We express the Masur–Veech volume and the area Siegel–Veech constant of the moduli space ${\mathcal{Q}}_{g,n}$ of genus g meromorphic quadratic differentials with at most n simple poles and no other poles as polynomials in the intersection numbers ${\int }_{{\bar{\mathcal{M}}}_{g^{\prime },n^{\prime }}}{\mathit{\psi }}_1^{d_1}\cdots {\mathit{\psi }}_{n^{\prime }}^{d_{n^{\prime }}}$ with explicit rational coefficients, where $g^{\prime }<g$ and $n^{\prime }<2g+n$. The formulas obtained in this article are derived from lattice point counts involving the Kontsevich volume polynomials $N_{g^{\prime },n^{\prime }}(b_1,\dots ,b_{n^{\prime }})$ that also appear in Mirzakhani’s recursion for the Weil–Petersson volumes of the moduli spaces ${\mathcal{M}}_{g^{\prime },n^{\prime }}(b_1,\dots ,b_{n^{\prime }})$ of bordered hyperbolic surfaces with geodesic boundaries of lengths $b_1,\dots ,b_{n^{\prime }}$. A similar formula for the Masur–Veech volume (but without explicit evaluation) was obtained earlier by Mirzakhani through a completely different approach. We prove a further result: the density of the mapping class group orbit ${\mathrm{Mod}}_{g,n}\cdot \mathit{\gamma }$ of any simple closed multicurve γ inside the ambient set ${\mathcal{ML}}_{g,n}(\mathbb{Z})$ of integral measured laminations, computed by Mirzakhani, coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to γ among all square-tiled surfaces in ${\mathcal{Q}}_{g,n}$. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case when $n=0$. In particular, we compute explicitly the asymptotic frequencies of separating and nonseparating simple closed geodesics on a closed hyperbolic surface of genus g for all small genera g, and we show that in large genera the separating closed geodesics are $\sqrt{\frac{2}{3\mathrm{\pi }g}}\cdot \frac{1}{4^g}$ times less frequent.

A cohomology class of a smooth complex variety of dimension n has coniveau at least c if it vanishes in the complement of a closed subvariety of codimension at least c, and it has strong coniveau at least c if it comes by proper pushforward from the cohomology of a smooth variety of dimension at most $n-c$. We show that these two notions differ in general, both for integral classes on smooth projective varieties and for rational classes on smooth open varieties.

We study the evolution of complete, noncompact, convex hypersurfaces in ${\mathbb{R}}^{n+1}$ by the inverse mean curvature flow. We establish the long-time existence of solutions, and we provide the characterization of the maximal time of existence in terms of the tangent cone at infinity of the initial hypersurface. Our proof is based on an a priori pointwise estimate on the mean curvature of the solution from below in terms of the aperture of a supporting cone at infinity. The strict convexity of convex solutions is shown by means of viscosity solutions. Our methods also give an alternative proof of a result by Huisken and Ilmanen on compact star-shaped solutions.

We give a new proof of the so-called Lie algebra version of the Jacquet–Rallis fundamental lemma for local non-Archimedean fields of characteristic 0. This proof is local and based on a previous result of Zhang on the compatibility of smooth transfer with a (partial) Fourier transform.