We construct a family of involutions on the space ${\mathfrak{g}\mathfrak{l}}_n^{\prime }(\mathbb{C})$ of $n\times n$ matrices with real eigenvalues interpolating the complex conjugation and the transpose. We deduce from it a stratified homeomorphism between the space ${\mathfrak{g}\mathfrak{l}}_n^{\prime }(\mathbb{R})$ of $n\times n$ real matrices with real eigenvalues and the space ${\mathfrak{p}}_n^{\prime }(\mathbb{C})$ of $n\times n$ symmetric matrices with real eigenvalues, which restricts to a real analytic isomorphism between individual ${\mathrm{GL}}_n(\mathbb{R})$-adjoint orbits and ${\mathrm{O}}_n(\mathbb{C})$-adjoint orbits. We also establish similar results in more general settings of Lie algebras of classical types and quiver varieties. To this end, we prove a general result about involutions on hyper-Kähler quotients of linear spaces. We provide applications to the (generalized) Kostant–Sekiguchi correspondence, singularities of real and symmetric adjoint orbit closures, and Springer theory for real groups and symmetric spaces.

We prove that any length metric space homeomorphic to a 2-manifold with boundary, also called a length surface, is the Gromov–Hausdorff limit of polyhedral surfaces with controlled geometry. As an application, using the classical uniformization theorem for Riemann surfaces and a limiting argument, we establish a general “one-sided” quasiconformal uniformization theorem for length surfaces with locally finite Hausdorff 2-measure. Our approach yields a new proof of the Bonk–Kleiner theorem characterizing Ahlfors 2-regular quasispheres.

We describe a conjectural formula via intersection numbers for the Masur–Veech volumes of strata of quadratic differentials with prescribed zero orders, and we prove the formula for the case when the zero orders are odd. For the principal strata of quadratic differentials with simple zeros, the formula reduces to computing the top Segre class of the quadratic Hodge bundle, which can be further simplified to certain linear Hodge integrals. An appendix proves that the intersection of this class with ψ-classes can be computed by Eynard–Orantin topological recursion.

As applications, we analyze numerical properties of Masur–Veech volumes, area Siegel–Veech constants, and sums of Lyapunov exponents of the principal strata for fixed genus and varying number of zeros, which settles the corresponding conjectures due to Grivaux and Hubert, Fougeron, and elaborated in Andersen et al. We also describe conjectural formulas for area Siegel–Veech constants and sums of Lyapunov exponents for arbitrary affine invariant submanifolds, and verify them for the principal strata.

For the two-dimensional random field Ising model where the random field is given by independent and identically distributed mean zero Gaussian variables with variance ${\mathit{\epsilon }}^2$, we study (one natural notion of) the correlation length, which is the critical size of a box at which the influences of the random field and of the boundary condition on the spin magnetization are comparable. We show that as $\mathit{\epsilon }\to 0$, at zero temperature the correlation length scales as $e^{\mathrm{\Theta }({\mathit{\epsilon }}^{-4∕3})}$ (and our upper bound applies for all positive temperatures).