数学前沿

We provide a rigorous derivation of Boltzmann’s kinetic equation from the hard sphere system for rarefied gas, which is valid for arbitrarily long times, as long as the (regular) solution to the Boltzmann equation exists. This extends Lanford’s landmark theorem [46], which justifies this derivation for a sufficiently short time. In a companion paper [29], we connect this derivation to existing literature on the hydrodynamic limits of Boltzmann’s equation. This executes the original program proposed in Hilbert’s Sixth Problem in 1900, which asked for the derivation of fluid equations from Newton’s laws, via Boltzmann’s kinetic theory. The general strategy follows the paradigm introduced by the first two authors for the long-time derivation of the wave kinetic equation in wave turbulence theory. This is based on propagating a long-time cumulant ansatz, which keeps memory of the full collision history of the relevant particles, by an important partial time expansion. The heart of the matter is proving the smallness of these cumulants in $L^1$, which can be reduced to combinatorial properties for the associated diagrams which we call _molecules_. These properties are then proved by devising an elaborate _cutting algorithm_, which is a major novelty of this work.

We construct area-minimizing submanifolds with fractal singular sets on compact Riemannian manifolds. Thus, we settle Problem 5.4 in [4] by Almgren and our answer is sharp dimensionwise. Furthermore, we can prescribe arbitrarily the strata in the Almgren stratification of the singular sets of our area-minimizing submanifolds, and our results hold in the category of integral currents, mod $v$ currents and stable stationary varifolds.

Let $H\to \mathbb{R}^4\to \mathbb{R}$ be any smooth function. This article introduces some arguments for extracting dynamical information about the Hamiltonian flow of $H$ from high-dimensional families of closed holomorphic curves. We work in a very general setting, without imposing convexity or contact-type assumptions. For any compact regular level set $Y$, we prove that the Hamiltonian flow admits an finite family of pairwise distinct, proper, compact invariant subsets whose union is dense in $Y$. This is a generalization of the Fish–Hofer theorem, which shows that $Y$ has at least one proper compact invariant subset. We then establish a global Le Calvez–Yoccoz property for almost every compact regular level set $Y$: any compact invariant subset containing all closed orbits is either equal to $Y$ or is not locally maximal. Next, we prove quantitative versions, in four dimensions, of the celebrated almost-existence theorem for Hamiltonian systems; such questions have been open for general Hamiltonians since the late 1980s. We prove that almost every compact regular level set of $H$ contains at least two closed orbits, a sharp lower bound. Under explicit and $C^\infty$-generic conditions on $H$, we prove almost-existence of infinitely many closed orbits.

We describe the obstruction to decomposing in degrees $\leq p$ the de Rham complex of a smooth variety over a perfect field $k$ of characteristic $p$ that lifts over $W_2(k)$, and show that there exist liftable smooth projective varieties of dimension $p+1$ whose Hodge-to-de Rham spectral sequence does not degenerate at the first page. We also describe the action of the Sen operator on the de Rham complex in degrees $\leq p$ and give examples of varieties with a non-semisimple Sen operator. Our methods rely on the commutative algebra structure on de Rham and Hodge-Tate cohomology, and are inspired by the properties of Steenrod operations on cohomology of cosimplicial commutative algebras. The example of a non-degenerate Hodge-to-de Rham spectral sequence relies on a non-vanishing result on cohomology of groups of Lie type. We give applications to other situations such as describing extensions in the canonical filtration on de Rham, Hodge, and étale cohomology of an abelian variety equipped with a group action. We also show that the de Rham complex of a smooth variety over $k$ is formal as an $E_{\infty}$-algebra if and only if the variety lifts to $W_2(k)$ together with its Frobenius endomorphism.

The Ramsey number $R(k)$ is the minimum $n \in \mathbb{N}$ such that every red-blue colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove that R(k) \le (4 – \varepsilon)^k for some constant $\varepsilon > 0$. This is the first exponential improvement over the upper bound of Erdős and Szekeres, proved in 1935.

For a prime number $p$ and a $p$-quasisyntomic commutative ring $R$, Bhatt–Morrow–Scholze defined motivic filtrations on the $p$-completions of $\mathrm{THH}(R)$, $\mathrm{TC}^-(R)$,$\mathrm{TP}(R)$, and $\mathrm{TC}(R)$, with the associated graded objects or $\mathrm{TP}(R)$ and $\mathrm{TC}(R)$ recovering the prismatic and syntomic cohomology of $R$, respectively. We give an alternate construction of thesee filtrations that applies also when $R$ is a well-behaved commutative ring spectrum; for example, we can take $R$ to be $\mathbb{S}$, $\mathrm{MU}$, $\mathrm{ku}$, $\mathrm{ko}$, or $\mathrm{tmf}$. We compute the $\mathrm{mod}(p,v_1)$ syntomic cohomology of the Adams summand $\ell$ and observe that, when $p\ge 3$, the motivic spectral sequenc for $V(1)_\ast \mathrm{TC}(\ell)$ collapses at the $\mathrm{E}_2$-page.

We prove that one-relator groups are coherent, solving a well-known problem of Gilbert Baumslag. Our proof strategy is readily applicable to many classes of groups of cohomological dimension two. We show that fundamental groups of two complexes with non-positive immersions are homologically coherent, we show that groups with staggered presentations and many Coxeter groups are coherent and we show that group algebras over fields of characteristic zero of groups with reducible presentations without proper powers are coherent.

Let $A$ be an abelian variety over a number field $\mathrm{E}\subset \mathbb{C}$ and let $\mathbf{G}$ denote the Mumford–Tate group of $A$. After replacing $\mathrm{E}$ by a finite extension, the action of the absolute Galois group $\mathrm{Gal}(\overline{\mathrm{E}}/\mathrm{E})$ on the $\ell$-adic cohomology $\mathrm{H}^1_{\mathrm{\acute{e}t}}(A_{\overline{\mathrm{E}}},\mathbb{Q}_\ell)$ factors through $\mathbf{G}(\mathbb{Q}_\ell)$. We show that for $v$ an odd prime of $\mathrm{E}$ where $A$ has good reduction, the conjugacy class of Frobenius $\mathrm{Frob}_v$ in $\mathbf{G}(\mathbb{Q}_\ell)$ is independent of $\ell$. Along the way, we prove that under certain hypotheses, every point in the $\mu$-ordinary locus of the special fiber of Shimura varieties has a special point lifting it.

Employing the Einstein-scalar field system, we demonstrate an approach for proving high co-dimensional nonlinear instability of naked-singularity solutions as constructed by Christodoulou in 1994. We further investigate the censorship of Christodoulou’s naked singularity and show that a tiny anisotropic perturbation arising from the outgoing characteristic initial data would lead to the emergence of an anisotropic apparent horizon, which covers and censors the naked singularity. Our approach advances the hyperbolic short-pulse method by not requiring the aid of additional large parameters, by permitting the use of initial perturbations for the shear tensor and the derivative of scalar field to be with finite $BV$ and $C^0$ norms, and by allowing the initial perturbation to be arbitrarily small in scale-critical norms. New elliptic arguments based on non-perturbative methods are also developed.

We explain that the Pontryagin product structure on the equivariant $K$-group of an affine Grassmannian considered in [Lam–Schilling–Shimozono, Compos. Math. **146** (2010)] coincides with the tensor structure on the equivariant $K$-group of a semi-infinite flag manifold considered in [K-Naito-Sagaki, Duke Math. **169** (2020)]. Then, we construct an explicit isomorphism between the equivariant $K$-group of a semi-infinite flag manifold and a suitably localized equivariant quantum $K$-group of the corresponding flag manifold. These exhibit a new framework to understand the ring structure of equivariant quantum $K$-groups and the Peterson isomorphism.

We show Poincaré Duality for ${\bf F}_p$-étale cohomology of a smooth proper rigid-analytic space over a non-archimedean field $K$ of mixed characteristic $(0, p)$. It positively answers the question raised by P. Scholze in his paper “$p$-adic Hodge theory for rigid-analytic varieties.” We prove duality via constructing Faltings’ trace map relating Poincaré Duality on the generic fiber to (almost) Grothendieck Duality on the mod-$p$ fiber of a formal model. We also formally deduce PoincaréDuality for $\mathbf{Z}/p^n\mathbf{Z}$, $\mathbf{Z}_p$, and $\mathbf{Q}_p$-coefficients.

We prove the endoscopic fundamental lemma for the Lie algebra of the symmetric variety $\mathrm{U}(2n)/\mathrm{U}(n)\times \mathrm{U}(n)$, where $\mathrm{U}(n)$ denotes a unitary group of rank $n$. This is the first major step in the stabilization of the relative trace formula associated to the $\mathrm{U}(n)\times \mathrm{U}(n)$-periods of automorphic forms on $\mathrm{U}(2n)$.

Faltings proved that there are finitely many abelian varieties of genus $g$ over a number field $K$, with good reduction outside a finite set of primes $S$. Fixing one of these abelian varieties $A$, we prove that there are finitely many smooth hypersurfaces in $A$, with good reduction outside $S$, representing a given ample class in the Néron–Severi group of $A$, up to translation, as long as the dimension of $A$ is at least four. Our approach builds on the approach of Lawrence and Venkatesh, which studies $p$-adic variations of Hodge structure to turn finiteness results for $p$-adic Galois representations into geometric finiteness statements. A key new ingredient is an approach to proving big monodromy for the variations of Hodge structure arising from the middle cohomology of these hypersurfaces using the Tannakian theory of sheaf convolution on abelian varieties. Ancillary files are available on the [arXiv](https://arxiv.org/src/2004.09046v5/anc) and [https://doi.org/10.4007/annals.2025.202.3.1.code](https://doi.org/10.4007/annals.2025.202.3.1.code)

We prove a conjecture of K. Marton, widely known as the polynomial Freiman–Ruzsa conjecture, in characteristic $2$. The argument extends to odd characteristic, with details to follow in a subsequent paper.

We derive a priori interior Hessian estimates and interior regularity for the $\sigma_2$ equation in dimension four. Our method provides both a new proof for the corresponding three dimensional results and a Hessian estimate for smooth solutions satisfying a dynamic semi-convexity condition in higher ($n\ge 5$) dimensions.

This is the second paper in a series studying the nonlinear stability of rarefaction waves in multi-dimensional gas dynamics. We construct initial data near singularities in the rarefaction wave region and, combined with the _a priori_ energy estimates from the first paper, demonstrate that any smooth perturbation of constant states on one side of the diaphragm in a shock tube can be connected to a centered rarefaction wave. We apply this analysis to study multi-dimensional perturbations of the classical Riemann problem for isentropic Euler equations. We show that the Riemann problem is structurally stable in the regime of two families of rarefaction waves.

We show that, if a closed, connected, and oriented Riemannian $n$-manifold $N$ admits a non-constant quasiregular mapping from the Euclidean $n$-space $\mathbb{R}^n$, then the de Rham cohomology algebra $H_{\mathrm {dR}}^*(N)$ of $N$ embeds into the exterior algebra $\bigwedge^*\mathbb{R}^n$. As a consequence, we obtain a homeomorphic classification of closed simply connected quasiregularly elliptic $4$-manifolds.

We study the resolution of discontinuous singularities in gas dynamics via rarefaction waves. The mechanism is well-understood in the one dimensional case. We will prove the nonlinear stability of the Riemann problem for multi-dimensional isentropic Euler equations in the regime of rarefaction waves. The proof relies on the new energy estimates _without loss of derivatives_. We also give a detailed geometric description of the rarefaction wave fronts. This is the first paper in the series which provides the _a priori_ energy bounds.

In this paper we consider the inductive Alperin–McKay condition for quasi-isolated $2$-blocks of exceptional groups of Lie type. Thereby, we complete the proof of the Alperin–McKay conjecture and Brauer’s height zero conjecture for the prime $2$.