Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Coarse density of subsets of $M_g$   Let $\mathcal{M}_g$ be the moduli space of genus $g$ Riemann surfaces. We
show that an algebraic subvariety of $\mathcal{M}_g$ is coarsely dense with
respect to the Teichm\"uller metric (or Thurston metric) if and only if it is
all of $\mathcal{M}_g$. We apply this to projections of
$\operatorname{GL}_2(\mathbb{R})$-orbit closures in the space of abelian
differentials. Moreover, we determine which strata of abelian differentials
have coarsely dense projection to $\mathcal{M}_g$.

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The categories of this paper are: physics
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The categories of this paper are: q-alg
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-th
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The categories of this paper are: math-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin
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The categories of this paper are: cs
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The categories of this paper are: funct-an