Showing papers in "Duke Mathematical Journal in 2014"

Metric measure spaces with Riemannian Ricci curvature bounded from below

Abstract: In this paper, we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov– Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm, and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, global-to-local, and local-to-global properties. In these spaces, which we call RCD(K,∞) spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry–Emery estimates and the L∞-Lip Feller regularization. We also prove that the distance induced by the Dirichlet form coincides with d, that the local energy measure has density given by the square of Cheeger’s relaxed slope, and, as a consequence, that the underlying Brownian motion has continuous paths. All these results are obtained independently of Poincare and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional ones.

Tropical combinatorics and Whittaker functions

Abstract: We establish a fundamental connection between the geometric Robinson–Schensted–Knuth (RSK) correspondence and GL(N,R)-Whittaker functions, analogous to the well-known relationship between the RSK correspondence and Schur functions. This gives rise to a natural family of measures associated with GL(N,R)-Whittaker functions which are the analogues in this setting of the Schur measures on integer partitions. The corresponding analogue of the Cauchy–Littlewood identity can be seen as a generalization of an integral identity for GL(N,R)-Whittaker functions due to Bump and Stade. As an application, we obtain an explicit integral formula for the Laplace transform of the law of the partition function associated with a 1-dimensional directed polymer model with log-gamma weights recently introduced by one of the authors.

Second-order estimates and regularity for fully nonlinear elliptic equations on riemannian manifolds

Abstract: We derive a priori second-order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under structure conditions which are close to optimal. We treat both equations on closed manifolds and the Dirichlet problem on manifolds with boundary without any geometric restrictions to the boundary. These estimates yield regularity and existence results, some of which are new even for equations in Euclidean space.

Universality of general β-ensembles

Abstract: We prove the universality of the -ensembles with convex analytic potentials and for any > 0, i.e. we show that the spacing distributions of log-gases at any inverse temperature coincide with those of the Gaussian -ensembles.

Kesten's theorem for Invariant Random Subgroups

Abstract: An invariant random subgroup of the countable group is a ran dom subgroup of whose distribution is invariant under conjuga tion by all elements of . We prove that for a nonamenable invariant random subgroup H, the spectral radius of every finitely supported random walk on i s strictly less than the spectral radius of the corresponding random walk on /H. This generalizes a result of Kesten who proved this for normal subgroups. As a byproduct, we show that for a Cayley graph G of a linear group with no amenable normal subgroups, any sequence of finite quotients of G that spectrally approximates G converges to G in Benjamini-Schramm convergence. In particular, this implies that infinite sequences of finite d-regular Ramanujan Schreier graphs have essentially large girth.

Hodge theory and derived categories of cubic fourfolds

Abstract: Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics—conjecturally, the ones that are rational—have specific K3 surfaces associated to them geometrically. Hassett has studied cubics with K3 surfaces associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with K3 surfaces associated to them at the level of derived categories. These two notions of having an associated K3 surface should coincide. We prove that they coincide generically: Hassett’s cubics form a countable union of irreducible Noether–Lefschetz divisors in moduli space, and we show that Kuznetsov’s cubics are a dense subset of these, forming a nonempty, Zariski-open subset in each divisor.

Decomposition rank of UHF-absorbing c* -algebras

Abstract: Let A be a unital separable simple C∗-algebra with a unique tracial state We prove that if A is nuclear and quasidiagonal, then A tensored with the universal uniformly hyperfinite (UHF) algebra has decomposition rank at most one We then prove that A is nuclear, quasidiagonal, and has strict comparison if and only if A has finite decomposition rank For such A, we also give a direct proof that A tensored with a UHF algebra has tracial rank zero Using this result, we obtain a counterexample to the Powers–Sakai conjecture

Two-weight inequality for the hilbert transform: A real variable characterization, I

Abstract: Let σ and w be locally finite positive Borel measures on R which do not share a common point mass. Assume that the pair of weights satisfy a Poisson A2 condition, and satisfy the testing conditions below, for the Hilbert transform H, ∫IH(σ1I)2dw≲σ(I),∫IH(w1I)2dσ≲w(I), with constants independent of the choice of interval I. Then H(σ⋅) maps L2(σ) to L2(w), verifying a conjecture of Nazarov, Treil, and Volberg. The proof uses basic tools of nonhomogeneous analysis with two components particular to the Hilbert transform. The first component is a global-to-local reduction which is a consequence of prior work by Lacey, Sawyer, Shen, and Uriarte-Tuero. The second component, an analysis of the local part, is the particular contribution of this article.

A necessary and sufficient condition for edge universality of Wigner matrices

Abstract: In this paper, we prove a necessary and sufficient condition for the Tracy–Widom law of Wigner matrices. Consider N×N symmetric Wigner matrices H with Hij=N−1/2xij whose upper-right entries xij (1≤i

Torus knots and the rational DAHA

Abstract: Author(s): Gorsky, E; Oblomkov, A; Rasmussen, J; Shende, V | Abstract: We conjecturally extract the triply graded Khovanov-Rozansky homology of the (m;n) torus knot from the unique finite-dimensional simple representation of the rational DAHA of type A, rank n - 1, and central character m/n. The conjectural differentials of Gukov, Dunfield, and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the doubly graded Khovanov-Rozansky homologies. We match our conjecture to previous conjectures of the first author relating knot homology to q; t-Catalan numbers and to previous conjectures of the last three authors relating knot homology to Hilbert schemes on singular curves.

From symplectic measurements to the Mahler conjecture

Abstract: In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains and Mahler’s conjecture on the volume product of centrally symmetric convex bodies. More precisely, we show that if for convex bodies of fixed volume in the classical phase space the Hofer–Zehnder capacity is maximized by the Euclidean ball, then a hypercube is a minimizer for the volume product among centrally symmetric convex bodies.

Algebraic trace functions over the primes

Abstract: We study sums over primes of trace functions of l-adic sheaves. Using an extension of our earlier results on algebraic twists of modular forms to the case of Eisenstein series and bounds for Type II sums based on similar applications of the Riemann hypothesis over finite fields, we prove general estimates with power saving for such sums. We then derive various concrete applications.

Supersingular K3 surfaces for large primes

Abstract: Given a K3 surface X over a field of characteristic p, Artin conjectured that if X is supersingular (meaning infinite height), then its Picard rank is 22. Along with work of Nygaard–Ogus, this conjecture implies the Tate conjecture for K3 surfaces over finite fields with p≥5. We prove Artin’s conjecture under the additional assumption that X has a polarization of degree 2d with p>2d+4. Assuming semistable reduction for surfaces in characteristic p, we can improve the main result to K3 surfaces which admit a polarization of degree prime to p when p≥5. The argument uses Borcherds’s construction of automorphic forms on O(2,n) to construct ample divisors on the moduli space. We also establish finite-characteristic versions of the positivity of the Hodge bundle and the Kulikov–Pinkham–Persson classification of K3 degenerations. In the appendix by A. Snowden, a compatibility statement is proven between Clifford constructions and integral p-adic comparison functors.

Bounded-rank tensors are defined in bounded degree

Abstract: Matrices of rank at most k are defined by the vanishing of polynomials of degree k+1 in their entries (namely, their ((k+1)×(k+1))-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this statement for tensors of arbitrary dimension, where matrices correspond to two-dimensional tensors. More specifically, we prove that for each k there exists an upper bound d=d(k) such that tensors of border rank at most k are defined by the vanishing of polynomials of degree at most d, regardless of the dimension of the tensor and regardless of its size in each dimension. Our proof involves passing to an infinite-dimensional limit of tensor powers of a vector space, whose elements we dub infinite-dimensional tensors, and exploiting the symmetries of this limit in crucial ways.

Localization for linearly edge reinforced random walks

Abstract: We prove that the linearly edge reinforced random walk (LRRW) on any graph with bounded degrees is recurrent for sufficiently small initial weights. In contrast, we show that for nonamenable graphs the LRRW is transient for sufficiently large initial weights, thereby establishing a phase transition for the LRRW on nonamenable graphs. While we rely on the equivalence of the LRRW to a mixture of Markov chains, the proof does not use the so-called magic formula which is central to most work on this model. We also derive analogous results for the vertex reinforced jump process.

Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets

Abstract: We consider the inverse problem to determine a smooth compact Riemannian manifold with boundary (M,g) from a restriction ΛS,R of the Dirichlet-to-Neumann operator for the wave equation on the manifold. Here S and R are open sets in ∂M and the restriction ΛS,R corresponds to the case where the Dirichlet data is supported on R+×S and the Neumann data is measured on R+×R. In the novel case where S‾∩R‾=∅, we show that ΛS,R determines the manifold (M,g) uniquely, assuming that the wave equation is exactly controllable from the set of sources S. Moreover, we show that the exact controllability can be replaced by the Hassell–Tao condition for eigenvalues and eigenfunctions, that is, λj≤C‖∂νϕj‖L2(S)2,j=1,2,…, where λj are the Dirichlet eigenvalues and where (ϕj)j=1∞ is an orthonormal basis of the corresponding eigenfunctions.

A cohomological classification of vector bundles on smooth affine threefolds

Abstract: We give a cohomological classification of vector bundles of rank 2 on a smooth affine threefold over an algebraically closed field having characteristic unequal to 2. As a consequence we deduce that cancellation holds for rank 2 vector bundles on such varieties. The proofs of these results involve three main ingredients. First, we give a description of the first nonstable A1-homotopy sheaf of the symplectic group. Second, these computations can be used in concert with F. Morel’s A1-homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques (stemming from a version of the Postnikov tower in A1-homotopy theory) to reduce the classification results to cohomology vanishing statements. Third, we prove the required vanishing statements.

Poincaré–Birkhoff–Witt bases and Khovanov–Lauda–Rouquier algebras

Abstract: We generalize Lusztig’s geometric construction of the Poincare–Birkhoff–Witt (PBW) bases of finite quantum groups of type ADE under the framework of Varagnolo and Vasserot. In particular, every PBW basis of such quantum groups is proven to yield a semi-orthogonal collection in the module category of the Khovanov–Lauda–Rouquier (KLR) algebras. This enables us to prove Lusztig’s conjecture on the positivity of the canonical (lower global) bases in terms of the (lower) PBW bases. In addition, we verify Kashiwara’s problem on the finiteness of the global dimensions of the KLR algebras of type ADE.

Homological properties of finite type Khovanov-Lauda-Rouquier algebras

Abstract: We give an algebraic construction of standard modules-infinite-dimensional modules categorifying the Poincare-Birkhoff-Witt basis of the underlying quantized enveloping algebra-for Khovanov-Lauda-Rouquier algebras in all finite types. This allows us to prove in an elementary way that these algebras satisfy the homological properties of an "affine quasihereditary algebra." In simply laced types these properties were established originally by Kato via a geometric approach. We also construct some Koszul-like projective resolutions of standard modules corresponding to multiplicity-free positive roots.

Sphere packing bounds via spherical codes

Abstract: The sphere packing problem asks for the greatest density of a packing of congruent balls in Euclidean space. The current best upper bound in all suciently high dimensions is due to Kabatiansky and Levenshtein in 1978. We revisit their argument and improve their bound by a constant factor using a simple geometric argument, and we extend the argument to packings in hyperbolic space, for which it gives an exponential improvement over the previously known bounds. Additionally, we show that the Cohn-Elkies linear programming bound is always at least as strong as the Kabatiansky-Levenshtein bound; this result is analogous to Rodemich's theorem in coding theory. Finally, we develop hyperbolic linear programming bounds and prove the analogue of Rodemich's theorem there as well.

Uniform rectifiability and harmonic measure, II: Poisson kernels in Lp imply uniform rectifiability

Abstract: We present the converse to a higher-dimensional, scale-invariant version of the classical F. and M. Riesz theorem, proved by the first two authors. More precisely, for n≥2, for an Ahlfors–David regular domain Ω⊂Rn+1 which satisfies the Harnack chain condition plus an interior (but not exterior) corkscrew condition, we show that absolute continuity of the harmonic measure with respect to the surface measure on ∂Ω, with scale-invariant higher integrability of the Poisson kernel, is sufficient to imply quantitative rectifiability of ∂Ω.

Holonomy reductions of Cartan geometries and curved orbit decompositions

Abstract: We develop a holonomy reduction procedure for general Cartan geometries. We show that, given a reduction of holonomy, the underlying manifold naturally decomposes into a disjoint union of initial submanifolds. Each such submanifold corresponds to an orbit of the holonomy group on the modeling homogeneous space and carries a canonical induced Cartan geometry. The result can therefore be understood as a “curved orbit decomposition.” The theory is then applied to the study of several invariant overdetermined differential equations in projective, conformal, and CR geometry. This makes use of an equivalent description of solutions to these equations as parallel sections of a tractor bundle. In projective geometry we study a third-order differential equation that governs the existence of a compatible Einstein metric, and in conformal geometry we discuss almost-Einstein scales. Finally, we discuss analogues of the two latter equations in CR geometry, which leads to invariant equations that govern the existence of a compatible Kahler–Einstein metric.

The twisted symmetric square L -function of GL(r)

Abstract: In this paper, we consider the (partial) symmetric square L-function LS(s,π,Sym2⊗χ) of an irreducible cuspidal automorphic representation π of GLr(A) twisted by a Hecke character χ. In particular, we will show that the L-function LS(s,π,Sym2⊗χ) is holomorphic for the region Re(s)>1−1/r with the exception that, if χrω2=1, a pole might occur at s=1, where ω is the central character of π. Our method of proof is essentially a (nontrivial) modification of the one by Bump and Ginzburg in which they considered the case χ=1.

L2-theory for the ?¯-operator on compact complex spaces

Abstract: Let X be a singular Hermitian complex space of pure dimension n. We use a resolution of singularities to give a smooth representation of the L2-∂¯-cohomology of (n,q)-forms on X. The central tool is an L2-resolution for the Grauert–Riemenschneider canonical sheaf KX. As an application, we obtain a Grauert–Riemenschneider-type vanishing theorem for forms with values in almost positive line bundles. If X is a Gorenstein space with canonical singularities, then we also get an L2-representation of the flabby cohomology of the structure sheaf OX. To understand also the L2-∂¯-cohomology of (0,q)-forms on X, we introduce a new kind of canonical sheaf, namely, the canonical sheaf of square-integrable holomorphic n-forms with some (Dirichlet) boundary condition at the singular set of X. If X has only isolated singularities, then we use an L2-resolution for that sheaf and a resolution of singularities to give a smooth representation of the L2-∂¯-cohomology of (0,q)-forms.

Moduli spaces of isoperiodic forms on Riemann surfaces

Abstract: This paper describes the intrinsic geometry of a leaf A(L) of the absolute period foliation of the Hodge bundle ΩM¯g: its singular Euclidean structure, its natural foliations, and its discretized Teichmuller dynamics. We establish metric completeness of A(L) for general g and then turn to a study of the case g=2. In this case the Euclidean structure comes from a canonical meromorphic quadratic differential on A(L)≅H whose zeros, poles, and exotic trajectories are analyzed in detail.

A refinement of Rasmussen’s S-invariant

Abstract: In a previous work, we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism maps on Khovanov homology. As a consequence we obtain a refinement of Rasmussen’s slice genus bound s for each stable cohomology operation. We show that in the case of the Steenrod square Sq2 our refinement is strictly stronger than s.

Arithmeticity of certain symplectic hypergeometric groups

Abstract: We give a sufficient condition on a pair of (primitive) polynomials that the associated hypergeometric group (monodromy group of the corresponding hypergeometric differential equation) is an arithmetic subgroup of the integral symplectic group.

Polarisations et isogénies

Abstract: Soit A une variete abelienne definie sur un corps de nombres k. Nous demontrons qu’il existe un faisceau inversible ample et symetrique sur A dont le degre est borne par une constante explicite qui depend seulement de la dimension de A, de sa hauteur de Faltings et du degre du corps de nombres k. Nous etablissons egalement des versions explicites du theoreme de Bertrand relatif au theoreme de reductibilite de Poincare et des theoremes d’isogenies de Masser et Wustholz entre varietes abeliennes. Les preuves reposent sur des arguments de geometrie des nombres dans les reseaux euclidiens constitues des morphismes entre varietes abeliennes munis des metriques de Rosati. Nous majorons les minima successifs de ces reseaux grâce au theoreme des periodes que nous avons demontre dans un article precedent.

On collapsing ring blow-up solutions to the mass supercritical nonlinear Schrödinger equation

Abstract: We consider the nonlinear Schrodinger equation i∂tu+Δu+u|u|p−1=0 in dimension N≥2 and in the mass supercritical and energy subcritical range 1+4N

Représentations lisses modulo ℓ de GLm(D)

Abstract: Let F be a non-Archimedean locally compact field of residue characteristic p, let D be a finite-dimensional central division F-algebra, and let R be an algebraically closed field of characteristic different from p. We classify all smooth irreducible representations of GLm(D) for m⩾1, with coefficients in R, in terms of multisegments, generalizing works by Zelevinski, Tadic, and Vigneras. We prove that any irreducible R-representation of GLm(D) has a unique supercuspidal support and thus get two classifications: one by supercuspidal multisegments, classifying representations with a given supercuspidal support, and one by aperiodic multisegments, classifying representations with a given cuspidal support. These constructions are made in a purely local way, with a substantial use of type theory.