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I think that going all the way to barcodes and persistent homology is a big leap, and probably not one where there will be something interesting. But maybe there are interesting things if you just go …

Here are some more details. As John Voight said, the quaternion algebra is kind of irrelevant here. If $\gamma$ is a regular semisimple element, then its centralizer is a torus ${\mathbf T}$ over ${ …

Consider a DVR $A$ with fraction field $K$ and residue field $k$. Assume $2 \in A^\times$. Let $Q: A^n \rightarrow A$ be a quadratic form defined over $A$. Then one has the (naively defined) specia …

This might shed some light on relationship between anisotropic quadratic forms in 10 variables and the desired forms of $E_7$, though it uses results more recent than Tits, and doesn't quite answer yo …

I'm having trouble finding a reference for something that I'm guessing the experts worked out long ago. Let's take a local or global field $F$ for this post, and fix a separable algebraic closure $\b …

Maybe this isn't an answer, but below is a photograph of the tablet BM15285 (British Museum catalog #15285). It's a series of geometry problems, from c.1800 BCE (+/- 200 years?). There are plenty mo …

I'll take a stab at answering this controversial question in a way that might satisfy the OP and benefit the mathematical community. I also want to give some opinions that contrast with or at least c …

The places to look are: Steinberg, "Endomorphisms of linear algebraic groups." Memoir AMS 80, (1968), and Gross, "The motive of a reductive group" Invent. math. 130, 287 ± 313 (1997). (I learne …

A no-nonsense construction, over $Z$, following work of Kostant and Chevalley, is given in Lusztig's paper "Twelve bridges from a reductive group to its Langlands dual". The heart of the construction …

In my opinion, instead of a "height" on the Weil-Chatelet group, one should consider a "depth", using the local duality between the points on an elliptic curve and the elements of the Weil-Chatelet gr …

Over any field $k$, $\hat A=Ext(A,G_m)$ in the abelian category (see "Is the category of commutative group schemes abelian" here on MO) of commutative group schemes of finite type over $k$. There is a …

Excuse the possible naivete of this question. Since reading a nice survey article by Daniel Biss a few years ago, I'm always worried about what $P^1(R)$ is, for a ring $R$. So that I stop worrying, …

Let $G$ be a connected reductive group over a (perfect, why not) field $F$. Let $m$, $pr_1$, $pr_2$ denote the multiplication, first, and second projection maps from $G \times G$ to $G$. Then I'm pr …

Let $X$ be a (smooth) compact complex manifold, and suppose that $H^1(X, \Theta_X) = 0$, where $\Theta_X$ is the tangent sheaf. In other words, suppose that $X$ is rigid. Suppose moreover that $X$ a …

Brian's comment does what you want, and describes the almost direct product caveat. A standard and excellent reference for all things of this nature is Demazure's "Sous-groupes Paraboliques des group …