“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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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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Last week I gave a talk on the Canadian Undergraduate Mathematics Conference 2022 on Schubert calculus and cohomology of Grassmannians. Here are the slides of my talk.

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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. Here are my slides.

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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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