数学前沿

We initiate a series of works where we study the interior of dynamical rotating vacuum black holes without symmetry. In the present paper, we take up the problem starting from appropriate Cauchy data for the Einstein vacuum equations defined on a hypersurface already within the black hole interior, representing the expected geometry just inside the event horizon. We prove that, for all such data, the maximal Cauchy evolution can be extended across a non-trivial piece of Cauchy horizon as a Lorentzian manifold with continuous metric. In subsequent work, we will retrieve our assumptions on data assuming only that the black hole event horizon geometry suitably asymptotes to a rotating Kerr solution. In particular,  _if the exterior region of the Kerr family is proven to be dynamically stable_—as is widely expected—then it will follow that the $C^0$-inextendibility formulation of Penrose’s celebrated strong cosmic censorship conjecture is in fact _false_. The proof suggests, however, that the $C^0$-metric Cauchy horizons thus arising are generically singular in an essential way, representing so-called “weak null singularities”, and thus that a revised version of strong cosmic censorship holds.

Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$ with $\max{|a_1|,\ldots,|a_n|}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such polynomials, as may be obtained by setting $a_n=0$. In 1936, van der Waerden conjectured that $O(H^{n-1})$ should in fact also be the correct upper bound for the count of uch polynomials. The conjecture has been known previously for degrees $n\leq 4$, due to work of van der Waerden and Chow and Dietmann. The purpose of this paper is to prove van der Waerden’s Conjecture for all degrees $n$.

In his seminal 1961 paper, Wirsing studied how well a given transcendental real number $\xi$ can be approximated by algebraic numbers $\alpha$ of degree at most $n$ for a given positive integer $n$, in terms of the so-called naive height $H(\alpha )$ of $\alpha$. He showed that the supremum $\omega ^*_n(\xi )$ of all $\omega$ for which infinitely many such $\alpha$ have $|\xi -\alpha | \le H(\alpha )^{-\omega -1}$ is at least $(n+1)/2$. He also asked if we could even have $\omega ^*_n(\xi ) \ge n$ as it is generally expected. Since then, all improvements on Wirsing’s lower bound were of the form $n/2+\mathcal {O}(1)$ until Badziahin and Schleischitz showed in 2021 that $\omega ^*_n(\xi ) \ge an$ for each $n\ge 4$, with $a=1/\sqrt {3}\simeq 0.577$. In this paper, we use a different approach partly inspired by parametric geometry of numbers and show that $\omega ^*_n(\xi ) \ge an$ for each $n\ge 2$, with $a=1/(2-\log 2)\simeq 0.765$.

We prove that a smooth embedded paper Moebius band must have aspect ratio greater than $\sqrt 3$. We also prove that any sequence of smooth embedded paper Moebius bands whose aspect ratio converges to $\sqrt 3$ must converge, up to isometry, to the triangular Moebius band. These results answer the minimum aspect ratio question discussed by W. Wunderlich in 1962 and prove the more specific conjecture of B. Halpern and C. Weaver from 1977.

We prove that if a fractal set in $\mathbb{R}^d$ avoids lines in a certain quantitative sense, which we call line porosity, then it has a fractal uncertainty principle. The main ingredient is a new higher-dimensional Beurling–Malliavin multiplier theorem.

It was conjectured by Milnor in 1968 that the fundamental group of a complete manifold with nonnegative Ricci curvature is finitely generated. The main result of this paper is a counterexample, which provides an example $M^7$ with $\text {Ric}\geq 0$ such that $\pi _1(M)= \mathbb{Q}/\mathbb{Z}$ is infinitely generated. There are several new points behind the result. The first is a new topological construction for building manifolds with infinitely generated fundamental groups, which can be interpreted as a smooth version of the fractal snowflake. The ability to build such a fractal structure will rely on a very twisted gluing mechanism. Thus the other new point is a careful analysis of the mapping class group $\pi _0\text {Diff}(S^3\times S^3)$ and its relationship to Ricci curvature. In particular, a key point will be to show that the action of $\pi _0\text {Diff}(S^3\times S^3)$ on the standard metric $g_{S^3\times S^3}$ lives in a path connected component of the space of metrics with $\text {Ric}>0$.

In this paper, we will prove an effective version of Ratner’s equidistribution theorem for unipotent orbits in $\mathrm{SL}(3,\mathbb{R})/{\mathrm{SL}}(3,\mathbb Z)$ with a natural Diophantine condition.

The characterization of global solutions to the obstacle problems in $\mathbb {R}^N$, or equivalently of null quadrature domains, has been studied for more than 90 years. In this paper, we give a conclusive answer to this problem by proving the following long-standing conjecture:  _The coincidence set of a global solution to the obstacle problem is either a half-space, an ellipsoid, a paraboloid, or a cylinder with an ellipsoid or a paraboloid as base._

This paper gives geometric characterizations of the Weil-Petersson class of rectifiable quasicircles, i.e., the closure of the smooth planar curves in the Weil-Petersson metric on universal Teichmüller space defined by Takhtajan and Teo. Although motivated by the planar case, many of our characterizations make sense for curves in $\mathbb{R}^n$ and remain equivalent in all dimensions. We prove that $\Gamma$ is Weil-Petersson if and only if it is well approximated by polygons in a precise sense, has finite Möbius energy or has arclength parametrization all over in $H^{3/2}(\mathbb{T})$. Other results say that a curve is Weil-Petersson if and only if local curvature is square integrable over all locations and scales, where local curvature is measured using various quantities such as Jones’s $\beta$-numbers, nonlinearity of conformal weldings, Menger curvature, the “thickness” of the hyperbolic convex hull of $\Gamma$, and the total curvature of minimal surfaces in hyperbolic space. Finally, we prove that planar Weil-Petersson curves are exactly the asymptotic boundaries of minimal surfaces in $\mathbb{H}^3$ with finite renormalized area.

Let $K$ be a number field not containing a CM subfield. For any smooth projective curve $Y/K$ of genus $\geq 2$, we prove that the image of the “Selmer” part of Grothendieck’s section set inside the $K_v$-rational points $Y(K_v)$ is finite for every finite place $v$. This gives an unconditional verification of a prediction of Grothendieck’s section conjecture. In the process of proving our main result, we also refine and extend the method of Lawrence and Venkatesh, with potential consequences for explicit computations.

We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in $\mathbb {Z}/2\mathbb {Z}$). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and then using a grid diagram for the link. We also give combinatorial descriptions of the mod 2 Ozsváth-Szabó mixed invariants of closed four-manifolds, also in terms of grid diagrams.

We show that for an arbitrary totally complex number field $L$ the (regularized) critical $L$-values of algebraic Hecke characters of $L$ divided by certain periods are algebraic integers. This relies on a new construction of an equivariant coherent cohomology class with values in the completion of the Poincaré bundle on an abelian scheme $\mathcal {A}$. From this we obtain a cohomology class for the automorphism group of a CM abelian scheme $\mathcal{A}$ with values in some canonical bundles, which can be explicitly calculated in terms of Eisenstein–Kronecker series. As a further consequence, using an infinitesimal trivialization of the Poincaré bundle, we construct a $p$-adic measure interpolating the critical $L$-values in the ordinary case. This generalizes previous results for CM fields by Damerell, Shimura and Katz and settles the algebraicity and $p$-adic interpolation in the remaining open cases of critical values of Hecke $L$-functions.