“And every science, when we understand it not as an instrument of power and domination but as an adventure in knowledge pursued by our species across the ages, is nothing but this harmony, more or less vast, more or less rich from one epoch to another, which unfurls over the course of generations and centuries, by the delicate counterpoint of all the themes appearing in turn, as if summoned from the void.”

Alexander Grothendieck

This post is about symmetric polynomials. A symmetric polynomial is \(f(x_1,\dots,x_n)\in \mathbb Z[x_1,\dots,x_n]\) such that for any permutation \(\sigma\in S_n\), we have \(\sigma f=f\) where \(\sigma\) acts by permuting the variables. The symmetric polynomials form a subring of \(\mathbb Z[x_1,\dots,x_n]\) which we denote as \(\Lambda_n\). One example of symmetric polynomials is the Newton power sums \(p_k=x_1^k+\cdots+x_n^k\) for \(k\ge 1\). Another example is elementary symmetric polynomials \(e_k=\sum_{1\le i_1<\cdots< i_k\le n}x_{i_1}\cdots x_{i_k}\) for \(1\le k\le n\). They arise in Vieta’s relations for a polynomial equation. Obviously \(\Lambda_n\) is a graded ring in degree. For a monomial \(u=ax^{k_1}_1\cdots x^{k_n}_n\), we define its weight as \(\mathrm{wt}(u)=k_1+2k_2+\cdots+nk_n\), and for a polynomial \(f\) define \(\mathrm{wt}(f)\) as the largest weight occuring among its monomials. We note that the weight of a polynomial is the same as the degree of \(f(e_1,\dots,e_n)\). One of the first results in symmetric polynomials is that the elementary symmetric polynomials forms a generating set of \(\Lambda_n\) as a (graded) algebra over \(\mathbb Z\). This is our first theorem.

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“What is mathematics? It is only a systematic effort of solving puzzles posed by nature.”

Shakuntala Devi

Recently I’ve been learning about Grothendieck-Witt rings because I want to know about \(\mathbb A^1\)-enumerative geometry. The Grothendieck-Witt ring of a field (of characteristic different from \(2\)) is formed using quadratic forms over it. Let \(k\) be a field of characteristic not \(2\). A quadratic form is a degree \(2\) homogeneous polynomial over \(k\), say \(f=\sum_{1\le i,j\le n}a_{i,j}x_{i}x_{j}\). Of course \(x_{i}x_{j}=x_{j}x_{i}\), so we can rewrite \(f=\sum_{1\le i,j\le n}a^\prime_{i,j}x_{i}x_{j}\) where \(a^\prime_{i,j}=\frac{1}{2}(a_{i,j}+a_{j,i})\), we can then associate to it a symmetric matrix \(M_f=(a^\prime_{i,j})_{1\le i,j\le n}\). We see that \(f=\mathbf{x}^{\mathrm T}M_f\mathbf{x}\) where \(\mathbf{x}=(x_i)_{1\le i\le n}\). Two \(n\)-ary quadratic forms \(f,g\) are said to be equivalent if there is \(A\in \mathrm{GL}_n(k)\) such that \(f(\mathbf{x})=g(A\mathbf{x})\). Equivalently their associated symmetric matrix are equivalent iff \(M_f=A^{\mathrm T}M_g A\) for some \(A\in \mathrm{GL}_n(k)\), i.e. congruence of matrices.

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“The pursuit of mathematics is a divine madness of the human spirit.”

Alfred North Whitehead

Yesterday I gave a talk on OGMC (Ontario Graduate Math Conference) at Waterloo on rigid analytic geometry. Here are the slides. I had a lot of fun at OGMC, and went to lots of great talks. The talks I liked the most was one on algebraic K-theory, and another one on categorical logic (eventhough I can’t say I understood most of them). They seem to be very interesting topics.

“God exists since mathematics is consistent, and the Devil exists since we cannot prove it.”

André Weil

I’ve recently finished my final project for my algebraic number theory course on adeles and ideles, which can be found here. Adeles is an object in algebraic number theory that solves the technical problem of doing analysis over \(\mathbb Q\) so to speak. It lets us work over all completions of a global field simultaneously. One of the applications of it is that certain compactness theorems on it proves the Dirichlet unit theorem and class number theorem. If I had more time, I would have elaborated further at the last part in my project but i unfortunately did not have enough time.

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“Mathematics is not the rigid and rigidity-producing schema that the layman thinks it is; rather, in it we find ourselves at that meeting point of constraint and freedom that is the very essence of human nature.”

Hermann Weyl

Dold-Kan correspondence is a basic result in homotopy theory that establishes correspondence between simplicial abelian groups and (connective) chain complexes of abelian groups. This is given by functors that form an equivalence of categories between the category of simplicial abelian groups \(\mathbf{SAb}\) and the category of connective chain complexes of abelian groups \(\mathrm{Ch}_{+}(\mathbf{Ab})\). Thus, this correspondence interpolates between homological algebra and (simplicial) homotopy theory.

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“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.

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“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.

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“Algebra is but written geometry and geometry is but figured algebra.”

Sophie Germain

Recall that for a topological space \(X\), a fibre bundle of fibres \(F\), where \(F\) is a topological space, is a topological space \(E\), with a surjective map \(\pi: E\rightarrow X\) such that for all \(x\in X\), exists an open nbhd \(x\in U\subseteq X\) such that there is a homeomorphism \(\varphi: \pi^{-1}(U)\rightarrow U\times F\) such that \(\mathrm{pr}_U\circ \varphi=\pi|_{\pi^{-1}(U)}\) where \(\mathrm{pr}_U:U\times F\rightarrow U\) is the projection onto \(U\). Fibre bundles generalize vector bundles and covering spaces. In this post, we interpret this algebraically and generalize this notion to schemes.

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“I am not saying that I believe in the law of the excluded middle, I am just saying that it isn’t not true. ”

Kavin Satheeskumar

Recently, I’ve been reading Cisinski’s Higher Categories and Homotopical Algebra. I wanted to write down some of the things I’ve learned. In this post, we start from a review of presheaves of sets and simplicial sets, and then build towards the definition of an \(\infty\)-category. Let \(A\) be a category, recall a presheaf on \(A\) is a contravariant functor \(X:A^{\mathrm{op}}\rightarrow \mathbf{Set}\), where we denote \(X_a=X(a)\) and \(u^*:b\rightarrow a\) the induced morphism for each \(u:a\rightarrow b\). And the category of presheaves on \(A\) is denoted as \(\widehat{A}\). The category of elements \(\int_AX\) (or \(\int X\)) is the category where objects are \((a,s)\) where \(a\in A\) and \(s\in X_a\), and a morphisms \(u:(a,s)\rightarrow (b,t)\) is a morphism \(u:a\rightarrow b\) where \(u^*(t)=s\). It comes equipped with a faithful functor \(\varphi_X:\int_A X\rightarrow \widehat{A}\) given by \((a,s)\mapsto \mathscr{H}_a\) on objects and \(u\mapsto \mathscr{H}(u)\) on morphisms, where \(\mathscr{H}:A\rightarrow \widehat{A}\) is the Yoneda embedding. In this post, we ignore all set-theoretic size issues.

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“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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