Showing papers in "Journal of the European Mathematical Society in 2017"

Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries

Abstract: Let $\M$ be a smooth connected manifold endowed with a smooth measure $\mu$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $\mu$. Associated with $L$ one has \textit{le carr\'e du champ} $\Gamma$ and a canonical distance $d$, with respect to which we suppose that $(M,d)$ be complete. We assume that $\M$ is also equipped with another first-order differential bilinear form $\Gamma^Z$ and we assume that $\Gamma$ and $\Gamma^Z$ satisfy the Hypothesis below. With these forms we introduce in \eqref{cdi} below a generalization of the curvature-dimension inequality from Riemannian geometry, see Definition \ref{D:cdi}. In our main results we prove that, using solely \eqref{cdi}, one can develop a theory which parallels the celebrated works of Yau, and Li-Yau on complete manifolds with Ricci bounded from below. We also obtain an analogue of the Bonnet-Myers theorem. In Section \ref{S:appendix} we construct large classes of sub-Riemannian manifolds with transverse symmetries which satisfy the generalized curvature-dimension inequality \eqref{cdi}. Such classes include all Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is bounded from below, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is bounded from below.

Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production

Abstract: We study the Neumann initial-boundary problem for the chemotaxis system    ut = ∆u−∇ · (u∇v), x ∈ Ω, t > 0, 0 = ∆v − μ(t) + w, x ∈ Ω, t > 0, τwt + δw = u, x ∈ Ω, t > 0, (⋆) in the unit disk Ω := B1(0) ⊂ R, where δ ≥ 0 and τ > 0 are given parameters and μ(t) := − ∫ Ω w(x, t)dx, t > 0. It is shown that this problem exhibits a novel type of critical mass phenomenon with regard to the formation of singularities, which drastically differs from the well-known threshold property of the classical Keller-Segel system, as obtained upon formally taking τ → 0, in that it refers to blow-up in infinite time rather than in finite time: Specifically, it is first proved that for any sufficiently regular nonnegative initial data u0 and w0, (⋆) possesses a unique global classical solution. In particular, this shows that in sharp contrast to classical Keller-Segel-type systems reflecting immediate signal secretion by the cells themselves, the indirect mechanism of signal production in (⋆) entirely rules out any occurrence of blow-up in finite time. However, within the framework of radially symmetric solutions it is next proved that • whenever δ > 0 and ∫ Ω u0 8πδ, one can find initial data such that ∫ Ω u0 = m, and such that for the corresponding solution we have ‖u(·, t)‖L∞(Ω) → ∞ as t → ∞.

Quantitative results on the corrector equation in stochastic homogenization

Abstract: We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions $d\ge 2$. In previous works we studied the model problem of a discrete elliptic equation on $\mathbb{Z}^d$. Under the assumption that a spectral gap estimate holds in probability, we proved that there exists a stationary corrector field in dimensions $d>2$ and that the energy density of that corrector behaves as if it had finite range of correlation in terms of the variance of spatial averages - the latter decays at the rate of the central limit theorem. In this article we extend these results, and several other estimates, to the case of a continuum linear elliptic equation whose (not necessarily symmetric) coefficient field satisfies a continuum version of the spectral gap estimate. In particular, our results cover the example of Poisson random inclusions.

Sparse recovery under weak moment assumptions

Abstract: We prove that iid random vectors that satisfy a rather weak moment assumption can be used as measurement vectors in Compressed Sensing, and the number of measurements required for exact reconstruction is the same as the best possible estimate – exhibited by a random Gaussian matrix. We then show that this moment condition is necessary, up to a log log factor. In addition, we explore the Compatibility Condition and the Restricted Eigenvalue Condition in the noisy setup, as well as properties of neighbourly random polytopes.

Spherical DG-functors

Abstract: For two DG-categories A and B we define the notion of a spherical Morita quasi-functor A → B. We construct its associated autoequivalences: the twist T ∈ Aut D(B) and the co-twist F ∈ Aut D(A). We give sufficiency criteria for a quasi-functor to be spherical and for the twists associated to a collection of spherical quasi-functors to braid. Using the framework of DG-enhanced triangulated categories, we translate all of the above to Fourier-Mukai transforms between the derived categories of algebraic varieties. This is a broad generalisation of the results on spherical objects in [ST01] and on spherical functors in [Ann07]. In fact, this paper replaces [Ann07], which has a fatal gap in the proof of its main theorem. Though conceptually correct, the proof was impossible to fix within the framework of triangulated categories.

Polynomial chaos and scaling limits of disordered systems

Abstract: Inspired by recent work of Alberts, Khanin and Quastel, we formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables (called polynomial chaos expansions) converges to a limiting random variable, given by a Wiener chaos expansion over the d-dimensional white noise. A key ingredient in our approach is a Lindeberg principle for polynomial chaos expansions, which extends earlier work of Mossel, O'Donnell and Oleszkiewicz. These results provide a unified framework to study the continuum and weak disorder scaling limits of statistical mechanics systems that are disorder relevant, including the disordered pinning model, the (long-range) directed polymer model in dimension 1+1, and the two-dimensional random field Ising model. This gives a new perspective in the study of disorder relevance, and leads to interesting new continuum models that warrant further studies.

Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on $\mathbb R^d$, $d=4$ and $5$

Abstract: We consider the energy-critical defocusing nonlinear wave equation (NLW) on $\mathbb{R}^d$, $d=4$ and $5$. We prove almost sure global existence and uniqueness for NLW with rough random initial data in $H^s(\mathbb{R}^d)\times H^{s-1}(\mathbb{R}^d)$, with $0< s\leq 1$ if $d=4$, and $0\leq s\leq 1$ if $d=5$. The randomization we consider is naturally associated with the Wiener decomposition and with modulation spaces. The proof is based on a probabilistic perturbation theory. Under some additional assumptions, for $d=4$, we also prove the probabilistic continuous dependence of the flow with respect to the initial data (in the sense proposed by Burq and Tzvetkov).

All functions are locally $s$-harmonic up to a small error

Abstract: We show that we can approximate every function $f\in C^{k}(\bar{B_1})$ with a $s$-harmonic function in $B_1$ that vanishes outside a compact set. That is, $s$-harmonic functions are dense in $C^{k}_{\rm{loc}}$. This result is clearly in contrast with the rigidity of harmonic functions in the classical case and can be viewed as a purely nonlocal feature.

Mixing and un-mixing by incompressible flows

Abstract: We consider the questions of efficient mixing and un-mixing by incompressible flows which satisfy periodic, no-flow, or no-slip boundary conditions on a square. Under the uniform-in-time constraint ‖∇u(·, t)‖p ≤ 1 we show that any function can be mixed to scale e in time O(| log e|1+νp), with νp = 0 for p < 3+ √ 5 2 and νp ≤ 13 for p ≥ 3+ √ 5 2 . Known lower bounds show that this rate is optimal for p ∈ (1, 3+ √ 5 2 ). We also show that any set which is mixed to scale e but not much more than that can be un-mixed to a rectangle of the same area (up to a small error) in time O(| log e|2−1/p). Both results hold with scale-independent finite times if the constraint on the flow is changed to ‖u(·, t)‖Ẇ s,p ≤ 1 with some s < 1. The constants in all our results are independent of the mixed functions and sets.

A semi-algebraic version of Zarankiewicz's problem

Abstract: A bipartite graph G is semi-algebraic in R^d if its vertices are represented by point sets P,Q ⊂ R^d and its edges are defined as pairs of points (p,q) ∈ P×Q that satisfy a Boolean combination of a fixed number of polynomial equations and inequalities in 2d coordinates. We show that for fixed k, the maximum number of edges in a K_(k,k)-free semi-algebraic bipartite graph G=(P,Q,E) in R^2 with |P|=m and |Q|=n is at most O((mn)^(2/3) + m + n), and this bound is tight. In dimensions d ≥ 3, we show that all such semi-algebraic graphs have at most C((mn)^(dd+1+ϵ) + m + n) edges, where here ϵ is an arbitrarily small constant and C=C(d,k,t,ϵ). This result is a far-reaching generalization of the classical Szemeredi-Trotter incidence theorem. The proof combines tools from several fields: VC-dimension and shatter functions, polynomial partitioning, and Hilbert polynomials. We also present various applications of our theorem. For example, a general point-variety incidence bound in R^d, an improved bound for a d-dimensional variant of the Erdos unit distances problem, and more.

On completeness of groups of diffeomorphisms

Abstract: We study completeness properties of the Sobolev diffeomorphism groups Ds(M) endowed with strong right-invariant Riemannian metrics when the underlying manifold M is ℝd or compact without boundary. The main result is that for dim M/2 + 1, the group Ds (M) is geodesically and metrically complete with a surjective exponential map. We also extend the result to its closed subgroups, in particular the group of volume preserving diffeomorphisms and the group of symplectomorphisms. We then present the connection between the Sobolev diffeomorphism group and the large deformation matching framework in order to apply our results to diffeomorphic image matching.

Entropy and a convergence theorem for Gauss curvature flow in high dimension

Abstract: In this paper we prove uniform regularity estimates for the normalized Gauss curvature flow in higher dimensions. The convergence of solutions in C∞-topology to a smooth strictly convex soliton as t approaches to infinity is obtained as a consequence of these estimates together with an earlier result of Andrews. The estimates are established via the study of an entropy functional for convex bodies.

A new characterization of chord-arc domains

Abstract: The first author was partially supported by NSF RTG grant 0838212. The second author was supported by NSF grants DMS-1101244 and DMS-1361701. The third author was supported in part by MINECO Grant MTM2010-16518, ICMAT Severo Ochoa project SEV-2011-0087. He also acknowledges that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT. The fourth author was partly supported by the Swedish research council VR. The last author was partially supported by the Robert R. & Elaine F. Phelps Professorship in Mathematics.

Strong minimality and the j-function

Abstract: We show that the order three algebraic differential equation over ${\mathbb Q}$ satisfied by the analytic $j$-function defines a non-$\aleph_0$-categorical strongly minimal set with trivial forking geometry relative to the theory of differentially closed fields of characteristic zero answering a long-standing open problem about the existence of such sets. The theorem follows from Pila's modular Ax-Lindemann-Weierstrass with derivatives theorem using Seidenberg's embedding theorem and a theorem of Nishioka on the differential equations satisfied by automorphic functions. As a by product of this analysis, we obtain a more general version of the modular Ax-Lindemann-Weierstrass theorem, which, in particular, applies to automorphic functions for arbitrary arithmetic subgroups of $SL_2 ({\mathbb Z})$. We then apply the results to prove effective finiteness results for intersections of subvarieties of products of modular curves with isogeny classes. For example, we show that if $\psi:{\mathbb P}^1 \to {\mathbb P}^1$ is any non-identity automorphism of the projective line and $t \in {\mathbb A}^1({\mathbb C}) \smallsetminus {\mathbb A}^1({\mathbb Q}^\text{alg})$, then the set of $s \in {\mathbb A}^1({\mathbb C})$ for which the elliptic curve with $j$-invariant $s$ is isogenous to the elliptic curve with $j$-invariant $t$ and the elliptic curve with $j$-invariant $\psi(s)$ is isogenous to the elliptic curve with $j$-invariant $\psi(t)$ has size at most $36^7$. In general, we prove that if $V$ is a Kolchin-closed subset of ${\mathbb A}^n$, then the Zariski closure of the intersection of $V$ with the isogeny class of a tuple of transcendental elements is a finite union of weakly special subvarieties. We bound the sum of the degrees of the irreducible components of this union by a function of the degree and order of $V$.

Weyl-type hybrid subconvexity bounds for twisted $L$-functions and Heegner points on shrinking sets

Abstract: We prove a Weyl-type subconvexity bound for the central value of the $L$-function of a Hecke-Maass form or a holomorphic Hecke eigenform twisted by a quadratic Dirichlet character, uniform in the archimedean parameter as well as the twisting parameter. A similar hybrid bound holds for quadratic Dirichlet $L$-functions, improving on a result of Heath-Brown. As a consequence of these new bounds, we obtain explicit estimates for the number of Heegner points of large odd discriminant in shrinking sets.

A direct approach to Plateau's problem

Abstract: We provide a compactness principle which is applicable to different formulations of Plateau's problem in codimension one and which is exclusively based on the theory of Radon measures and elementary comparison arguments. Exploiting some additional techniques in geometric measure theory, we can use this principle to give a different proof of a theorem by Harrison and Pugh and to answer a question raised by Guy David. © European Mathematical Society 2017.

Boundedness of the space of stable logarithmic maps

Abstract: We prove that the moduli space of stable logarithmic maps with fixed numerical invariants, from logarithmic curves to a fixed projective target logarithmic scheme with fine and saturated logarithmic structure, is a proper algebraic stack. This was previously known only with further restrictions on the logarithmic structure of the target.

Intrinsic scaling properties for nonlocal operators

Abstract: We study integrodifferential operators and regularity estimates for solutions to integrodifferential equations. Our emphasis is on kernels with a critically low singularity which does not allow for standard scaling. For example, we treat operators that have a logarithmic order of differentiability. For corresponding equations we prove a growth lemma and derive a priori estimates. We derive these estimates by classical methods developed for partial differential operators. Since the integrodifferential operators under consideration generate Markov jump processes, we are able to offer an alternative approach using probabilistic techniques.

Triple Massey products and Galois theory

Abstract: We show that any triple Massey product with respect to prime 2 contains 0 whenever it is defined over any field. This extends the theorem of M. J. Hopkins and K. G. Wickelgren, from global fields to any fields. This is the first time when the vanishing of any $n$-Massey product for some prime $p$ has been established for all fields. This leads to a strong restriction on the shape of relations in the maximal pro-2-quotients of absolute Galois groups, which was out of reach until now. We also develop an extension of Serre's transgression method to detect triple commutators in relations of pro-$p$-groups, where we do not require that all cup products vanish. We prove that all $n$-Massey products, $n\\geq 3$, vanish for general Demushkin groups. We formulate and provide evidence for two conjectures related to the structure of absolute Galois groups of fields. In each case when these conjectures can be verified, they have some interesting concrete Galois theoretic consequences. They are also related to the Bloch-Kato conjecture.

Higher genus quasimap wall-crossing for semipositive targets

Abstract: In previous work (arXiv:1304.7056) we have conjectured wall-crossing formulas for genus zero quasimap invariants of GIT quotients and proved them via localization in many cases. We extend these formulas to higher genus when the target is semi-positive, and prove them for semi-positive toric varieties, in particular for toric local Calabi-Yau targets. The proof also applies to local Calabi-Yau's associated to some non-abelian quotients.

The motivic Steenrod algebra in positive characteristic

Abstract: Let S be an essentially smooth scheme over a field and l a prime number invertible on S. We show that the algebra of bistable operations in the mod l motivic cohomology of smooth S-schemes is generated by the motivic Steenrod operations. This was previously proved by Voevodsky for S a field of characteristic zero. We follow Voevodsky's proof but remove its dependence on characteristic zero by using etale cohomology instead of topological realization and by replacing resolution of singularities with a theorem of Gabber on alterations.

Poisson algebras via model theory and differential-algebraic geometry

Abstract: Brown and Gordon asked whether the Poisson Dixmier–Moeglin equivalence holds for any complex affine Poisson algebra, that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier–Moeglin equivalence in finite GK dimension. A weaker version of the Poisson Dixmier–Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero.

On base sizes for algebraic groups

Abstract: Let $G$ be a permutation group on a set $\Omega$. A subset of $\Omega$ is a base for $G$ if its pointwise stabilizer is trivial; the base size of $G$ is the minimal cardinality of a base. In this paper we initiate the study of bases for algebraic groups defined over an algebraically closed field. In particular, we calculate the base size for all primitive actions of simple algebraic groups, obtaining the precise value in almost all cases. We also introduce and study two new base measures, which arise naturally in this setting. We give an application concerning the essential dimension of simple algebraic groups, and we establish several new results on base sizes for the corresponding finite groups of Lie type. The latter results are an important contribution to the classical study of bases for finite primitive permutation groups. We also indicate some connections with generic stabilizers for representations of simple algebraic groups.

Classical solutions and higher regularity for nonlinear fractional diffusion equations

Abstract: We study the regularity properties of the solutions to the nonlinear equation with fractional diffusion $$ \partial_tu+(-\Delta)^{\sigma/2}\varphi(u)=0, $$ posed for $x\in \mathbb{R}^N$, $t>0$, with $0 \sigma$, and $\varphi'(u)>0$ for every $u\in\mathbb{R}$, we prove that bounded weak solutions are classical solutions for all positive times. We also explore sufficient conditions on the non-linearity to obtain higher regularity for the solutions, even $C^\infty$ regularity. Degenerate and singular cases, including the power nonlinearity $\varphi(u)=|u|^{m-1}u$, $m>0$, are also considered, and the existence of classical solutions in the power case is proved.

Triple Massey products and absolute Galois groups

Abstract: Let $p$ be a prime number, $F$ a field containing a root of unity of order $p$, and $G_F$ the absolute Galois group. Extending results of Hopkins, Wickelgren, Minac and Tan, we prove that the triple Massey product $H^1(G_F)^3\to H^2(G_F)$ contains $0$ whenever it is nonempty. This gives a new restriction on the possible profinite group structure of $G_F$.

The family Floer functor is faithful

Abstract: Family Floer theory yields a functor from the Fukaya category of a symplectic manifold admitting a Lagrangian torus fibration to a (twisted) category of perfect complexes on the mirror rigid analytic space. This functor is shown to be faithful by a degeneration argument involving moduli spaces of annuli.

The effective cone of the moduli space of sheaves on the plane

Abstract: We compute the cone of effective divisors on any moduli space of semistable sheaves on the plane. The computation hinges on finding a good resolution of a general stable sheaf. This resolution is determined by Bridgeland stability and arises from a well-chosen Beilinson spectral sequence. The existence of a good choice of spectral sequence depends on remarkable number-theoretic properties of the slopes of exceptional bundles.

Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems

Abstract: Friedland’s Lower Matching Conjecture asserts that if G is a d–regular bipartite graph on v(G) = 2n vertices, and mk(G) denotes the number of matchings of size k, then mk(G) ≥ ( n k )2( d− p d )n(d−p) (dp), where p = k n . When p = 1, this conjecture reduces to a theorem of Schrijver which says that a d–regular bipartite graph on v(G) = 2n vertices has at least ( (d− 1)d−1 dd−2 )n perfect matchings. L. Gurvits proved an asymptotic version of the Lower Matching Conjecture, namely he proved that lnmk(G) v(G) ≥ 1 2 ( p ln ( d p ) + (d− p) ln ( 1− p d ) − 2(1− p) ln(1− p) ) + ov(G)(1). In this paper, we prove the Lower Matching Conjecture. In fact, we will prove a slightly stronger statement which gives an extra cp √ n factor compared to the conjecture if p is separated away from 0 and 1, and is tight up to a constant factor if p is separated away from 1. We will also give a new proof of Gurvits’s and Schrijver’s theorems, and we extend these theorems to (a, b)–biregular bipartite graphs.

Integral transforms for coherent sheaves

Abstract: The theory of integral, or Fourier-Mukai, transforms between derived categories of sheaves is a well established tool in noncommutative algebraic geometry. General "representation theorems" identify all reasonable linear functors between categories of perfect complexes (or their "large" version, quasi-coherent sheaves) on schemes and stacks over some fixed base with integral kernels in the form of sheaves (of the same nature) on the fiber product. However, for many applications in mirror symmetry and geometric representation theory one is interested instead in the bounded derived category of coherent sheaves (or its "large" version, ind-coherent sheaves), which differs from perfect complexes (and quasi-coherent sheaves) once the underlying variety is singular. In this paper, we give general representation theorems for linear functors between categories of coherent sheaves over a base in terms of integral kernels on the fiber product. Namely, we identify coherent kernels with functors taking perfect complexes to coherent sheaves, and kernels which are coherent relative to the source with functors taking all coherent sheaves to coherent sheaves. The proofs rely on key aspects of the "functional analysis" of derived categories, namely the distinction between small and large categories and its measurement using $t$-structures. These are used in particular to correct the failure of integral transforms on Ind-coherent sheaves to correspond to such sheaves on a fiber product. The results are applied in a companion paper to the representation theory of the affine Hecke category, identifying affine character sheaves with the spectral geometric Langlands category in genus one.