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....
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....
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....
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....
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....
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....
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....
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!...