The goal of this paper is to express the extended Goldman symplectic structure on the $SL(n)$ character variety of a punctured Riemann surface in terms of Fock–Goncharov coordinates. The associated symplectic form has integer coefficients expressed via the inverse of the Cartan matrix. The main technical tool is a canonical two-form associated to a flat graph connection. We discuss the relationship between the extension of the Goldman Poisson structure and the Poisson structure defined by Fock and Goncharov. We elucidate the role of the Rogers’ dilogarithm as generating function of the symplectomorphism defined by a graph transformation.

Let $(M,g)$ be a smooth, compact Riemannian manifold and ${\lbrace \phi \lambda \rbrace}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-\Delta_g \phi_\lambda = \lambda^2 \phi_\lambda$. Given a smooth submanifold $H \subset M$ of codimension $k \geq 1$, we find conditions on the pair $(M,H)$, even when $H = {\lbrace x \rbrace}$, for which\[{\left\lvert \int_H \phi_\lambda d \sigma_H \right\rvert} = 0 \left( \dfrac{\lambda^\frac{k-1}{2}}{\sqrt{\log \lambda}} \right) \qquad \textrm{or} \qquad \left\lvert \phi_\lambda (x) \right\lvert = \left( \dfrac{\lambda^\frac{n-1}{2}}{\sqrt{\log \lambda}} \right)\]as $\lambda \to\infty$. These conditions require no global assumption on the manifold $M$ and instead relate to the structure of the set of recurrent directions in the unit normal bundle to $H$. Our results extend all previously known conditions guaranteeing improvements on averages, including those on sup-norms. For example, we show that if $(M,g)$ is a surface with Anosov geodesic flow, then there are logarithmically improved averages for any $H \subset M$. We also find weaker conditions than having no conjugate points which guarantee $\sqrt{\log \lambda}$ improvements for the $L^\infty$ norm of eigenfunctions. Our results are obtained using geodesic beam techniques, which yield a mechanism for obtaining general quantitative improvements for averages and sup-norms.

In this paper, we solve two problems dealing with the homogenization of random media. We show that a random quasiconformal mapping is close to an affine mapping, while a circle packing of a random Delauney triangulation is close to a conformal map, confirming a conjecture of K. Stephenson. We also show that on a Riemann surface equipped with a conformal metric, a random Delauney triangulation is close to being circle packed.

We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our construction recovers a family of harmonic maps originally studied by Wolf.

We employ these maps to solve a domination problem for representations. In particular, following ideas laid out by Deroin‑Tholozan, we prove that any representation from a finitely generated free group to the isometry group of a $\textrm{CAT}(-1)$ Hadamard manifold is strictly dominated in length spectrum by a large collection of Fuchsian ones. As an intermediate step in the proof, we obtain a result of independent interest: parametrizations of certain Teichmüller spaces by holomorphic quadratic differentials. The main consequence of the domination result is the existence of a new collection of anti‑de Sitter $3$-manifolds. We also present an application to the theory of maximal immersions into the Grassmanian of timelike planes in $\mathbb{R}^{2,2}$.