“The pursuit of mathematics is a divine madness of the human spirit.”
Alfred North Whitehead
In this post, we prove the uniformization theorem for elliptic curves. The theorem states that every elliptic curve over the complex numbers arose from the complex plane modulo a lattice, and vice versa. In fact there is an isomorphism between them which is both complex analytic and algebraic. This shows that elliptic curves over complex numbers is a torus.
“It’s to that being inside of you who knows how to be alone, it is to this infant that I wish to speak, and no-one else. ”
Alexander Grothendieck
In this post, we present a proof of van Kampen theorem in algebraic topology that is different from the standard proof in most texts. This is a proof due to Grothendieck and it generalizes better into algebraic geometry, which is an algebraic analogue known as the étale fundamental group. This proof is shorter, more conceptual (uses universal properties without invoking concrete generators and relations), and uses covering spaces.
“The mathematics are usually considered as being the very antipodes of Poesy. Yet Mathesis and Poesy are of the closest kindred, for they are both works of the imagination.”
Thomas Hill
The Frobenius endomorphism of schemes confuses the hell out of me. There’s the absolute Frobenius, the relative Frobenius, the arithmetic Frobenius, the geometric Frobenius… It’s a huge mess of concepts. So, I think it’ll probably benefit me to write an article elucidating these ideas. Our task is to generalize the Frobenius endomorphism \(\mathrm{Frob}_R:R\rightarrow R\) defined by \(x\mapsto x^p\) for a commutative ring \(R\) of characteristic \(p\) (recall \(\mathrm{Frob}_{\mathbb F_{p^n}}\) generates the Galois group \(\mathrm{Gal}(\mathbb F_{p^n}/\mathbb F_p)\)), to the more general case of an endomorphism of a scheme of characteristic \(p\). We begin by defining the absolute Frobenius, and show that it has some undesirable properties, which prompts us to make modifications.
“Epistemology without contact with science becomes an empty scheme. Science without epistemology is – insofar as it is thinkable at all – primitive and muddled.”
Albert Einstein
Epistemology is a branch of philosophy which studies knowledge. It concerns problems such as: how is knowledge defined (what does it mean to know something)? What is the value of knowledge? What is the structure of knowledge? I became interested in epistemology recently and have started reading Duncan Pritchard’s introductory text What is This Thing Called Knowledge? I can’t help but want to write down some of the things I’ve learned from my reading. In this post, I will talk about some key ideas from Pritchard’s book.
“The rules of go are so elegant, organic, and rigorously logical that if intelligent life forms exist elsewhere in the universe, they almost certainly play go. “
Edward Lasker
The game of Go is an ancient and profound abstract strategy board game that I, albeit losing most of my games, enjoy playing very much. If you have not heard of Go before, see here for a quick guide to the rules of Go. I’ve been recently designing my own variant of Go, which I call “Terrain Go” tentatively for lack of a better name. This variant is inspired by Bobby Fischer’s Fischer random chess, which is a chess variant that, at the outset of the game, randomizes the initial positions of pieces at the players’ home ranks in a certain way that preserves the dynamic nature of the game. The variant is designed to “eliminate the complete dominance of openings preparation in classical chess, replacing it with creativity and talent.” In a similar spirit, we introduce randomness at the outset of Go by way of “terrain”, in a way that preserves the dynamic nature of Go.
“The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps.”
Alexander Grothendieck
Recently, I came across a paper by Dan Isaksen, which I find very interesting. It shows that even incredibly basic math, such as the way we learned to add integers in elementary school, could be seen from a much deeper and beautifully illuminating light of group cohomology. The original paper is very accessibe, but I still wanted to share this with you in my own exposition. We begin by revisiting the classic algorthim for adding and subtracting non-negative whole numbers: we align the numbers vertically, we do the operation digit-by-digit, and carrying a \(1\) when the digits add to \(10\) or more. The “carrying” here is the essence, we will see that this could be interpreted as 2-cocycles in a certain group cohomology.
“Why did the two algebraic geometers get arrested at the airport? Because they were talking about blowing up six points on the plane.”
Anonymous
I spent my last week in Quebec attending the Canadian Undergraduate Mathematics Conference 2022. It was really an intense week of learning. I learned a lot and met many smart and interesting people. So I wanted to write about the things I learned and the people I met during the conference. There were so many excellent talks but unfortunately I could only attend some of them. In particular, there are some very good ones in geometry, topology, representation theory, algebra, logic theory, etc.
Last week I gave a talk on the Canadian Undergraduate Mathematics Conference 2022 on Schubert calculus and cohomology of Grassmannians. The Grassmannian is a space that parameterises linear subspaces, and the cohomology ring of the Grassmannian, the Chow ring, encodes data useful for intersection theory. In my talk, I introduced the definition of a Grassmannian as a manifold (via the Stiefel manifold or as a homogeneous space) and as a projective variety (via the Plücker embedding). Then I introduced the decomposition of Grassmannian into Schubert cells and Schubert varieties (by CW-decomposition and affine stratification). After that, I introduced the Chow ring. Here are the slides of my talk.
It is a well known fact that there is no general formula for quintic equations or algebraic equations of any higher degree, and a typical proof of this fact uses heavy machinery from Galois theory. However, there is a far more elementary but much less well known proof by V.I. Arnold using nothing more than basic knowledge of complex numbers and topology. Last week, I gave a talk on the Short Attention Span Math Seminars organized by the Pure Math Club at University of Waterloo. In my talk, I explained the main idea of his proof: moving the coefficients along loops to induce permutations of the roots. I also talked about how it could be turned into a rigorous proof using Riemann surfaces of algebraic functions and their monodromy groups, as well as its connections to Galois theory. In the end, I also discussed briefly some ideas related to this proof, such as topological Galois theory and Grothendieck’s Galois theory. Here are my slides.
I’m in a good mood today because it’s my birthday 🎉 and I finally finished recording my final presentation for PMATH 965! For my final presentation, I did an exposition on Galois categories and étale fundamental groups. Here’s the Youtube video of my presentation, and you can download the slides here.
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