

November 8, 2021 · 15 min · Torgeir Aambø

The homotopy groups of the spheres. Part 1

Recently I gave a talk about the homotopy groups of spheres, and as usual, I try to collect my thoughts on this blog before (or after) presenting. The homotopy groups of spheres have featured several times on this blog, and we have made some effort into calculating them for some small dimensions. In the talk I wanted to showcase some methods used to calculate these groups, as well as doing some of the “calculations”....

October 26, 2021 · 10 min · Torgeir Aambø

The stable homotopy category

A little while ago we discussed the definition of a tensor triangulated category, and in that post we mentioned an example that we didn’t explicitly define, namely the stable homotopy category. The goal for todays post is to fix this. There are many ways of defining it, and some are actually better than others. As the name suggests, the stable homotopy category is a homotopy category, which we have discussed before in the fibration series....

October 1, 2021 · 12 min · Torgeir Aambø

Stable homotopy

For the last few posts we have covered some theory surrounding cohomology theories, and today we want to do something else, namely again look at some homotopy theory. It’s been a long time since we have covered homotopy groups, but today we return once again. In particular I want to cover a theorem and its consequences — the Freudenthal suspension theorem. This is one of the central theorems in the homotopy theory of topological spaces, and is one of the more important theorems we left out from the fibration series....

September 24, 2021 · 9 min · Torgeir Aambø