数学前沿

We construct functorial Igusa stacks for all Hodge-type Shimura varieties, proving a conjecture of Scholze and extending earlier results of the fourth-named author for PEL-type Shimura varieties. Using the Igusa stack, we construct a sheaf on $\mathrm{Bun}_{G}$ that controls the cohomology of the corresponding Shimura variety. We use this sheaf and the spectral action of Fargues–Scholze to prove a compatibility between the cohomology of Shimura varieties of Hodge type and the semisimple local Langlands correspondence of Fargues–Scholze and to generalize the Eichler–Shimura relation conjectured by Blasius–Rogawski to Iwahori level at $p$. When the given Shimura variety is proper, we show moreover that the sheaf is perverse, which allows us to prove new torsion vanishing results for the cohomology of Shimura varieties.

We construct non-trivial weak solutions $\theta\in C_{t}^{0}C_{x}^{0-}$ to the surface quasi-geostrophic (SQG) equations, which have compact support in time and thus violate the conservation of the Hamiltonian. The result is sharp in view of the fact that such a conservation law holds for all weak solutions in the class $C_{t,x}^{0} \subset L_{t,x}^{3}$ (Isett and Vicol (2015)) and resolves the Onsager conjecture for SQG equations. The construction is achieved by means of a Nash iteration together with the linear decoupling method recently introduced by Giri and Radu (2023).

We introduce a Laplacian separation principle for the eikonal equation on Markov chains. As application, we prove an isoperimetric concentration inequality for reversible Markov chains with non-negative Ollivier curvature. That is, every single point from the concentration profile yields an estimate for every point of the isoperimetric profile. Applying this to exponential and Gaussian concentration, we obtain affirmative answers to two open questions by Erbar and Fathi. Moreover, we prove that the modified log-Sobolev constant is at least the minimal Ollivier Ricci curvature, assuming non-negative Ollivier sectional curvature, i.e., the Ollivier Ricci curvature when replacing the $\ell_{1}$ by the $\ell_{\infty}$ Wasserstein distance. This settles a recent open problem by Pedrotti. We give a simple example showing that non-negative Ollivier sectional curvature is necessary to obtain a modified log-Sobolev inequality via a positive Ollivier Ricci bound. This provides a counterexample to a conjecture by Peres and Tetali.

Let $s \in [0,1]$. We show that a Borel set $N \subset \mathbb{R}^{2}$ each of whose points is linearly accessible by an $s$-dimensional family of lines has Hausdorff dimension at most $2 - s$. This is a sharp bound, due to a construction of A. Máthé.

We prove an interpolation theorem for nonlinear functionals defined on scales of Banach spaces that generalize Besov spaces. It applies to functionals defined only locally, requiring only some weak Lipschitz conditions, extending those introduced by Lions and Peetre. Our analysis is self-contained and independent of any previous results in interpolation theory. It depends solely on the concepts of Friedrichs’ mollifiers, seen through the formalism introduced by Hamilton, combined with the frequency envelopes introduced by Tao and used recently by two of the present authors and others to study the Cauchy problem for various quasilinear evolutions in partial differential equations. Inspired by this latter work, our main application states that, for an abstract flow map of a quasilinear problem, both the continuity of the flow as a function of time and the continuity of the data-to-solution map follow automatically from the estimates that are usually proven when establishing the existence of solutions: propagation of regularity via tame _a priori_ estimates for higher regularities and contraction for weaker norms.

We study the categories $\mathrm{Perv}_{\mathrm{thin}}$, $\mathrm{Perv}_{\mathrm{thick}}$ of Iwahori-equivariant perverse sheaves on the thin and thick affine flag varieties associated to a split reductive group $G$. An earlier work of the first author describes $\mathrm{Perv}_{\mathrm{thin}}$ in terms of bimodules over the so-called noncommutative Springer resolution. We partly extend this result to $\mathrm{Perv}_{\mathrm{thick}}$, providing a similar description for its anti-spherical quotient. The long intertwining functor realizes $\mathrm{Perv}_{\mathrm{thick}}$ as the Ringel dual of $\mathrm{Perv}_{\mathrm{thin}}$; we point out that it shares some exactness properties with the similar functor acting on perverse sheaves on the finite-dimensional flag variety. We use this result to resolve a conjecture of Arkhipov and the first author, proving that the image in the Iwahori–Whittaker category of any convolution-exact perverse sheaf on the affine flag variety is tilting.

We obtain density theorems for exceptional cuspidal automorphic representations of $\mathrm{GL}_{n}$ over $\mathbb{Q}$ that fail the generalized Ramanujan conjecture at some place. We depart from approaches based on trace formulae, and instead rely on properties of families of Rankin–Selberg $L$-functions. This improves previous density results near the threshold of the pointwise bounds.

We study the asymptotic distribution of the Galois orbits of generic sequences of algebraic points of small height in a projective variety over a number field. Our main result is a generalization of Yuan’s equidistribution theorem that applies to heights for which Zhang’s lower bound for the essential minimum is not necessarily an equality. It extends to all projective varieties a theorem of Burgos Gil, Philippon, Rivera-Letelier and the second author for toric varieties. It also applies to sums of canonical heights for an algebraic dynamical system, and in particular it recovers Kühne’s semiabelian equidistribution theorem. We also generalize previous work of Chambert-Loir and Thuillier to obtain new logarithmic equidistribution results. Finally we extend our main result to the quasi-projective setting recently introduced by Yuan and Zhang.

We consider the long-time behavior of a diffusion process on $\mathbb{R}^{d}$ advected by a stationary random vector field which is assumed to be divergence-free, dihedrally symmetric in law and have a log-correlated potential. A special case includes $\nabla ^{\perp }$ of the Gaussian free field in two dimensions. We show the variance and the second moment of the displacement at a large time $t$ behave like $2dc_{*}^{\nicefrac 12} t (\log t)^{\nicefrac 12}$, in a quenched sense and with a precisely determined prefactor $c_{*}= c_{*}( \mathbb{P} )>0$, independent of $\nu$. We also prove a quenched invariance principle under this superdiffusive scaling. The proof is based on a rigorous renormalization group argument in which we inductively analyze coarse-grained diffusivities, scale-by-scale. Our analysis leads to sharp homogenization and large-scale regularity estimates on the infinitesimal generator, which are subsequently transferred into quantitative information on the process.

Given an oligomorphic group $G$ and a measure $\mu$ for $G$ (in a sense that we introduce), we define a rigid tensor category $\operatorname{\underline{Perm}}(G; \mu )$ of “permutation modules,” and, in certain cases, an abelian envelope $\operatorname{\underline{\textup{Rep}}}(G; \mu )$ of this category. When $G$ is the infinite symmetric group, this recovers Deligne’s interpolation category. Other choices for $G$ lead to fundamentally new tensor categories. For example, we construct the first known semi-simple pre-Tannakian categories in positive characteristic with super-exponential growth. One interesting aspect of our construction is that, unlike previous work in this direction, our categories are concrete: the objects are modules over a ring, and the tensor product receives a universal bi-linear map. Central to our constructions is a novel theory of integration on oligomorphic groups, which could be of more general interest. Classifying the measures on an oligomorphic group appears to be a difficult problem, which we solve in only a few cases.

Let $\Gamma$ be a non-elementary subgroup of the mapping class group of a surface. In this paper, we study the dynamics of the $\Gamma$-action on the space of measured laminations. We prove that on the space $\mathcal{R}_{\Gamma }$ of $\Gamma$-recurrent measured laminations, there is at most one $\Gamma$-invariant Radon measure (up to scaling). In particular, when $\Gamma$ is of divergence type, we show that the $\Gamma$-action on $\mathcal{R}_{\Gamma }$ is uniquely ergodic, by explicitly constructing the ergodic measure. This generalizes Lindenstrauss–Mirzakhani’s result and Hamenstädt’s result for the full mapping class group, in which case the ergodic measure is the Thurston measure. When $\Gamma$ is convex cocompact, we also prove that every invariant ergodic Radon measure on the space of all measured laminations is either the unique measure on $\mathcal{R}_{\Gamma }$, or a counting measure on the orbit of a non-recurrent measured lamination. Our method is geometric and does not rely on continuous or homogeneous flows on the ambient space or a dynamical system associated with a finite measure space. This leads to a unifying approach for various metric spaces, including Teichmüller spaces and partially $\operatorname{CAT}(-1)$ spaces. In the appendix, we prove non-arithmeticity of the Teichmüller length spectrum of a non-elementary subgroup of the mapping class group. We prove this by introducing the notion of cross-ratios on $\mathcal{MF}$ and $\mathcal{PMF}$, and studying their geometric and dynamical properties, despite the lack of negatively curved features of the Teichmüller space and of conformal geometry on $\mathcal{PMF}$. As an application, we show topological mixing of the Teichmüller geodesic flow for every non-elementary subgroup.

For a free action $G \curvearrowright X$ of an amenable group on a compact metrizable space, we study the Uniform Rokhlin Property (URP) and the conjunction of Uniform Rokhlin Property and comparison (URPC). We give several equivalent formulations of the latter and show that it passes to extensions. We introduce technical conditions called property FCSB and property FCSB in measure, both of which reduce to the marker property if $G$ is abelian. Our first main result is that for any amenable group $G$ property FCSB in measure is equivalent to URP, and for a large class of amenable groups property FCSB is equivalent to URPC. In the latter case it follows that if the action is moreover minimal then the C∗-crossed product $C(X) \rtimes G$ has stable rank one, satisfies the Toms-Winter conjecture, and is classifiable if ${\mathrm{mdim}}(G \curvearrowright X) = 0$. Our second main result is that if a system $G \curvearrowright X$ has URPC and ${\mathrm{mdim}}(G \curvearrowright X) < M/2$, then it is embeddable into the $M$-cubical shift $\left ([0, 1]^{M}\right )^{G}$. Combined with the first main result, we recover the Gutman-Qiao-Tsukamoto sharp shift embeddability theorem as a special case. Notably, the proof avoids the use of either Euclidean geometry or signal analysis and directly extends the theorem to all abelian groups. Finally, we show that if $G$ is a nonamenable group that contains a free subgroup on two generators and $G \curvearrowright X$ is a topologically amenable action, then it is embeddable into $[0, 1]^{G}$.

Let $\mu$ and $\nu$ be compactly supported probability measures on ℝ, and let $\mu \boxplus \nu$ denote their additive free convolution. We show that for all sufficiently large real $z$, $$\int _{-\infty }^{\infty }\log (z - x) \, (\mu \boxplus \nu )( \mathrm{d}x) = \sup _{\Pi } \left \{ \mathbf{E}_{\Pi }[\log (z - (X+Y))] - H(\Pi \mid \mu \otimes \nu ) \right \} ,$$ where the supremum is taken over all couplings $\Pi$ of $\mu$ and $\nu$. Analogous formulas hold for multiplicative free convolution $\mu \boxtimes \nu$ and free compression $[\mu ]_{\tau }$. In this way, integrals of a log-potential against free convolutions can be expressed as entropic optimal transport problems. The corresponding optimal couplings admit explicit formulas, from which the standard $R$- and $S$-transform descriptions of additive and multiplicative free convolution are recovered. In particular, the optimizer $\Pi _{z}$ in (0.1) encodes the subordination equations via couplings of random variables. Our approach is based on a large deviation principle on the symmetric group, combined with the quadrature method of Marcus–Spielman–Srivastava.

We prove that there exists a bound $N'_{L}(W)$ for a positively weighted Coxeter group $(W, S, L)$ of finite rank. In particular, Lusztig’s $\boldsymbol{a}$-function of $(W, S, L)$ is bounded.

We establish the scalar curvature and distance bounds, extending Perelman’s work on the Fano Kähler–Ricci flow to general finite time solutions of the Kähler–Ricci flow. These bounds are achieved by our Li–Yau type and Harnack estimates for weighted Ricci potential functions of the Kähler–Ricci flow. We further prove that the Type I blow-ups of the finite time solution always sub-converge in Gromov–Hausdorff sense to an ancient solution on a family of analytic normal varieties with suitable choices of base points. As a consequence, the Type I diameter bound is obtained for a fibre over every relatively compact open subset of the regular base of the fibration. We also apply our estimates to show that every solution of the Kähler–Ricci flow with Calabi symmetry must develop Type I singularities, including both cases of high codimensional contractions and fibre collapsing.

We prove that for finitely generated abelian groups $A$ and $B$, the space of $\mathbb{E}_{\infty }$-ring maps between the spherical groups rings $\mathbb{S}[A] \to \mathbb{S}[B]$ is equivalent to the discrete set of group homomorphisms $A \to B$. We also prove generalizations where the sphere is replaced by other ring spectra, e.g. we give a formula for the strict units in group rings of the form $R[A]$ for $A$ a finite $p$-group and $R$ $p$-completely chromatically complete.

The complement of the union of a collection of disjoint open disks in the 2-sphere is called a Schottky set. We prove that a subset $S$ of the 2-sphere is quasiconformally equivalent to a Schottky set if and only if every pair of complementary components of $S$ can be mapped to a pair of open disks with a uniformly quasiconformal homeomorphism of the sphere. Our theorem applies to Sierpiński carpets and gaskets, yielding for the first time a general quasiconformal uniformization result for gaskets. Moreover, it contains Bonk’s uniformization result for carpets as a special case and does not rely on the condition of uniform relative separation that is used in relevant works.

We prove the $K(n)$-local analogue of the Hahn-Wilson conjecture on fp-spectra, which states that the truncated Brown-Peterson spectra generate the category of fp-spectra as a thick subcategory. As a corollary, we deduce the original conjecture at height 1. Along the way, we prove the existence of $K(n)$-local finite complexes with particularly regular rings of homotopy groups.

This is the first in a sequence of four papers, where we prove the arithmetic Siegel–Weil formula in co-rank 1 for Kudla–Rapoport special cycles on exotic smooth integral models of unitary Shimura varieties of arbitrarily large even arithmetic dimension. Our arithmetic Siegel–Weil formula implies that degrees of Kudla–Rapoport arithmetic special 1-cycles are encoded in near-central first derivatives of unitary Eisenstein series Fourier coefficients. The crucial input is a new local limiting method at all places. In this paper, we formulate and prove the key local theorems at all non-Archimedean places. On the analytic side, the limit relates local Whittaker functions on different groups. On the geometric side at nonsplit non-Archimedean places, the limit relates degrees of 0-cycles on Rapoport–Zink spaces and local contributions to heights of 1-cycles in mixed characteristic.

We prove the Schinzel–Zassenhaus conjecture, in the following explicit form. If $P(X) \in \mathbf{Z}[X]$ is an integer polynomial of degree $n$ and having $P(0) = 1$, then either $P(X)$ is a product of cyclotomic polynomials, or else at least one of the complex roots of $P$ belongs to the disc $|X| \leq 2^{ – 1 / (4n) }$. We also obtain a relative version of this result over the compositum $\mathbf{Q}^{\mathrm{ab}} \cdot \mathbf{Q}^{\mathrm{t}.p}$ of all abelian and all totally $p$-adic extensions of $\mathbf{Q}$, for any fixed prime $p$, and apply it to prove a $\mathbf{Q}^{\mathrm{ab}} \cdot \mathbf{Q}^{\mathrm{t}.p}$-relative canonical height lower bound on the multiplicative group. Another extension is given to a uniform positive height lower bound, inverse-proportional to the total number of singular points, on holonomic power series in $\mathbf{Q} [[X ]]$ and not of the form $p(X) / (X^l-1)^m$, where $p(X) \in \mathbf{Q}[X]$. We derive applications to upper bounds on the number of noncyclotomic irreducible factors of a lacunary polynomial, and a further application to the existence of a small critical value for certain rational functions.