“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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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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“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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“In mathematics the art of proposing a question must be held of higher value than solving it.”

George Cantor

This week I finally finished my final project for my PMATH 499 reading course in arithmetic geometry. My project is called “Galois cohomology and weak Mordell–Weil theorem”, which is available here. Galois cohomology is the application of group cohomology to Galois groups. In particular, we know that for a perfect field \(k\) with an algebraic closure \(K\), the Galois group \(\mathrm{Gal}(K/k)\) is isomorphic to an inverse limit of topological Galois groups \(\mathrm{Gal}(L/k)\) ranging over the finite Galois extensions \(L/k\) with the natural projections. This fact makes \(G\) a profinite group, and the Krull topology, the topology of \(G\) endowed by the inverse limit process by viewing each finite Galois group as a topological group with the discrete topology, is a topology where a basis for the neighborhood at the identity is the collection of normal subgroups having finite index in \(G\). This is a motivation to use homological algebra in such situation.

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I’ve been playing with the Manim library lately. Here’s a test animation I made demonstrating the nine point circle and part of Feuerbach’s theorem. It should be noted that Feuerbach’s Theorem also tells us that the nine point circle of a triangle is tangent to the three excircles of that triangle, but I did not include that in the video.