“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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“Teaching is not about information. It’s about having an honest intellectual relationship with your students.”

Paul Lockhart

It’s been 14 years since the publication of Paul Lockhart’s celebrated Lament, in which he envisioned a pedagogy of mathematics that emphasizes aesthetics, inquiry, and intuition. Lockhart’s critical insights resonated with me profoundly, however, after more than a decade, his vision has not materialized. In this post, I wish to delineate more concretely what I have in mind a curriculum in his spirit, which includes mathematics as well as other subjects, would look like if it were to be implemented.

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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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“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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