Many interesting spaces—including all positroid strata and wild character varieties—are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact Lagrangian fillings of Legendrian links in the standard contact three space.

In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank $2$. Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest. We prove that these representations are holonomies of certain geometric structures, recovering results of Guichard and Wienhard. We also prove that their length spectrum is uniformly bigger than that of a suitably chosen Fuchsian representation, extending a previous work of the second author. Finally, we show that these representations preserve a unique minimal surface in the symmetric space, extending a theorem of Labourie for Hitchin representations in rank $2$.

For an abelian variety $A$ over a number field $F$, we prove that the average rank of the quadratic twists of $A$ is bounded, under the assumption that the multiplication-by-$3$-isogeny on $A$ factors as a composition of $3$-isogenies over $F$. This is the first such boundedness result for an absolutely simple abelian variety $A$ of dimension greater than $1$. In fact, we exhibit such twist families in arbitrarily large dimension and over any number field. In dimension $1$, we deduce that if $E/F$ is an elliptic curve admitting a $3$-isogeny, then the average rank of its quadratic twists is bounded. If $F$ is totally real, we moreover show that a positive proportion of twists have rank $0$ and a positive proportion have $3$-Selmer rank $1$. These results on bounded average ranks in families of quadratic twists represent new progress toward Goldfeld’s conjecture—which states that the average rank in the quadratic twist family of an elliptic curve over $\mathbb{Q}$ should be $1/2$—and the first progress toward the analogous conjecture over number fields other than $\mathbb{Q}$. Our results follow from a computation of the average size of the $\varphi$-Selmer group in the family of quadratic twists of an abelian variety admitting a $3$-isogeny $\varphi$.