Blog posts

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. We have already met half of this structure during our several encounters of monoidal categories, but the other half remains, as well as how to glue them together into a cohesive joint structure. ...

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. ...

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. Today we will rectify this situation. I have my algebraic geometry exam this week, so this is both an explainer-post, and a “making sure I understand the course material”-post. These types of posts have in fact become common on this blog, but hopefully that is ok. ...

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. We saw it produced the same structure, so we weakened to looking at transferring it through a homotopy equivalence instead. We then got an operation that was associative only up to homotopy, which we studied a bit through the so called Stasheff associahedra. We then introduced H-spaces and saw that some of these, in particular loop spaces, had the same type of homotopy associative operation. In the latest edition we looked at a more algebraic situation, still heavily motivated by the topology we had discussed earlier. In this situation we explicitly described a ternary operation that we proved was the associating homotopy and saw that we got a certain relation involving the associator and the boundary of 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. To be more precise I am talking about the claim that a map we denoted by $m_3$ was a homotopy between $m_2(id\otimes m_2)$ and $m_2(m_2 \otimes id)$, where $m_2$ was a product induced through a homotopy equivalence. Don’t worry if you don’t recall the definitions and this problem, we will go through it again shortly. Today we in fact upgrade this earlier homotopy equivalence slightly such as to have a bit more to work with. As said we also take a turn away from standard topology and make our choice of “space” for this post to be chain complexes of vector spaces. I will not cover in detail why this is a reasonable thing to do but I will mention that the de Rham complex of a manifold, the rational singular cochains on a topological space and the rational cohomology of a topological space are all such structures. So, if we believe that algebraic topology is a nice way to study spaces, then studying these should be highly relevant. ...

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. ...

Imagine we have a system of two topological spaces $f:T\longrightarrow G$. We are often interested in knowing if a certain property on the space $G$ can be transferred through f such that we have the same property on $T$. If f is a nice enough morphism an example could be a topological invariant of $G$, for example its Euler characteristic. In this post we are more interested in transferring other things than invariants, more specifically structures. If $G$ has an algebraic structure, for example a group structure, can we then transfer the same or some other similar structure onto $T$ through $f$? ...

For those that don’t know I am a fifth year mathematics student at NTNU, meaning I am finishing my masters degree after this semester. During my time at NTNU I have had some wonderful classes, and some wonderful teachers. Since most I post about on this blog is related to topology, it is very safe to assume that some of my most memorable courses are exactly the topology courses. I very recently looked at my notes from my first topology course, or rather one of the two first, as I took two in parallel during my fourth semester. The course was focused on differential topology and the study of smooth manifolds. It was taught by my now supervisor, but on a couple of the last lectures we had some guest appearances from the other topology professors at NTNU. One of these guest lectures is the focus of todays blog post. ...

Last year I posted a blog post where we looked at a way to use elementary homotopy theory to hang a picture on the wall in a stupid way. The task was to hang a picture on two nails in such a way that if we pull one of the nails out, the picture falls down. “That is stupid” I hear you say, but premise is as premise does, or something similar quoted from Forrest Gump. I remarked then that I had seen the problem a couple years earlier, and I actually recently found were, which is how this blog post got made. I initially came across the problem from a video by a YouTube channel called GoldPlatedGoof. I watched the video again recently and decided to look around math-YouTube for other videos on the problem. There I came across one similar, and a bit less rigorous video by Matt Parker and Steve Mould. They solved it similarly to the original one, i.e. by using commutators, which is formalized by using the fundamental groups we did last time when posting about this. More interestingly I came across a video by Tom Scott and Jade Tan-Holmes which used a completely different (yet actually the same) method for solving it. Jade used knots and braid diagrams to produce a solution for the problem, which inspired me to make this post. We are also going to solve the problem using knots. The overall tactic will be roughly the same as Jades, but the method and the proof will be a bit different. The post has turned out to be quite long, but there are many pictures, and not that much text! ...

Something I have been looking into a bit lately, due to it sadly not being taught at my university is knot theory. This is something I have always known to be a part of topology, and have known to have interesting applications in physics, medicine, chemistry and more. So to rectify the situation I thought I would prove that knots exist. It was also nice to take a brake from all the higher category theory we have been looking into lately! ...