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Question about differential operators in a completely non-integrable distribution

Consider the following example: On $\mathbb{R}^4$ with coordinates $u^1,u^2,u^3, z^1$, define $z^2 = u^2 - z^1 u^1$ and $z^3 = u^3 - z^1u^2$. We have that $U\cap Z$ is spanned by $\mathrm{d}u^2-z^1\,\...
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Why are we interested in spectral gaps for Laplacian operators

A spectral gap gives information on geometry of the manifold via Cheeger's inequality, https://en.wikipedia.org/wiki/Cheeger_constant See also Buser's inequality discussed there. More directly, a ...
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Harmonic polynomials on the sphere

I view this as a concatenation of two facts: Fact 1: Let $k$ be a field, let $I$ be an ideal of $k[x_1, \ldots, x_n]$ and let $J$ be associated graded ideal of $I$, meaning that a degree $d$ ...
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Harmonic polynomials on the sphere

There is a similar separation of the variables for ellipsoids $x^2/a^2+y^2/b^2+z^2/c^2=1$ related to ellipsoidal harmonic functions, see Whittaker Watson, Chap. 23. There is a generalization to ...
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