Consider a Riemannian manifold with bounded Ricci curvature $|\mathrm{Ric}|\leq n-1$ and the noncollapsing lower volume bound $\mathrm{Vol}(B_1(p))>\mathrm{v}>0$. The first main result of this paper is to prove that we have the $L^2$ curvature bound $\int_{B_1(p)}|\mathrm{Rm}|^2(x)\, dx < C(n,\mathrm{v})$,which proves the $L^2$ conjecture. In order to prove this, we will need to first show the following

Let $k$ be a non-archimedean local field with residual characteristic $p$. Let $G$ be a connected reductive group over $k$ that splits over a tamely ramified field extension of $k$. Suppose $p$ does not divide the order of the Weyl group of $G$. Then we show that every smooth irreducible complex representation of $G(k)$ contains an $\mathfrak {s}$-type of the form constructed by Kim–Yu and that every

We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm {PSL}(2,\mathbb {Z}[\omega ])$ with $\omega ^2+\omega +1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in ${\rm {PSL}}(2

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 Painlevé irreducibility of certain Schwarzian equations, and considerable

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

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

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 $e(\rho ) \geq 4\pi a \rho ^2 (1+ \frac {128}{15\sqrt {\pi }} \sqrt {\rho a^3} +o(\sqrt {\rho a^3}))$, thereby proving the Lee-Huang-Yang formula for the energy density.

We show that there exist absolute constants $\Delta > \delta > 0$ such that, for all $n \geqslant 2$, there exists a polynomial $P$ of degree\nonbreakingspace $n$, with coefficients in $\{-1,1\}$, such that \[ \delta \sqrt {n} \leqslant |P(z)| \leqslant \Delta \sqrt {n} \] for all $z\in \mathbb {C}$ with $|z|=1$. This confirms a conjecture of Littlewood from\nonbreakingspace 1966.

In this paper, we consider the CM line bundle on the $\mathrm {K}$-moduli space, i.e., the moduli space parametrizing $\mathrm {K}$-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly $\mathrm {K}$-stable Fano varieties that conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable $\mathrm

It is shown that if $A$ is a separable, exact C$^*$-algebra which satisfies the Universal Coefficient Theorem (UCT) and has a faithful, amenable trace, then $A$ admits a trace-preserving embedding into a simple, unital AF-algebra with a unique trace. Modulo the UCT, this provides an abstract characterization of C$^*$-subalgebras of simple, unital AF-algebras. As a consequence, for a countable, discrete

In this paper we consider the classification of closed non-collapsed ancient solutions to the Mean Curvature Flow ($n\geq 2$) that are uniformly two-convex. We prove that they are either contracting spheres or they must coincide up to translations and scaling with the rotationally symmetric closed ancient non-collapsed solution first constructed by Brian White, and later by Robert Haslhofer and Or

We prove a Weyl-exponent subconvex bound for any Dirichlet $L$-function of cube-free conductor. We also show a bound of the same strength for certain $L$-functions of self-dual $\mathrm {GL}_2$ automorphic forms that arise as twists of forms of smaller conductor.

We construct small models of number fields and deduce a better bound for the number of number fields of given degree and bounded discriminant.

We prove for the $N$-body problem the existence of hyperbolic motions for any prescribed limit shape and any given initial configuration of the bodies. The energy level $h>0$ of the motion can also be chosen arbitrarily. Our approach is based on the construction of global viscosity solutions for the Hamilton-Jacobi equation $H(x,d_xu)=h$. We prove that these solutions are fixed points of the associated

We prove a sharp square function estimate for the cone in $\mathbb {R}^3$ and consequently the local smoothing conjecture for the wave equation in $2+1$ dimensions.

We show that the decomposition matrix of unipotent $\ell $-blocks of a finite reductive group $\mathbf {G}(\mathbb {F}_q)$ has a unitriangular shape, assuming $q$ is a power of a good prime and $\ell $ is very good for $\mathbf {G}$. This was conjectured by Geck in 1990 as part of his PhD thesis. We establish this result by constructing projective modules using a modification of generalised Gelfand–Graev

A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. We prove that Rademacher type and Enflo type coincide, settling a long-standing open problem in Banach space theory. The

When is a topological branched self-cover of the sphere equivalent to a post-critically finite rational map on $\mathbb {C}\mathbb {P}^1$? William Thurston gave one answer in 1982, giving a negative criterion (an obstruction to a map being rational). We give a complementary, positive criterion: the branched self-cover is equivalent to a rational map if and only if there is an elastic graph spine for

We prove affirmatively the one-dimensional case of a conjecture of Stein regarding the $L^p$-boundedness of the Polynomial Carleson operator for $1\lt p\lt \infty $. \par Our proof relies on two new ideas: (i) we develop a framework for \emph higher-order wave-packet analysis that is consistent with the time-frequency analysis of the (generalized) Carleson operator, and (ii) we introduce a \emph local

We prove an explicit formula for the Poincaré polynomials of parabolic character varieties of Riemann surfaces with semisimple local monodromies, which was conjectured by Hausel, Letellier and Rodriguez-Villegas. Using an approach of Mozgovoy and Schiffmann the problem is reduced to counting pairs of a parabolic vector bundle and a nilpotent endomorphism of prescribed generic type. The generating function

The Glauberman correspondence is a fundamental bijection in the character theory of finite groups. In 1994, Hartley and Turull established a degree-divisibility property for characters related by that correspondence, subject to a congruence condition which should hold for the Green functions of finite groups of Lie type, as defined by Deligne and Lusztig. Here, we present a general argument for completing

Let $\psi :\mathbb {N}\to \mathbb {R}_{\geqslant 0}$ be an arbitrary function from the positive integers to the non-negative reals. Consider the set $\mathcal {A}$ of real numbers $\alpha $ for which there are infinitely many reduced fractions $a/q$ such that $|\alpha -a/q|\leqslant \psi (q)/q$. If $\sum _{q=1}^\infty \psi (q)\varphi (q)/q=\infty $, we show that $\mathcal {A}$ has full Lebesgue measure