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.
We solve a well-known open problem in Ricci flow: Strong rigidity of cylinders. Strong rigidity is an illustration of a shrinker principle that uniqueness radiates out from a compact set. It implies that if one tangent flow at a future singular point is a cylinder, then all tangent flows are. At the heart of this problem in Ricci flow is comparing and recognizing metrics. This can be rather complicated because of the group of diffeomorphisms. Two metrics, that could even be the same, could look completely different in different coordinates. This is the gauge problem. Often it can be avoided if one uses some additional structure of the particular situation. The gauge problem is subtle for non-compact spaces without additional structure. We solve this gauge problem by solving a nonlinear system of PDEs. The PDE produces a diffeomorphism that fixes an appropriate gauge in the spirit of the slice theorem for group actions. We then show optimal bounds for the displacement function of the diffeomorphism. Strong rigidity relies on gauge fixing and several other new ideas. One of these is "propagation of almost splitting", another is quadratic rigidity in the right gauge, and a third is an optimal polynomial growth bound for PDEs that holds in great generality.
We establish the curious Lefschetz property for generic character varieties of Riemann surfaces conjectured by Hausel, Letellier and Rodriguez-Villegas. Our main tool applies directly in the case when there is at least one puncture where the local monodromy has distinct eigenvalues. We pass to a vector bundle over the character variety, which is then stratified into strata which look like vector bundles over varieties associated to positive braids. These varieties are in turn stratified into strata that look like 𝐂^*d-2k×𝐂^k . The curious Lefschetz property is shown to hold on each stratum, and therefore holds for the character variety. To deduce the general case, we introduce a fictitious puncture with trivial monodromy, and show that the cohomology of the character variety where one puncture has trivial monodromy is isomorphic to the sign component of the S_n action on the cohomology for the character variety where trivial monodromy is replaced by regular semisimple monodromy. This involves an argument with the Grothendieck-Springer sheaf, and analysis of how the cohomology of the character variety varies when the eigenvalues are moved around.
Let $K$ K be a finite extension of $\mathbf {Q}_{p}$ Q p , and $\rho $ ρ be an $n$ n -dimensional (non-critical generic) crystabelline representation of the absolute Galois group of $K$ K of regular Hodge-Tate weights. We associate to $\rho $ ρ an explicit locally $\mathbf {Q}_{p}$ Q p -analytic representation $\pi _{1}(\rho )$ π 1 ( ρ ) of $\operatorname{\mathrm {G}L}_{n}(K)$ GL n ( K ) , which encodes some $p$ p -adic Hodge parameters of $\rho $ ρ . When $K=\mathbf {Q}_{p}$ K = Q p , it encodes the full information hence reciprocally determines $\rho $ ρ . When $\rho $ ρ is associated to $p$ p -adic automorphic representations, we show under mild hypotheses that $\pi _{1}(\rho )$ π 1 ( ρ ) is a subrepresentation of the $\operatorname{\mathrm {G}L}_{n}(K)$ GL n ( K ) -representation globally associated to $\rho $ ρ .
RésuméNous démontrons que le feuilletage de Jouanolou de degré 2 sur le plan projectif complexe est structurellement stable. De plus, son ensemble de Fatou est une fibration holomorphe sur la quartique de Klein ayant une structure de fibré lisse localement trivial en disques. En particulier, aucune feuille de $\mathcal{J}_{2}$ J 2 n’est dense dans $\mathbf{P}^{2}$ P 2 .
We prove that uniformly locally finite metric spaces with isomorphic Roe algebras must be coarsely equivalent. As an application, we also prove that the outer automorphism group of the Roe algebra of such a metric space is canonically isomorphic to the group of coarse equivalences of the space up to closeness.
Let K be a finite extension of 𝐐_p , and ρ be an n -dimensional (non-critical generic) crystabelline representation of the absolute Galois group of K of regular Hodge-Tate weights. We associate to ρ an explicit locally 𝐐_p -analytic representation π _1(ρ ) of GL_n(K) , which encodes some p -adic Hodge parameters of ρ . When K=𝐐_p , it encodes the full information hence reciprocally determines ρ . When ρ is associated to p -adic automorphic representations, we show under mild hypotheses that π _1(ρ ) is a subrepresentation of the GL_n(K) -representation globally associated to ρ .
In this paper, we prove the generalised Andr\'e-Pink-Zannier conjecture (an important case of the Zilber-Pink conjecture) for all Shimura varieties of abelian type. Questions of this type were first asked by Y. Andr\'e in 1989. We actually prove a general statement for all Shimura varieties, subject to certain assumptions that are satisfied for Shimura varieties of abelian type and are expected to hold in general. We also prove another result, a p-adic Kempf-Ness theorem, on the relation between good reduction of homogeneous spaces over p-adic integers with Mumford stability property in p-adic geometric invariant theory.
We show that the Jouanolou foliation of degree 2 on the complex projective plane is structurally stable. Moreover, its Fatou set is a fibration on the Klein quartic with the structure of a smooth fiber bundle in disks. In particular, there is no dense leaf.
We explicitly construct a pair of immersed tori in three dimensional Euclidean space that are related by a mean curvature preserving isometry. These Bonnet pair tori are the first examples of compact Bonnet pairs. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique smooth compact immersion. Moreover, we prove these isometric tori are real analytic. This resolves a second longstanding open problem on whether real analyticity of the metric already determines a unique compact immersion. Our construction uses the relationship between Bonnet pairs and isothermic surfaces. The Bonnet pair tori arise as conformal transformations of an isothermic torus with one family of planar curvature lines. We classify such isothermic tori in our companion paper (arXiv:2312.14956). The above approach stems from computational investigations of a 5x7 quad decomposition of a torus using a discrete differential geometric analog of isothermic surfaces and Bonnet pairs.
We consider integral area-minimizing $2$-dimensional currents $T$ in $U\subset \mathbb R^{2+n}$ with $\partial T = Q[\![\Gamma]\!]$, where $Q\in \mathbb N \setminus \{0\}$ and $\Gamma$ is sufficiently smooth. We prove that, if $q\in \Gamma$ is a point where the density of $T$ is strictly below $\frac{Q+1}{2}$, then the current is regular at $q$. The regularity is understood in the following sense: there is a neighborhood of $q$ in which $T$ consists of a finite number of regular minimal submanifolds meeting transversally at $\Gamma$ (and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for $Q=1$. As a corollary, if $\Omega\subset \mathbb R^{2+n}$ is a bounded uniformly convex set and $\Gamma\subset \partial \Omega$ a smooth $1$-dimensional closed submanifold, then any area-minimizing current $T$ with $\partial T = Q [\![\Gamma]\!]$ is regular in a neighborhood of $\Gamma$.
For both the cubic Nonlinear Schrödinger Equation (NLS) as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension we consider the set 𝐌_N of pure N -soliton states, and their associated multisoliton solutions. We prove that (i) the set 𝐌_N is a uniformly smooth manifold, and (ii) the 𝐌_N states are uniformly stable in H^s , for each s>-1/2 . One main tool in our analysis is an iterated Bäcklund transform, which allows us to nonlinearly add a multisoliton to an existing soliton free state (the soliton addition map) or alternatively to remove a multisoliton from a multisoliton state (the soliton removal map). The properties and the regularity of these maps are extensively studied.
In this article we construct many examples of properly convex irreducible domains divided by Zariski dense relatively hyperbolic groups in every dimension at least 3. This answers a question of Benoist. Relative hyperbolicity and non-strict convexity are captured by a family of properly embedded cones (convex hulls of points and ellipsoids) in the domain. Our construction is most flexible in dimension 3 where we give a purely topological criterion for the existence of a large deformation space of geometrically controlled convex projective structures with totally geodesic boundary on a compact 3-manifold.
We study the Fröhlich polaron model in 𝐑^3 , and prove a lower bound on its ground state energy as a function of the total momentum. The bound is asymptotically sharp at large coupling. In combination with a corresponding upper bound proved earlier (Mitrouskas et al. in Forum Math. Sigma 11:1–52, 2023), it shows that the energy is approximately parabolic below the continuum threshold, and that the polaron’s effective mass (defined as the semi-latus rectum of the parabola) is given by the celebrated Landau–Pekar formula. In particular, it diverges as α ^4 for large coupling constant α .
Poincaré’s work more than one century ago, or Laskar’s numerical simulations from the 1990’s on, have irrevocably impaired the long-held belief that the Solar System should be stable. But mathematical mechanisms explaining this instability have remained mysterious. In 1968, Arnold conjectured the existence of “Arnold diffusion” in celestial mechanics. We prove Arnold’s conjecture in the planetary spatial 4-body problem as well as in the corresponding hierarchical problem (where the bodies are increasingly separated), and show that this diffusion leads, on a long time interval, to some large-scale instability. Along the diffusive orbits, the mutual inclination of the two inner planets is close to π /2 , which hints at why even marginal stability in planetary systems may exist only when inner planets are not inclined. More precisely, consider the normalised angular momentum of the second planet, obtained by rescaling the angular momentum by the square root of its semimajor axis and by an adequate mass factor (its direction and norm give the plane of revolution and the eccentricity of the second planet). It is a vector of the unit 3-ball. We show that any finite sequence in this ball may be realised, up to an arbitrary precision, as a sequence of values of the normalised angular momentum in the 4-body problem. For example, the second planet may flip from prograde nearly horizontal revolutions to retrograde ones. As a consequence of the proof, the non-recurrent set of any finite-order secular normal form accumulates on circular motions – a weak form of a celebrated conjecture of Herman.
