-
作者：Zhou, Xin
摘要：We prove that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves. We prove that in a bumpy metric each volume spectrum is realized by the min-max value of certain relative homotopy class of sweepouts of boundaries of Caccioppoli sets. The main result follows by approximating such...
-
作者：Casale, Guy; Freitag, James; Nagloo, Joel
摘要：We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear differential equations, the study of algebraic and Liouvillian solutions, differential algebraic work of Nishioka towards the Painleve irreducibility of certain Schwarzian equations, and considerable machinery from the model theory of differentially closed fields. Our techniques allow f...
-
作者：Alper, Jarod; Hall, Jack; Rydh, David
摘要：We prove that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, is (tale-locally a quotient stack in a neighborhood of a point with a linearly reductive stabilizer group. The proof uses an equivariant version of Artin's algebraization theorem proved in the appendix. We provide numerous applications of the main theorems.
-
作者：Branden, Petter; Huh, June
摘要：We study the class of Lorentzian polynomials. The class contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. We prove that the Hessian of a nonzero Lorentzian polynomial has exactly one positive eigenvalue at any point on the positive orthant. This property can be seen as an analog of the Hodge-Riemann relations for Lorentzian polynomials. Lorentzian polynomials are intimately connected to matroid theory and negative dependence proper...
-
作者：DeMarco, Laura; Krieger, Holly; Ye, Hexi
摘要：We introduce a general strategy for proving quantitative and uniform bounds on the number of common points of height zero for a pair of inequivalent height functions on P-1((Q) over bar). We apply this strategy to prove a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds on the number of common torsion points of elliptic curves in the case of two Legendre curves over C. As a consequence, we obtain two uniform bounds for a two-dimensional family of genus 2 curves: a uniform Ma...
-
作者：Goette, S.; Kerin, M.; Shankar, K.
摘要：In this article, a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO(3)-invariant metric of non-negative curvature.
-
作者：Balister, Paul; Bollobas, Bela; Morris, Robert; Sahasrabudhe, Julian; Tiba, Marius
摘要：We show that there exist absolute constants Delta > delta > 0 such that, for all n >= 2, there exists a polynomial P of degree n, with coefficients in { -1, 1}, such that delta root n <= vertical bar P(z)vertical bar <= Delta root n for all z is an element of C with vertical bar z vertical bar = 1. This confirms a conjecture of Littlewood from 1966.
-
作者：Browning, Tim; Sawin, Will
摘要：We develop a geometric version of the circle method and use it to compute the compactly supported cohomology of the space of rational curves through a point on a smooth affine hypersurface of sufficiently low degree.
-
作者：Fournais, Soren; Solovej, Jan Philip
摘要：For a dilute system of non-relativistic bosons interacting through a positive, compactly supported, L-1-potential v with scattering length a we prove that the ground state energy density satisfies the bound e(rho) >= 4 pi a rho(2)(1 + 128/15 root pi root rho a(3) + o(root rho a(3))), thereby proving the Lee-Huang-Yang formula for the energy density.
-
作者：Xu, Chenyang; Zhuang, Zhiquan
摘要：In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties that conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective. During the course of proof, we develop a new invariant for filtrations that can be used to tes...
