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.

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).

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.

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.

Un des résultats principaux de ce travail est l’absence de torsion dans la ${\bar{\mathbb{Z}}}_l$-cohomologie de la tour de Lubin–Tate lorsque $l>2$. Comme dans mon article à Inventiones en 2009, la stratégie est globale et repose sur l’étude des diverses filtrations de stratification du faisceau pervers des cycles évanescents de certaines variétés de Shimura de type Kottwitz–Harris–Taylor. On montre que les gradués, pour la filtration utilisant les morphismes d’adjonction $j_{!}{j}^{\ast }⟶Id$, se décrivent comme la p-extension intermédiaire de systèmes locaux construits par Harris et Taylor dans leur livre. Le point crucial consiste à décrire la différence entre les p et $p+$ extensions de ces systèmes locaux. Pour ce faire, on utilise la théorie des dérivées pour les représentations du groupe mirabolique ainsi que des filtrations construites à partir de la stratification de Newton.

One of the main results of this work is the freeness in the ${\bar{\mathbb{Z}}}_l$-cohomology of the Lubin–Tate tower when $l>2$. As in my 2009 Inventiones paper, the strategy is of global nature and relies on studying the various filtration of stratification of the perverse sheaf of vanishing cycles of some Shimura varieties of Kottwitz–Harris–Taylor types. We show that these graded parts, when considering only the adjunction maps $j_{!}{j}^{\ast }⟶Id$, are given by the p-intermediate extensions of the local systems constructed by Harris and Taylor in their book. The crucial point relies on the study of the difference between the p and $p+$ intermediate extensions of these local systems. The main ingredients use the theory of derivative for representations of the mirabolic group and filtrations given by the Newton stratification.

We prove a quantitative estimate with a power saving error term for the number of points in a mapping class group orbit of Teichmüller space that lie within a Teichmüller metric ball of given center and large radius. Estimates of the same kind are also proved for sector and bisector counts. These estimates effectivize asymptotic counting results of Athreya, Bufetov, Eskin, and Mirzakhani.