“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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“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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“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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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.

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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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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