Annals of Mathematics最新文献

We analyze the Poincaré and Log-Sobolev constants of logconcave densities in $mathbb{R}^{n}$. For the Poincaré constant, we give an improved estimate of $O(sqrt{n})$ for any isotropic logconcave density. For the Log-Sobolev constant, we prove a bound of $Omega (1/D)$, where $D$ is the diameter of the support of the density, and show that this is asymptotically the best possible bound, resolving a question posed by Frieze and Kannan in 1997. These bounds have several interesting consequences. Improved bounds on the thin-shell and Cheeger/KLS constants are immediate. The ball walk to sample an isotropic logconcave density in $mathbb{R}^{n}$ converges in $O^{*}(n^{2.5})$ steps from a warm start, and the speedy version of the ball walk, studied by Kannan, Lov’aasz and Simonovits mixes in $O^{*}(n^{2}D)$ proper steps from any start, also a tight bound. As another consequence, we obtain a unified and improved large deviation inequality for the concentration of any $L$-Lipshitz function over an isotropic logconcave density (studied by many), generalizing bounds of Paouris and Guedon-E. Milman. Our proof technique is a development of stochastic localization, first introduced by Eldan.

For appropriate Gaussian processes, as a corollary of the majorizing measure theorem, Michel Talagrand (1987) proved that the event that the supremum is significantly larger than its expectation can be covered by a set of half-spaces whose sum of measures is small. We prove a conjecture of Talagrand that is the analog of this result in the Bernoulli-$p$ setting, and answer a question of Talagrand on the analogous result for general positive empirical processes.

We prove the nonlinear inviscid damping for a class of monotone shear flows in $mathbb{T}times [0,1]$ for initial perturbation in Gevrey-$frac{1}{s}$ class ($1lt frac{1}{s}<2$) with compact support. The main new idea of the proof is to construct and use the wave operator of a slightly modified Rayleigh operator in a well-chosen coordinate system.

We prove that the Borel space of torsion-free abelian groups with domain $omega$ is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open problem in descriptive set theory, which dates back to the seminal paper on Borel reducibility of Friedman and Stanley from 1989.

We compute the number of $mathbb{F}_q$-points on $overline{mathcal{M}}_{4,n}$ for $n leq 3$ and show that it is a polynomial in $q$, using a sieve based on Hasse–Weil zeta functions. As an application, we prove that the rational singular cohomology group $H^k (overline{mathcal{M}}_{g,n})$ vanishes for all odd $k leq 9$. Both results confirm predictions of the Langlands program, via the conjectural correspondence with polarized algebraic cuspidal automorphic representations of conductor $1$, which are classified in low weight. Our vanishing result for odd cohomology resolves a problem posed by Arbarello and Cornalba in the 1990s.

Let $f$ be a $C^{1+alpha}$ diffeomorphism of a compact manifold $M$ preserving a smooth measure $mu$. We show that if $fcolon (M,mu)to (M,mu)$ is exponentially mixing, then it is Bernoulli.

The Nadler–Zaslow correspondence famously identifies the finite-dimensional Floer homology groups between Lagrangians in cotangent bundles with the finite-dimensional Hom spaces between corresponding constructible sheaves. We generalize this correspondence to incorporate the infinite-dimensional spaces of morphisms “at infinity,” given on the Floer side by Reeb trajectories (also known as “wrapping”) and on the sheaf side by allowing unbounded infinite rank sheaves which are categorically compact. When combined with existing sheaf theoretic computations, our results confirm many new instances of homological mirror symmetry. par More precisely, given a real analytic manifold $M$ and a subanalytic isotropic subset $Lambda$ of its co-sphere bundle $S^*M$, we show that the partially wrapped Fukaya category of $T^*M$ stopped at $Lambda$ is equivalent to the category of compact objects in the unbounded derived category of sheaves on $M$ with microsupport inside $Lambda$. By an embedding trick, we also deduce a sheaf theoretic description of the wrapped Fukaya category of any Weinstein sector admitting a stable polarization.

We prove pointwise convergence for the scattering data of a Dirac system of differential equations. Equivalently, we prove an analog of Carleson’s theorem on almost everywhere convergence of Fourier series for a version of the non-linear Fourier transform. Our proofs are based on the study of resonances of Dirac systems using families of meromorphic inner functions, generated by a Riccati equation corresponding to the system.

We establish a prime number theorem for all uniquely ergodic, analytic skew products on the $2$-torus $mathbb{T}^2$. More precisely, for every irrational $alpha$ and every $1$-periodic real analytic $g:mathbb{R}tomathbb{R}$ of zero mean, let $T_{alpha,g} : mathbb{T}^2 rightarrow mathbb{T}^2$ be defined by $(x,y) mapsto (x+alpha,y+g(x))$. We prove that if $T_{alpha, g}$ is uniquely ergodic then, for every $(x,y) in mathbb{T}^2$, the sequence ${T_{alpha, g}^p(x,y)}$ is equidistributed on $mathbb{T}^2$ as $p$ traverses prime numbers. This is the first example of a class of natural, non-algebraic and smooth dynamical systems for which a prime number theorem holds. We also show that such a prime number theorem does not necessarily hold if $g$ is only continuous on $mathbb{T}$.