If $\Gamma <PSL(2,\mathbb{C})$ is a lattice, we define an invariant of a representation $\Gamma \to PSL(n,\mathbb{C})$ using the Borel class $\beta \left(n\right)\in {\mathrm{H}}_{\mathrm{c}}^3(PSL(n,\mathbb{C}),\mathbb{R})$. We show that this invariant satisfies a Milnor–Wood type inequality and its maximal value is attained precisely by the representations conjugate to the restriction to $\Gamma$ of the irreducible complex $n$-dimensional representation of $PSL(2,\mathbb{C})$ or its complex conjugate. Major ingredients of independent interest are the study of our extension to degenerate configurations of flags of a cocycle defined by Goncharov, as well as the identification of ${\mathrm{H}}_{\mathrm{b}}^3(SL(n,\mathbb{C}),\mathbb{R})$ as a normed space.

We prove that the coefficients of the generating function of Hausel, Letellier, and Rodriguez-Villegas and its recent generalization by Carlsson and Rodriguez-Villegas, which according to various conjectures should compute mixed Hodge numbers of character varieties and moduli spaces of Higgs bundles of curves of genus $g$ with $n$ punctures, are polynomials in $q$ and $t$ with integer coefficients for any $g,n\ge 0$.

We consider the cubic nonlinear Schrödinger (NLS) equation as well as the modified Korteweg–de Vries (mKdV) equation in one space dimension. We prove that for each $s>-\frac{1}{2}$ there exists a conserved energy which is equivalent to the $H^s$-norm of the solution. For the Korteweg–de Vries (KdV) equation, there is a similar conserved energy for every $s\ge -1$.