“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.

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“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.

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“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.

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“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.

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“My first impression on seeing him (Grothendieck) lecture was that he had been transported from an advanced alien civilization in some distant solar system to visit ours in order to speed up our intellectual evolution.”

Marvin Greenberg

I was very hesitant about writing this article, for the fear that my ignorance of logic and philosophy might prompt me to say something stupid and embarrass myself (which I probably will anyway). Nevertheless, I couldn’t resist the temptation. These ideas have circulated in my mind for too long. Therefore, I ask the reader to indulge me in spilling some nonsense, which, in my defense, even if worthless mathematically, may still serve as a fun sci-fi project or thought experiment. The question I’m interested in is the following. Imagine an alien civilization on a distant planet in a solar system millions of lightyears away which is advanced enough to have developed their own “mathematics” (whatever it means). What would their mathematics be like? To break this down into several smaller questions: firstly, what would their formal logical system (to make things interesting, we assume they do use one) be like; secondly, what would the mathematical objects they study be like; and thirdly, what would their theorems be like?

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Recently, I gave another talk on the Short Attention Span Math Seminars which is a friendly introduction to category theory. Here are my slides, and here are the slides for the first five minutes. In my talk, I first talked about the idea of abstractions, then I talked about the categorical way of thing and along the way talked about many ideas like universal properties, functors, natural transformations. In the end, this culminated in the Yoneda lemma or the fundamental theorem of category theory. I also discussed a little an application of it to algebraic geometry.

“Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate.”

David Mumford

Continuing last week, we are finally going to define the derived category \(D(\mathcal A)\) for an abelian category \(\mathcal A\). Before we actually do that, we need to introduce localization of categories, and we will define the derived category in terms of a localization of the homotopy category of complexes. Localization of categories is very analogous to localization of rings or modules. Given any category (with no assumption of abelian, additive, triangulated, etc), we formally invert a class of morphisms. We will give one definition in terms of a localization construction and show that it is unsatisfactory in that it does not have a lot of useful and desirable properties. We will then develop a better description of derived categories in terms of roofs.

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“It is impossible to be a mathematician without being a poet in soul.”

Sofia Kovalevskaya

Continuing last week where we defined the homotopy category of complexes, we will take a look at triangulated categories. This will help us build towards the definition of derived categories. In some sense, triangulated categories are approximations of abelian categories. They are not strictly speaking a weaker version of abelian categories, and they don’t really imply each other. Triangulated categories takes a different approach to abelian categories. Instead of kernels, cokernels, and strict morphisms, triangulated categories bypass these notions and start out with “distinguished triangles”, which are similar to exact sequences.

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“A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.”

Stefan Banach

I have always been curious about derived categories. In particular, I read from the wikipedia page on homological mirror symmetry that the derived category of coherent sheaves is a popular research topic in algebraic geometry. Also, I’ve been told by a professor that derived categories are related to something called geometric Satake correspondance, which is also something I totally don’t understand but it sounds very cool. Therefore, I decided to study a little about derived categories, and write about my progress here. I want this post to be the first part in several parts of my notes. In this first part, we will focus on abelian categoires, derived functors, and homotopy category of complexes. We begin by reviewing abelian categories and their properties. In particular, the (co)chain complexes in an abelian category and their related notions. After that, we define derived functors. Eventually, we finish by defining the homotopy category of complexes \(K(\mathcal A)\).

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“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.

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