Articles

“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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“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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“Mathematics requires a small dose, not of genius, but of an imaginative freedom which, in a larger dose, would be insanity. And if mathematicians tend to burn out early in their careers, it is probably because life has forced them to acquire too much common sense, thereby rendering them too sane to work. But by then they are sane enough to teach, so a use can still be found for them.”

Angus K. Rodgers

For this week, I’ll write about the theory of root systems, especially things related to Weyl groups, weight spaces and etc, for the purpose of my URA. This is because one of the objectives of my URA is to generalise my previous blog post about a fraction of an \(S_n\)-orbit on a hyperplane to Weyl groups. First, a Lie algebra is simple if it is of degree greater than \(1\) (so we are excluding the one dimensional abelian Lie algebras) and has no proper ideals, and a semisimple Lie algebra is a direct sum of simple Lie algebras. Equivalently, a Lie algebra \(\mathfrak{g}\) is semisimple if \(\mathrm{rad}(\mathfrak{g})\), the sum of all solvable ideals (or the maximal solvable ideal), is trivial. Any semisimple Lie algebra \(\mathfrak{g}\) is the Lie algebra of an algebraic group \(G\). Let \(\mathrm{ad}:\mathfrak{g}\rightarrow \mathfrak{gl}(\mathfrak{g})\) by \((\mathrm{ad} X)(Z)=[X,Z]\) be the adjoint representation. Let \(B:\mathfrak{g}\times \mathfrak{g}\rightarrow \mathbf C\) be the symmetric bilinear form defined by \(B(X,Y)=\mathrm{tr}(\mathrm{ad}X\mathrm{ad} Y)\), called the Killing form which is invariant in the sense of \(B([X,Y],Z)=B(X,[Y,Z])\). Cartan’s criterion for semisimplicity says that a Lie algebra is semisimple iff the associated Killing form is nondegenerate. From now on, we let \(\mathfrak{g}\) be a semisimple Lie algebra over \(\mathbf C\). We start by discussing the root–space decomposition \(\mathfrak{g}=\mathfrak{h}\oplus \bigoplus_{\alpha\in\Phi}\mathfrak{g}_{\alpha}\) of semisimple Lie algebras.

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“Mathematics is the music of reason.”

James J. Sylvester

This week is the first week of the spring term and also my first week of undergraduate research with Prof. Satriano, so I decided to write about a research paper I’ve given to read, this one. In a nutshell, the paper proves a combinatorical conjecture posed by McKinnon, Satriano and Huang, about orbits on a hyperplane under permutation. Specifically, suppose that \(\sigma\in S_n\) acts on \(v\in\mathbf R^n\) by permutation, and let \(\mathcal{O}(v, w)=\{\sigma\in S_n: w\cdot \sigma v=0\}\). We prove a best bound for the number of vectors obtained by permutation of coordinates that are contained in a hyperplane through the origin except for \(\sum_ix_i=0\), where \(n\ge 3\). In notations, we have the following theorem.

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“Serious mathematics (contrary to a popular misconception) is not ‘about’ proofs and logic any more than serious literature is ‘about’ grammar, or music is ‘about’ notes.”

Ethan D. Bloch

In this week we discuss fppf sites (recall that we defined \((\mathbf{Sch}/X)_{\mathrm{fppf}}\) in last week) and faithfully flat descent. This will help us to build our way to the definition of a stack. The idea of faifully flat descent is that a certain property of schemes can be descended via a cartesian square with a fppf side. The important result we are building towards is that for any scheme \(X\), the functor of points \(\mathscr{H}_X\) is a sheaf on the fppf site of \(X\). After this, we can talk about fibred categories. Fibred categories plus descents is categorical stacks, plus geometry is algebraic stacks.

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“Complex analysis is the good twin and real analysis the evil one: beautiful formulas and elegant theorems seem to blossom spontaneously in the complex domain, while toil and pathology rule the reals.”

Charles Pugh

This week I plan to start writing about what I’ve learned in PMATH 965 on algebraic stacks. In the end of this semaster hopefully I will have written enough about them in this blog so that I can compile them into actual typesetted notes. We start by revisiting étale morphisms and introducing Grothendieck topologies and sites, and we define presheaves and sheaves on sites and show that they work almost the same as the usual presheaves and sheaves we are accostomed to. Then, we will define important sites such as the big and small Zariski sites and étale sites as well as the fppf site.

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