Formal group laws

Recently we have covered a lot of heavy topology and abstract mathematics, so today I thought we would cover something else — something maybe a bit easier to grasp. We will introduce the concept of formal group laws, and a bit on why they are interesting. Introduction and definition To not just spew out the definition straight away, we look at a situation where formal group laws arise very naturally. Let $G$ be a one-dimensional commutative Lie group (Think here of the real numbers $\mathbb{R}$ or the circle group $S^1$)....

September 3, 2021 · 13 min · Torgeir Aambø

A first look at spectra

Even though this blog is not centered around a specific topic, we have during the last year looked more frequently at certain topics than others, such as (co)homology theory, homotopy theory and category theory. We will continue this trend today as we will try to find a solid reason for a particular object to exist. These objects were briefly mentioned in the earlier post on tensor triangulated categories, namely spectra. These objects are hugely important to the field of algebraic topology, one reason being that they are intimately linked to cohomology....

August 20, 2021 · 10 min · Torgeir Aambø

Traveling

How can we use travel-journaling to study some highly abstract and highly complicated mathematical machinery? Can we get anything out of such an analogy? In this post we do just that, study the homotopy hypothesis and infinity groupoids through the lens of a travel ledger.

July 21, 2021 · 10 min · Torgeir Aambø

On formal DG-algebras

I have recently handed in and defended my master thesis in mathematics, so I though I would go through its abstract and try to explain what it’s all about. We look at formality of DG-algebras, Massey products, A_infinity-algebras and how we can use these to some interesting results.

July 19, 2021 · 14 min · Torgeir Aambø

Tensor triangulated categories

For the last five years mathematics has been my passion, as well as my main focus in life. This passion for mathematics will hopefully not diminish, as I am now heading into four more years of studies and research through a PhD in mathematics at NTNU. I am joining a project, called Tensor triangulated geometry in Trondheim, so today I thought I would explore the definition of one of the main players in this theory, namely tensor triangulated categories....

June 28, 2021 · 11 min · Torgeir Aambø

Updated geometric intuition

The first post on this blog is titled “geometric intuition”, and discusses the geometry behind Noether’s normalization lemma. When I wrote it I didn’t yet understand all the pieces, as I was not very comfortable working with algebraic geometry. One year later, I’m still not comfortable, but a bit more than last year. So, I thought I would update last years post with my new knowledge, as well as generalize the intuition to schemes - which we introduced in the last post....

May 20, 2021 · 13 min · Torgeir Aambø

Schemes

The first two posts ([1],[2]) I ever did on this blog - now over a year ago - were posts about algebraic geometry. In particular we explored the geometric implications of some of the algebraic results I was learning in my commutative algebra class. Last summer I also wrote a post about sheaves, and left it off by claiming to soon write about schemes. If you scroll through the blog we have covered a bunch of different topics, but the blog post on schemes, seems to have fallen through the cracks....

May 18, 2021 · 15 min · Torgeir Aambø

Sha-algebras

This is part four in a sort of connected story about operations in mathematics that are associative up to homotopy. It will probably be beneficial to read part 1, part 2 and part 3 in advance of this, but it is not required in theory. These previous posts do however build up some intuition and motivation for the object we are looking at today. To quickly recap what we already seen in these previous posts we recall that we started out by looking at how to transfer a group structure on a topological space through an isomorphism....

April 19, 2021 · 8 min · Torgeir Aambø

The associating homotopy

In the two last posts we have been discussing operations that are associative up to homotopy, and where such operations might arise naturally in topology. One claim I made, which I later realized was maybe a bit unmotivated and in need of some clarification was how some higher arity maps actually defined (or were defined by) homotopies between combinations of the lower arity maps. We also purely looked at this in a topological setting, but in algebraic topology we often translate to algebraic structures, so I also wanted to see clearly that the same constructions hold in that setting....

April 3, 2021 · 12 min · Torgeir Aambø

Spaces with operations

In the most recent blog post we discussed homotopy associativity and how to transfer algebraic structures on topological spaces. There we in particular used topological groups, which are topological spaces with group structures. That said, any group is a topological group by equipping it with the discrete topology. So if we want to study some actual topology, and not just glorified group theory, we need to look at where multiplications and binary operations arise naturally in topology....

March 4, 2021 · 6 min · Torgeir Aambø