We resolve an open problem posed by Alexeev and Knutson on the projectivity of the moduli of branchvarieties in the equidimensional case. As an application, we construct projective moduli spaces of reduced equidimensional varieties equipped with ample linear series and subject to a semistability condition.

We describe the structure of the Whittaker or Gelfand–Graev module on an n-fold metaplectic cover of a p-adic group G at both the Iwahori and spherical level. Following a conjecture of Gaitsgory and Lurie, we express our answer in terms of the representation theory of a quantum group at a root of unity attached to the Langlands dual group of G. To do so, we introduce an algebro-combinatorial model for our p-adic Whittaker modules and develop for them a Kazhdan–Lusztig theory involving several generic parameters. These parameters can either be specialized to Gauss sums to recover the p-adic theory or to the grading parameter in the representation theory of quantum groups. As applications, we deduce geometric Casselman–Shalika type results for metaplectic covers as well as a variant of Savin’s local Shimura type correspondences at the Whittaker level.

We prove the Zilber–Pink conjecture for curves in Y(1)n whose Zariski closure in (P1)n passes through the point (∞,…,∞), going beyond the asymmetry condition of Habegger and Pila. Our proof is based on a height bound following André’s G-functions method. The principal novelty is that we exploit relations between evaluations of G-functions at unboundedly many non-Archimedean places.

We describe a general method for giving p-adic interpretations of G-functions arising from degenerating periods of smooth projective algebraic varieties. Using this, we are able to implement a strategy due to André for bounding heights of moduli points where period functions acquire unusual algebraic relations. This leads to new results on Galois lower bounds for special moduli, and new cases of the Zilber–Pink conjecture. In particular, we establish the first Galois-orbit lower bounds on CM moduli in non-Shimura settings. As a more technical contribution, we introduce a streamlined Pila–Zannier strategy capable of handling Zilber–Pink-type atypical intersection problems in arbitrary dimension and for arbitrary smooth projective families.

It has long been conjectured that the classical Lorenz attractor supports a unique measure of maximal entropy. In this article, we give a positive answer to this conjecture and its higher-dimensional counterpart by considering the uniqueness of equilibrium states for Hölder continuous functions on a sectional-hyperbolic attractor Λ. We prove that in a C1 open and dense family of vector fields (including the classical Lorenz attractor), if the point masses at singularities are not equilibrium states, then there exists a unique equilibrium state supported on Λ. In particular, there exists a unique measure of maximal entropy for the flow X|Λ.

We introduce a framework for proving mixing bounds for Markov chains which connects ideas from two seemingly unrelated techniques: the framework of spectral independence for the mixing of Markov chains on discrete space and the stochastic localization for mixing of Markov chains on log-concave measures and the discrete hypercube. In the center of our framework is the concept of a localization scheme which, to every probability measure on some space, assigns a martingale of probability measures which localize in space as time evolves. To every such scheme corresponds a Markov chain, and many chains can be derived through localization schemes. This viewpoint provides tools for deriving mixing bounds for Markov chains through the analysis of the corresponding localization process. Generalizations of concepts of spectral independence and entropic independence naturally arise from our definitions, and in particular we recover the main theorems in the spectral and entropic independence frameworks via simple martingale arguments bypassing the need to use the theory of high-dimensional expanders. We demonstrate the strength of our proposed framework by deriving new mixing bounds and giving simple proofs to many existing bounds in the recent literature. In particular, we (i) give the first O(nlogn) bound for mixing time of the hardcore model of arbitrary degree in the tree-uniqueness regime, via Glauber dynamics; (ii) give the first optimal mixing bounds for Ising models in the uniqueness regime under any external fields; (iii) prove a Kullback–Leibler (KL) divergence decay bound for log-concave sampling via the restricted Gaussian oracle, which achieves optimal mixing under any exp(n)-warm start; and (iv) prove a log-Sobolev inequality for near-critical ferromagnetic Ising models, recovering in a simple way a variant of a recent result by Bauerschmidt and Dagallier.

We show that algebraic equivalence of images of stable maps of curves lifts to deformation equivalence of the stable maps. The main applications concern A1(X), the group of 1-cycles modulo algebraic equivalence, for smooth, separably rationally connected varieties. If K∕k is an algebraic extension, then the kernel of A1(Xk)→A1(XK) is at most Z∕2Z. If k is finite, then the image equals the subgroup of Galois invariant cycles.