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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 …

A geometry question that I thought about more seriously a few years ago... thought it'd be a good first question for MO. I'm aware that there are a number of Torelli type theorems now proven for comp …

I don't know anything about quantum groups, but the Peter-Weyl theorem for compact groups generalizes nicely to Type I second-countable locally compact topological groups, a result of Segal and Mautne …

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 …

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 …

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 …

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 …

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 …

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 …

Though I'm not an expert on motives, by any measure, I think that an answer to your question can be given by considering periods. As Kontsevich and Zagier recall in their paper "Periods", publ. IHES, …

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 …

Near the bottom of Jacob Lurie's homepage, you can find an exposition of the Borel-Weil-Bott theorem from an algebro-geometric standpoint. It is "easily readable" if you're familiar with the things l …

I'll try to answer both questions, though I will change the first question somewhat. Let's work in the setting of a real reductive algebraic group $G$ and a closed subgroup $H \subset G$. Your fir …

Maybe I can sketch an argument for your first question. Let $P$ be the ring of effective "formal" periods, generated by quadruples $[X,D,\omega,\gamma]$ consisting of a smooth projective $Q$-variet …

One suggestion: "Period mappings and Period Domains", by Carlson, Muller-Stach, and Peters, in the Cambridge studies in advanced mathematics series. It's a very nice read, and each chapter comes wit …