We determine the Lyapunov spectrum of ball quotients arising from cyclic coverings. The computations are performed by rewriting the sum of Lyapunov exponents as ratios of intersection numbers and by the analysis of the period map near boundary divisors.

As a corollary, we complete the classification of commensurability classes of all presently known nonarithmetic ball quotients.

We study the vertex algebras associated with modular invariant representations of affine Kac–Moody algebras at fractional levels, whose simple highest weight modules are classified by Joseph’s characteristic varieties. We show that an irreducible highest weight representation of a nontwisted affine Kac–Moody algebra at an admissible level $k$ is a module over the associated simple affine vertex algebra if and only if it is an admissible representation whose integral root system is isomorphic to that of the vertex algebra itself. This in particular proves the conjecture of Adamović and Milas on the rationality of admissible affine vertex algebras in the category $\mathcal{O}$.

The problem of estimating the multiplicity of the zero of a polynomial when restricted to the trajectory of a nonsingular polynomial vector field, at one or several points, has been considered by authors in several different fields. The two best (incomparable) estimates are due to Gabrielov and Nesterenko.

In this paper we present a refinement of Gabrielov’s method which simultaneously improves these two estimates. Moreover, we give a geometric description of the multiplicity function in terms of certain naturally associated polar varieties, giving a topological explanation for an asymptotic phenomenon that was previously obtained by elimination-theoretic methods in the works of Brownawell, Masser, and Nesterenko. We also give estimates in terms of Newton polytopes, strongly generalizing the classical estimates.

The period-doubling Cantor sets of strongly dissipative Hénon-like maps with different average Jacobian are not smoothly conjugated, as was shown previously. The Jacobian rigidity conjecture says that the period-doubling Cantor sets of two-dimensional Hénon-like maps with the same average Jacobian are smoothly conjugated. This conjecture is true for average Jacobian zero, for example, the one-dimensional case. The other extreme case is when the maps preserve area, for example, when the average Jacobian is one. Indeed, the main result presented here is that the period-doubling Cantor sets of area-preserving maps in the universality class of the Eckmann–Koch–Wittwer renormalization fixed point are smoothly conjugated.

Building on the theory of parity sheaves due to Juteau, Mautner, and Williamson, we develop a formalism of “mixed modular perverse sheaves” for varieties equipped with a stratification by affine spaces. We then give two applications: (1) a “Koszul-type” derived equivalence relating a given flag variety to the Langlands dual flag variety and (2) a formality theorem for the modular derived category of a flag variety (extending a previous result of Riche, Soergel, and Williamson).