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We establish the -boundedness, for , 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 . 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 -functions averaged over Hilbert modular forms of a fixed weight and level. We apply these formulas to the subconvexity problem for these -functions. We also establish an equidistribution result for the Hecke eigenvalues weighted by these -values
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