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 Apr 25 comment Why is $W(V)\simeq D(k[X_1,\dots,X_n])$? @Kally what is your working definition of polynomial differential operators, if not the algebra generated by $x_i, \partial_j$ subject to the above relations? There are several equivalent definitions, so the answer depends on which you take as fundamental. Apr 25 answered Reference request for “Hodge Theorem” Apr 15 answered Symplectic positive definite matrix. Apr 14 comment Embedding/Submersion Properties of Cotangent Maps (Pullbacks) Yes, $d\phi$ (or $\phi_\ast$, or $T\phi$) is the differential of $\phi$. I think (4) should be true as well, since $\phi$ will be a diffeomorphism on the total spaces and it is linear on the fibers. Apr 14 answered Embedding/Submersion Properties of Cotangent Maps (Pullbacks) Apr 13 comment Why is $W(V)\simeq D(k[X_1,\dots,X_n])$? Updated to include a sketch of the proof. You can find the details in the lecture notes by Cannas da Silva. Apr 13 revised Why is $W(V)\simeq D(k[X_1,\dots,X_n])$? added 1002 characters in body Apr 13 answered formal glueing of schemes Apr 12 comment Exactness Axiom of Homology Theory $C_k(\cdot)$ are the chains, with boundary map $\partial: C_k(\cdot) \to C_{k-1}(\cdot)$, so that $H_k = \ker \partial_k / \mathrm{im} \partial_{k+1}$. They depend on what homology theory you are considering, but they are always there in the background somewhere. For the singular homology of a space $X$ they are formal linear combinations of maps $\Delta^k \to X$ where $\Delta^k$ is the standard $k$-simplex. For simplicial and cellular homology there are analogous definitions. Whenever you encounter any kind of homology you should think of it as the homology of some underlying chain complex. Apr 9 answered subgroup problem in abstract algebra Apr 8 comment Center of Clifford Algebra depending on the parity of $\dim V$? The parity issue is discussed in detail in E. Meinrenken's Notes math.toronto.edu/mein/teaching/clif_main.pdf . See Proposition 2.6 in particular, which gives the result over $\mathbb{C}$. Another good reference is Chevalley's book. Apr 8 revised Why is $W(V)\simeq D(k[X_1,\dots,X_n])$? small commment added Apr 8 answered Why is $W(V)\simeq D(k[X_1,\dots,X_n])$? Apr 7 answered Exactness Axiom of Homology Theory Mar 16 answered Structure Sheaf on Scheme Feb 13 answered Fourier transform of gaussian times polynomial to a high power Feb 9 comment Non-closed subgroups of Lie groups Irrational flows on the torus are well-studied (and famous) examples in noncommutative geometry. So it might be worth looking at the noncommutative geometry literature to see if there is anything like what you're asking. Feb 4 comment Expressing Differential Form in Different Coordinates @Confused I edited it to include a few more details of the evaluation of the pushforward. Feb 4 awarded Editor Feb 4 revised Expressing Differential Form in Different Coordinates added some extra detail