Showing papers in "Annals of Mathematics in 2018"
TL;DR: For any α < 1/3, Daneri and Szekelyhidi as discussed by the authors constructed weak solutions to the 3D incompressible Euler equations in the class C_tC_^xα that have nonempty, compact support in time on R × T^3 and therefore fail to conserve the total kinetic energy.
Abstract: For any α < 1/3, we construct weak solutions to the 3D incompressible Euler equations in the class C_tC_^xα that have nonempty, compact support in time on R × T^3 and therefore fail to conserve the total kinetic energy. This result, together with the proof of energy conservation for α < 1/3 due to [Eyink] and [Constantin, E, Titi], solves Onsager's conjecture that the exponent α = 1/3 marks the threshold for conservation of energy for weak solutions in the class L_t^∞C_x^α. The previous best results were solutions in the classC_tC_x^α for α < 1/5, due to [Isett], and in the class L_t^1C_x^α for α < 1/3 due to [Buckmaster, De Lellis, Szekelyhidi], both based on the method of convex integration developed for the incompressible Euler equations by [De Lellis, Szekelyhidi]. The present proof combines the method of convex integration and a new “Gluing Approximation” technique. The convex integration part of the proof relies on the “Mikado flows” introduced by [Daneri, Szekelyhidi] and the framework of estimates developed in the author's previous work.
275 citations
TL;DR: The notion of matroids was introduced by Rota as discussed by the authors, who defined a matroidM as a closure operator defined on all subsets of an afinite set E satisfying the Steinitz-MacLaneexchange property.
Abstract: The combinatorial theory of matroids starts with Whitney [Whi35], who introduced matroids as models for independence in vector spaces and graphs. By definition, a matroidM is given by a closure operator defined on all subsets of afinite set E satisfying the Steinitz-MacLaneexchange property: For every subset I of E and every element a not in the closure of I, if a is in the closure of I ∪ {b}, then b is in the closure of I ∪ {a}. The matroid is called loopless if the empty subset of E is closed, and is called a combinatorial geometry if in addition all single element subsets of E are closed. A closed subset of E is called a flat of M, and every subset of E has a well-defined rank and corank in the poset of all flats of M. The notion of matroid played a fundamental role in graph theory, coding theory, combinatorial optimization, and mathematical logic; we refer to [Wel71] and [Oxl11] for general introduction. As a generalization of the chromatic polynomial of a graph [Bir12, Whi32], Rota defined for an arbitrary matroid M the characteristic polynomial
179 citations
TL;DR: In this article, the authors prove global existence of appropriate weak solutions for the compressible Navier-Stokes equations for more general stress tensors than those covered by P. Lions and E. Feireisl's theory.
Abstract: We prove global existence of appropriate weak solutions for the compressible Navier–Stokes equations for more general stress tensor than those covered by P.–L. Lions and E. Feireisl’s theory. More precisely we focus on more general pressure laws which are not thermodynamically stable; we are also able to handle some anisotropy in the viscous stress tensor. To give answers to these two longstanding problems, we revisit the classical compactness theory on the density by obtaining precise quantitative regularity estimates: This requires a more precise analysis of the structure of the equations combined to a novel approach to the compactness of the continuity equation. These two cases open the theory to important physical applications, for instance to describe solar events (virial pressure law), geophysical flows (eddy viscosity) or biological situations (anisotropy).
136 citations
TL;DR: For a Riemannian manifold with (possibly empty) boundary, the authors showed that its volume spectrum satisfies a Weyl law that was conjectured by Gromov, and showed that it is possible to construct a manifold with a volume spectrum that satisfies this property.
Abstract: Given $M$ a Riemannian manifold with (possibly empty) boundary, we show that its volume spectrum $\{\omega_p(M)\}_{p\in\mathbb{N}}$ satisfies a Weyl law that was conjectured by Gromov.
99 citations
TL;DR: In this article, it was shown that for any compact Riemannian manifold (without boundary) of dimension n, there exists a positive constant ε ≥ 0 such that ε = 0 for any Laplace eigenfunction on the manifold, which corresponds to the eigenvalue ε.
Abstract: Let $u$ be a harmonic function in the unit ball $B(0,1) \subset \mathbb{R}^n$, $n \geq 3$, such that $u(0)=0$. Nadirashvili conjectured that there exists a positive constant $c$, depending on the dimension $n$ only, such that $H^{n-1}(\{u=0 \}\cap B) \geq c$. We prove Nadirashvili's conjecture as well as its counterpart on $C^\infty$-smooth Riemannian manifolds. The latter yields the lower bound in Yau's conjecture. Namely, we show that for any compact $C^\infty$-smooth Riemannian manifold $M$ (without boundary) of dimension $n$ there exists $c>0$ such that for any Laplace eigenfunction $\varphi_\lambda$ on $M$, which corresponds to the eigenvalue $\lambda$, the following inequality holds: $c \sqrt \lambda \leq H^{n-1}(\{\varphi_\lambda =0\})$.
95 citations
TL;DR: In this article, it was shown that the volume of the nodal sets in terms of the frequency and the doubling index of the Laplace eigenfunction on a compact Riemannian manifold can be characterized by propagation of smallness.
Abstract: Let $\mathbb{M}$ be a compact $C^\infty$-smooth Riemannian manifold of dimension $n$, $n\geq 3$, and let $\varphi_\lambda: \Delta_M \varphi_\lambda + \lambda \varphi_\lambda = 0$ denote the Laplace eigenfunction on $\mathbb{M}$ corresponding to the eigenvalue $\lambda$. We show that $$H^{n-1}(\{ \varphi_\lambda=0\}) \leq C \lambda^{\alpha},$$ where $\alpha>1/2$ is a constant, which depends on $n$ only, and $C>0$ depends on $\mathbb{M}$ . This result is a consequence of our study of zero sets of harmonic functions on $C^\infty$-smooth Riemannian manifolds. We develop a technique of propagation of smallness for solutions of elliptic PDE that allows us to obtain local bounds from above for the volume of the nodal sets in terms of the frequency and the doubling index.
93 citations
TL;DR: In this paper, it was shown that the union of all closed, smooth, embedded minimal hypersurfaces is dense, which implies that there are infinitely many such surfaces and thus proving a conjecture of Yau (1982) for generic metrics.
Abstract: For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that the union of all closed, smooth, embedded minimal hypersurfaces is dense. This implies there are infinitely many minimal hypersurfaces thus proving a conjecture of Yau (1982) for generic metrics.
93 citations
TL;DR: In this paper, exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices of the almost Mathieu operators for all frequencies in the localization regime were derived. But the authors did not consider the non-uniformity properties of hyperbolic cocycles.
Abstract: We determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices of the almost Mathieu operators for all frequencies in the localization regime. This uncovers a universal structure in their behavior, governed by the continued fraction expansion of the frequency, explaining some predictions in physics literature.
In addition it proves the arithmetic version of the frequency transition conjecture. Finally, it leads to an explicit description of several non-regularity phenomena in the corresponding non-uniformly hyperbolic cocycles, which is also of interest as both the first natural example of some of those phenomena and, more generally, the first non-artificial model where non-regularity can be explicitly studied.
77 citations
TL;DR: In this paper, a complete local version of the Birkhoff conjecture was proved: a small integrable perturbation of an elliptical table must be an ellipse.
Abstract: The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in [3], where nearly circular domains were considered. One of the crucial ideas in the proof is to extend action-angle coordinates for elliptic billiards into complex domains (with respect to the angle), and to thoroughly analyze the nature of their complex singularities. As an application, we are able to prove some spectral rigidity results for elliptic domains.
75 citations
TL;DR: In this paper, the authors show that the Liouville function has super-linear block growth and that all uniquely ergodic systems with zero topological entropy have no irrational spectrum and their building blocks are infinite-step nil systems and Bernoulli systems.
Abstract: The Mobius disjointness conjecture of Sarnak states that the Mobius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero topological entropy. We verify the logarithmically averaged variant of this conjecture for a large class of systems, which includes all uniquely ergodic systems with zero entropy. One consequence of our results is that the Liouville function has super-linear block growth. Our proof uses a disjointness argument and the key ingredient is a structural result for measure preserving systems naturally associated with the Mobius and the Liouville function. We prove that such systems have no irrational spectrum and their building blocks are infinite-step nilsystems and Bernoulli systems. To establish this structural result we make a connection with a problem of purely ergodic nature via some identities recently obtained by Tao. In addition to an ergodic structural result of Host and Kra, our analysis is guided by the notion of strong stationarity which was introduced by Furstenberg and Katznelson in the early 90's and naturally plays a central role in the structural analysis of measure preserving systems associated with multiplicative functions.
71 citations
TL;DR: For all convex co-compact hyperbolic surfaces, the existence of an essential spectral gap has been shown in this article, which is a strip beyond the unitarity axis in which the Selberg zeta function has only finitely many zeros.
Abstract: For all convex co-compact hyperbolic surfaces, we prove the existence of an essential spectral gap, that is a strip beyond the unitarity axis in which the Selberg zeta function has only finitely many zeroes. We make no assumption on the dimension $\delta$ of the limit set, in particular we do not require the pressure condition $\delta\leq {1\over 2}$. This is the first result of this kind for quantum Hamiltonians.
Our proof follows the strategy developed by Dyatlov-Zahl [arXiv:1504.06589]. The main new ingredient is the fractal uncertainty principle for $\delta$-regular sets with $\delta<1$, which may be of independent interest.
TL;DR: In this article, Bruinier-Kudla-Yang conjecture on the intersection multiplicities of special divisors and big CM points on M to the central derivatives of certain L-functions was proved.
Abstract: Let M be the Shimura variety associated with the group of spinor similitudes of a quadratic space over Q of signature (n,2). We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on M to the central derivatives of certain L-functions. As an application of this result, we prove an averaged version of Colmez’s conjecture on the Faltings heights of CM abelian varieties.
TL;DR: In this article, it was shown that Ricci flows with bounded scalar curvature and entropy converge smoothly away from a singular set of codimension in the Riemannian case.
Abstract: In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of codimension $\geq 4$. We also establish a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case.
These results are based on a compactness theorem for Ricci flows with bounded scalar curvature, which states that any sequence of such Ricci flows converges, after passing to a subsequence, to a metric space that is smooth away from a set of codimension $\geq 4$. In the course of the proof, we will also establish $L^{p < 2}$-curvature bounds on time-slices of such flows.
TL;DR: In this paper, a new class of groups of Burnside type, called simple groups of intermediate growth, is described. But the latter is restricted to groups with linearly repetitive Schreier graphs, and not to groups of infinite finitely generated periodic groups.
Abstract: We describe a new class of groups of Burnside type, giving a procedure transforming an arbitrary non-free minimal action of the dihedral group on a Cantor set into an orbit-equivalent action of an infinite finitely generated periodic group. We show that if the associated Schreier graphs are linearly repetitive, then the group is of intermediate growth. In particular, this gives first examples of simple groups of intermediate growth.
TL;DR: The proof of this "vertical versus horizontal isoperimetric inequality" uses a new structural result that decomposes sets of finite perimeter in the Heisenberg group into pieces that admit an "intrinsic corona decomposition."
Abstract: The discrete Heisenberg group $\mathbb{H}_{\mathbb{Z}}^{2k+1}$ is the group generated by $a_1,b_1,\ldots,a_k,b_k,c$, subject to the relations $[a_1,b_1]=\ldots=[a_k,b_k]=c$ and $[a_i,a_j]=[b_i,b_j]=[a_i,b_j]=[a_i,c]=[b_i,c]=1$ for every distinct $i,j\in \{1,\ldots,k\}$. Denote $S=\{a_1^{\pm 1},b_1^{\pm 1},\ldots,a_k^{\pm 1},b_k^{\pm 1}\}$. The horizontal boundary of $\Omega\subset \mathbb{H}_{\mathbb{Z}}^{2k+1}$, denoted $\partial_{h}\Omega$, is the set of all $(x,y)\in \Omega\times (\mathbb{H}_{\mathbb{Z}}^{2k+1}\setminus \Omega)$ such that $x^{-1}y\in S$. The horizontal perimeter of $\Omega$ is $|\partial_{h}\Omega|$. For $t\in \mathbb{N}$, define $\partial^t_{v} \Omega$ to be the set of all $(x,y)\in \Omega\times (\mathbb{H}_{\mathsf{Z}}^{2k+1}\setminus \Omega)$ such that $x^{-1}y\in \{c^t,c^{-t}\}$. The vertical perimeter of $\Omega$ is defined by $|\partial_{v}\Omega|= \sqrt{\sum_{t=1}^\infty |\partial^t_{v}\Omega|^2/t^2}$. It is shown here that if $k\ge 2$, then $|\partial_{v}\Omega|\lesssim \frac{1}{k} |\partial_{h}\Omega|$. The proof of this "vertical versus horizontal isoperimetric inequality" uses a new structural result that decomposes sets of finite perimeter in the Heisenberg group into pieces that admit an "intrinsic corona decomposition." This allows one to deduce an endpoint $W^{1,1}\to L_2(L_1)$ boundedness of a certain singular integral operator from a corresponding lower-dimensional $W^{1,2}\to L_2(L_2)$ boundedness. The above inequality has several applications, including that any embedding into $L_1$ of a ball of radius $n$ in the word metric on $\mathbb{H}_{\mathbb{Z}}^{5}$ incurs bi-Lipschitz distortion that is at least a constant multiple of $\sqrt{\log n}$. It follows that the integrality gap of the Goemans--Linial semidefinite program for the Sparsest Cut Problem on inputs of size $n$ is at least a constant multiple of $\sqrt{\log n}$.
TL;DR: In this paper, the positivity of quantum cluster coefficients for all skew-symmetric quantum cluster algebras was proved via a proof of a conjecture first suggested by Kontsevich on the purity of mixed Hodge structures arising in the theory of cluster mutation of spherical collections in 3-Calabi-Yau categories.
Abstract: Building on work by Kontsevich, Soibelman, Nagao and Efimov, we prove the positivity of quantum cluster coefficients for all skew-symmetric quantum cluster algebras, via a proof of a conjecture first suggested by Kontsevich on the purity of mixed Hodge structures arising in the theory of cluster mutation of spherical collections in 3-Calabi-Yau categories. The result implies positivity, as well as the stronger Lefschetz property conjectured by Efimov, and also the classical positivity conjecture of Fomin and Zelevinsky, recently proved by Lee and Schiffler. Closely related to these results is a categorified "no exotics" type theorem for cohomological Donaldson-Thomas invariants, which we discuss and prove in the appendix.
TL;DR: In this paper, the algebraic hull of the Kontsevich-Zorich cocycle over any GL^+_2(R) invariant subvariety of the Hodge bundle is derived.
Abstract: We compute the algebraic hull of the Kontsevich-Zorich cocycle over any GL^+_2(R) invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.
TL;DR: In this paper, a new form of approximate L monotonicity was identified for the linear solutions that holds whenever the background Kasner solution is sufficiently close to the Friedmann-Lemaitre-Robertson-Walker (FLRW) solution.
Abstract: We linearize the Einstein-scalar field equations, expressed relative to constant mean curvature (CMC)-transported spatial coordinates gauge, around members of the well-known family of Kasner solutions on (0,∞)×T. The Kasner solutions model a spatially uniform scalar field evolving in a (typically) spatially anisotropic spacetime that expands towards the future and that has a “Big Bang” singularity at {t = 0}. We place initial data for the linearized system along {t = 1} ≃ T and study the linear solution’s behavior in the collapsing direction t ↓ 0. Our main result is the identification of a new form of approximate L monotonicity for the linear solutions that holds whenever the background Kasner solution is sufficiently close to the Friedmann-Lemaitre-Robertson-Walker (FLRW) solution. Using the approximate monotonicity, we derive sharp information about the asymptotic behavior of the linear solution as t ↓ 0. In particular, we show that some of its time-rescaled components converge to regular functions defined along {t = 0}. In addition, we motivate the preferred direction of the approximate monotonicity by showing that the CMCtransported spatial coordinates gauge can be realized as a limiting version of a family of parabolic gauges for the lapse variable. An approximate L monotonicity inequality also holds in the parabolic gauges, but the corresponding parabolic PDEs are locally well-posed only in the direction t ↓ 0. In a companion article, we use the linear stability results to prove a stable singularity formation result for the nonlinear equations. Specifically, we show that the FLRW solution is globally nonlinearly stable in the collapsing direction t ↓ 0 under small perturbations of its data at {t = 1}.
TL;DR: In this paper, it was shown that the Gopakumar-Vafa conjecture holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi Yau 3-folds.
Abstract: The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa formula holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds.
TL;DR: In this article, it was shown that the Hausdorff dimensions of intersections of Markov and Lagrange spectra with half-lines always coincide, and may assume any real value in the interval $[0, 1].
Abstract: We prove several results on (fractal) geometric properties of the classical Markov and Lagrange spectra. In particular, we prove that the Hausdorff dimensions of intersections of both spectra with half-lines always coincide, and may assume any real value in the interval $[0, 1]$.
TL;DR: Gross conjectured a formula for the expected leading term at 0 for the Deligne-Ribet model of the abelian extension of a real field as mentioned in this paper, and proved Gross's conjecture.
Abstract: In 1980, Gross conjectured a formula for the expected leading term at $s=0$ of the Deligne--Ribet $p$-adic $L$-function associated to a totally even character $\psi$ of a totally real field $F$. The conjecture states that after scaling by $L(\psi \omega^{-1}, 0)$, this value is equal to a $p$-adic regulator of units in the abelian extension of $F$ cut out by $\psi \omega^{-1}$. In this paper, we prove Gross's conjecture.
TL;DR: The Colmez conjecture as discussed by the authors is a conjecture expressing the Faltings height of a CM abelian variety in terms of some linear combination of logarithmic derivatives of Artin L-functions.
Abstract: The Colmez conjecture, proposed by Colmez, is a conjecture expressing the Faltings height of a CM abelian variety in terms of some linear combination of logarithmic derivatives of Artin L-functions. The aim of this paper to prove an averaged version of the conjecture.
TL;DR: In this paper, it was shown that the 2-primary Brown-Peterson spectrum does not admit the structure of an E_n-algebra for any n greater than or equal to 12, answering a question of May in the negative.
Abstract: The dual Steenrod algebra has a canonical subalgebra isomorphic to the homology of the Brown-Peterson spectrum. We will construct a secondary operation in mod-2 homology and show that this canonical subalgebra is not closed under it. This allows us to conclude that the 2-primary Brown-Peterson spectrum does not admit the structure of an E_n-algebra for any n greater than or equal to 12, answering a question of May in the negative.
TL;DR: In this article, the authors construct positive-genus real Gromov-Witten invariants of real orientable manifolds of odd "complex" dimensions and study the orientations on the moduli spaces of real maps used in constructing them.
Abstract: The first part of this work constructs positive-genus real Gromov-Witten invariants of realorientable symplectic manifolds of odd “complex” dimensions; the second part studies the orientations on the moduli spaces of real maps used in constructing these invariants. The present paper applies the results of the latter to obtain quantitative and qualitative conclusions about the invariants defined in the former. After describing large collections of real-orientable symplectic manifolds, we show that the genus 1 real Gromov-Witten invariants of sufficiently positive almost Kahler threefolds are signed counts of real genus 1 curves only and thus provide direct lower bounds for the counts of these curves in such targets. We specify real orientations on the real-orientable complete intersections in projective spaces; the real Gromov-Witten invariants they determine are in a sense canonically determined by the complete intersection itself, (at least) in most cases. We also obtain equivariant localization data that computes the real invariants of projective spaces and determines the contributions from many torus fixed loci for other complete intersections. Our results confirm Walcher’s predictions for the vanishing of these invariants in certain cases and for the localization data in other cases.
TL;DR: In this article, the moduli functor of stable varieties is shown to be semipositive in the sense of Kollar's projectivity criterion for higher-dimensional stable varieties.
Abstract: We prove some semipositivity theorems for singular varieties coming from graded polarizable admissible variations of mixed Hodge structure. As an application, we obtain that the moduli functor of stable varieties is semipositive in the sense of Kollar. This completes Kollar's projectivity criterion for the moduli spaces of higher-dimensional stable varieties.
TL;DR: Turner's conjecture was proved in this article, which describes the blocks of the symmetric groups up to derived equivalence as certain explicit Turner double algebras, which are Schur-algebra-like 'local' objects.
Abstract: We prove Turner's conjecture, which describes the blocks of the Hecke algebras of the symmetric groups up to derived equivalence as certain explicit Turner double algebras. Turner doubles are Schur-algebra-like `local' objects, which replace wreath products of Brauer tree algebras in the context of the Brou\'e abelian defect group conjecture for blocks of symmetric groups with non-abelian defect groups. The main tools used in the proof are generalized Schur algebras corresponding to wreath products of zigzag algebras and imaginary semicuspidal quotients of affine KLR algebras.
TL;DR: In this article, the Ricci flow was extended to higher dimensions by a surgery procedure in the spirit of Hamilton and Perelman's neck-like curvature pinching estimate.
Abstract: We present a new curvature condition which is preserved by the Ricci flow in higher dimensions. For initial metrics satisfying this condition, we establish a higher dimensional version of Hamilton's neck-like curvature pinching estimate. Using this estimate, we are able to prove a version of Perelman's Canonical Neighborhood Theorem in higher dimensions. This makes it possible to extend the flow beyond singularities by a surgery procedure in the spirit of Hamilton and Perelman. As a corollary, we obtain a classification of all diffeomorphism types of such manifolds in terms of a connected sum decomposition. In particular, the underlying manifold cannot be an exotic sphere.
Our result is sharp in many interesting situations. For example, the curvature tensors of $\mathbb{CP}^{n/2}$, $\mathbb{HP}^{n/4}$, $S^{n-k} \times S^k$ ($2 \leq k \leq n-2$), $S^{n-2} \times \mathbb{H}^2$, $S^{n-2} \times \mathbb{R}^2$ all lie on the boundary of our curvature cone. Another borderline case is the pseudo-cylinder: this is a rotationally symmetric hypersurface which is weakly, but not strictly, two-convex. Finally, the curvature tensor of $S^{n-1} \times \mathbb{R}$ lies in the interior of our curvature cone.
TL;DR: In this paper, it was shown that every quasigeodesic flow on a closed hyperbolic 3-manifold has closed orbits and Calegari's conjecture is false.
Abstract: We prove Calegari's conjecture that every quasigeodesic flow on a closed hyperbolic 3-manifold has closed orbits.