We prove a priori bounds for Feigenbaum quadratic polynomials, i.e., infinitely renormalizable polynomials $f_c: z\mapsto z^2+c$ of bounded type. It implies local connectivity of the corresponding Julia sets $J(f_c)$ and MLC (local connectivity of the Mandelbrot set) at the corresponding parameters $c$. It also yields the scaling Universality, dynamical and parameter, for the corresponding combinatorics. The MLC Conjecture was open for the most classical period-doubling Feigenbaum parameter as well as for the complex tripling renormalizations. Universality for the latter was conjectured by Goldberg-Khanin-Sinai in the early 1980s.
In a previous paper we constructed $\textit{higher}$ theta series for unitary groups over function fields, and conjectured their modularity properties. Here we prove the generic modularity of the $\ell$-adic realization of higher theta series in cohomology. The proof debuts a new type of Fourier transform, occurring on the Borel-Moore homology of moduli spaces for shtuka-type objects, that we call the $\textit{arithmetic Fourier transform}$. Another novelty in the argument is a $\textit{sheaf-cycle correspondence}$ extending the classical sheaf-function correspondence, which facilitates the deployment of sheaf-theoretic methods to analyze algebraic cycles. Although the modularity property is a statement within classical algebraic geometry, the proof relies on derived algebraic geometry, especially a nascent theory of $\textit{derived Fourier analysis}$ on derived vector bundles, which we develop.
An endomorphism f:ℙ^k→ℙ^k of degree d≥2 is said to be postcritically finite (or PCF) if its critical set Crit(f) is preperiodic, i.e. if there are integers m>n≥0 such that f^m(Crit(f))⊆ f^n(Crit(f)). When k≥2, it was conjectured by Ingram, Ramadas and Silverman that, in the space End_d^k of all endomorphisms of degree d of ℙ^k, such endomorphisms are not Zariski dense. We prove this conjecture. Further, in the space Poly_d^2 of all regular polynomial endomorphisms of degree d≥2 of the affine plane 𝔸^2, we construct a dense and Zariski open subset where we have a uniform bound on the number of preperiodic points lying in the critical set. The proofs are a combination of the theory of heights in arithmetic dynamics and methods from real dynamics to produce open subsets with maximal bifurcation.
We prove that singularities of area minimizing hypersurfaces can be perturbed away in ambient dimensions 9 and 10.
The Mordell–Lang conjecture for abelian varieties states that the intersection of an algebraic subvariety X with a subgroup of finite rank is contained in a finite union of cosets contained in X. In this article, we prove a uniform version of this conjecture, meaning that that the number of cosets necessary does not depend on the ambient abelian variety. To achieve this, we prove a general gap principle on algebraic points that extends the gap principle for curves embedded into their Jacobians, previously obtained by Dimitrov–Gao–Habegger and Kühne. Our new gap principle also implies the full uniform Bogomolov conjecture in abelian varieties.
