We establish the $L^p$-boundedness, for $1<p\le \infty$, of maximal averages over sets defined by a finite number of monomial inequalities. The proof requires a detailed study of the geometric properties of these monomial polyhedra

We generalise the McShane-Mirzakhani identities from hyperbolic geometry to arbitrary cross ratios. We define and study Hitchin representations of open surface groups to $\mathrm{PSL}(n,\mathbb{R})$. We associate cross ratios to these representations and then give explicit expressions for our generalised identities in terms of (a suitable choice of) Fock-Goncharov coordinates

We use the relative trace formula to obtain exact formulas for central values of certain twisted quadratic base change $L$-functions averaged over Hilbert modular forms of a fixed weight and level. We apply these formulas to the subconvexity problem for these $L$-functions. We also establish an equidistribution result for the Hecke eigenvalues weighted by these $L$-values