Showing papers in "Duke Mathematical Journal in 2016"
TL;DR: In this paper, the authors study the asymmetric six-vertex model in the quadrant with parameters on the stochastic line and show that the random height function converges to an explicit deterministic limit shape as the mesh size tends to 0.
Abstract: We study the asymmetric six-vertex model in the quadrant with parameters on the stochastic line. We show that the random height function of the model converges to an explicit deterministic limit shape as the mesh size tends to 0. We further prove that the one-point fluctuations around the limit shape are asymptotically governed by the GUE Tracy–Widom distribution. We also explain an equivalent formulation of our model as an interacting particle system, which can be viewed as a discrete time generalization of ASEP started from the step initial condition. Our results confirm a 1992 prediction of Gwa and Spohn that this system belongs to the KPZ universality class.
143 citations
TL;DR: Gamma conjectures for Fano manifolds have been proposed in this paper, which can be thought of as a square root of the index theorem, and they can be seen as a refinement of Dubrovin's Gamma conjecture II.
Abstract: We propose Gamma conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class $A_{F}$ to a Fano manifold $F$. We say that $F$ satisfies Gamma conjecture I if $A_{F}$ equals the Gamma class $\widehat{\Gamma}_{F}$. When the quantum cohomology of $F$ is semisimple, we say that $F$ satisfies Gamma conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by $\widehat{\Gamma}_{F}\operatorname{Ch}(E_{i})$ for an exceptional collection $\{E_{i}\}$ in the derived category of coherent sheaves $\mathcal{D}^{b}_{\mathrm{coh}}(F)$. Gamma conjecture II refines a part of a conjecture by Dubrovin. We prove Gamma conjectures for projective spaces and Grassmannians.
124 citations
TL;DR: In this article, the authors studied fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order 2s, with s ∈ (0, 1).
Abstract: We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order 2s, with s ∈ (0,1). We consider the class of nonlocal operators L∗ ⊂L 0, which consists of all the infinitesimal generators of stable Levy processes belonging to the class L0 of Caffarelli-Silvestre. For fully nonlinear operators I elliptic with respec tt oL∗ ,w e prove that solutions to Iu = f in Ω, u = 0 in R n \ Ω, satisfy u/d s ∈ C s−ϵ (Ω) for all ϵ> 0, where d is the distance to ∂ Ωa ndf ∈ L ∞ . We expect the Holder exponent s − ϵ to be optimal (or almost optimal) for general right hand sides f ∈ L ∞ . Moreover, we also expect the class L∗ to be the largest scale invariant subclass of L0 for which this result is true. In this direction, we show that the class L0 is too large for all solutions to behave like d s . The constants in all the estimates in this paper remain bounded as th eo rder of the equation approaches 2.
120 citations
TL;DR: In this article, the scaling limit of diffusion-limited aggregation (DLA) in the plane has been studied, and a scaling limit candidate called quantum Loewner evolution (QLE(γ2,η) has been proposed.
Abstract: What is the scaling limit of diffusion-limited aggregation (DLA) in the plane? This is an old and famously difficult question. One can generalize the question in two ways: first, one may consider the dielectric breakdown model η-DBM, a generalization of DLA in which particle locations are sampled from the ηth power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider η-DBM on random graphs known or believed to converge in law to a Liouville quantum gravity (LQG) surface with parameter γ∈[0,2]. In this generality, we propose a scaling limit candidate called quantum Loewner evolution, QLE(γ2,η). QLE is defined in terms of the radial Loewner equation like radial stochastic Loewner evolution, except that it is driven by a measure-valued diffusion νt derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of νt using a stochastic partial differential equation. For each γ∈(0,2], there are two or three special values of η for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of νt. We also explain discrete versions of our construction that relate DLA to loop-erased random walks and the Eden model to percolation. A certain “reshuffling” trick (in which concentric annular regions are rotated randomly, like slot-machine reels) facilitates explicit calculation. We propose QLE(2,1) as a scaling limit for DLA on a random spanning-tree-decorated planar map and QLE(8/3,0) as a scaling limit for the Eden model on a random triangulation. We propose using QLE(8/3,0) to endow pure LQG with a distance function, by interpreting the region explored by a branching variant of QLE(8/3,0), up to a fixed time, as a metric ball in a random metric space.
101 citations
TL;DR: In this paper, the existence of convex global rotating solutions for the generalized surface quasigeostrophic equation was shown to exist for all ε 2 (0, 2).
Abstract: Motivated by the recent work of Hassainia and Hmidi [Z. Hassainia, T. Hmidi - On the Vstates for the generalized quasi-geostrophic equations,arXiv preprint arXiv:1405.0858], we close the question of the existence of convex global rotating solutions for the generalized surface quasigeostrophic equation for � 2 [1,2). We also show C 1 regularity of their boundary for all � 2 (0,2).
85 citations
TL;DR: In this paper, it was shown that for a quasiprojective variety X with only Kawamata log terminal singularities, there exists a Galois cover Y→X, ramified only over the singularities of X, such that the etale fundamental groups of Y and of Yreg agree.
Abstract: Given a quasiprojective variety X with only Kawamata log terminal singularities, we study the obstructions to extending finite etale covers from the smooth locus Xreg of X to X itself. A simplified version of our main results states that there exists a Galois cover Y→X, ramified only over the singularities of X, such that the etale fundamental groups of Y and of Yreg agree. In particular, every etale cover of Yreg extends to an etale cover of Y. As a first major application, we show that every flat holomorphic bundle defined on Yreg extends to a flat bundle that is defined on all of Y. As a consequence, we generalize a classical result of Yau to the singular case: every variety with at worst terminal singularities and with vanishing first and second Chern class is a finite quotient of an abelian variety. As a further application, we verify a conjecture of Nakayama and Zhang describing the structure of varieties that admit polarized endomorphisms.
83 citations
TL;DR: In this paper, it was shown that the defocusing, cubic nonlinear Schrodinger initial value problem is globally well posed and scattering for u 0 ∈ L 2 R 2.
Abstract: In this article we prove that the defocusing, cubic nonlinear Schrodinger initial value problem is globally well posed and scattering for u0∈L2(R2). The proof uses the bilinear estimates of Planchon and Vega and a frequency-localized interaction Morawetz estimate similar to the high-frequency estimate of Colliander, Keel, Staffilani, Takaoka, and Tao and especially the low-frequency estimate of Dodson.
73 citations
TL;DR: In this article, growing modes of the linearized Navier-Stokes equations about generic stationary shear flows of the boundary layer type in a regime of a sufficiently large Reynolds number were constructed.
Abstract: In this paper, we construct growing modes of the linearized Navier–Stokes equations about generic stationary shear flows of the boundary layer type in a regime of a sufficiently large Reynolds number: R→∞. Notably, the shear profiles are allowed to be linearly stable at the infinite Reynolds number limit, and so the instability presented is purely due to the presence of viscosity. The formal construction of approximate modes is well documented in physics literature, going back to the work of Heisenberg, C. C. Lin, Tollmien, Drazin, and Reid, but a rigorous construction requires delicate mathematical details, involving, for instance, a treatment of primitive Airy functions and singular solutions. Our analysis gives exact unstable eigenvalues and eigenfunctions, showing that the solution could grow slowly at the rate of et/R. The proof follows the general iterative approach introduced in our companion paper, avoiding having to deal with matching inner and outer asymptotic expansions, but instead involving a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators. Unlike in the channel flows, the spatial domain in the boundary layers is unbounded and the iterative scheme is likely to diverge due to the linear growth in the vertical variable. We introduce a new iterative scheme to simultaneously treat the singularity near critical layers and the asymptotic behavior of solutions at infinity. The instability of generic boundary layers obtained in this paper is linked to the emergence of Tollmien–Schlichting waves in describing the early stage of the transition from laminar to turbulent flows.
72 citations
TL;DR: In this paper, it was shown that an irreducible highest weight representation of a nontwisted affine Kac-Moody algebra at an admissible level k is a module over the associated simple affine vertex algebra if and only if it is an affine representation whose integral root system is isomorphic to that of the vertex algebra itself.
Abstract: We study the vertex algebras associated with modular invariant representations of affine Kac–Moody algebras at fractional levels, whose simple highest weight modules are classified by Joseph’s characteristic varieties. We show that an irreducible highest weight representation of a nontwisted affine Kac–Moody algebra at an admissible level k is a module over the associated simple affine vertex algebra if and only if it is an admissible representation whose integral root system is isomorphic to that of the vertex algebra itself. This in particular proves the conjecture of Adamovic and Milas on the rationality of admissible affine vertex algebras in the category O.
69 citations
TL;DR: In this article, the authors construct time quasiperiodic solutions to the energy supercritical nonlinear Schrodinger equations on the torus in arbitrary dimensions, which could have general applicability.
Abstract: We construct time quasiperiodic solutions to the energy supercritical nonlinear Schrodinger equations on the torus in arbitrary dimensions. This introduces a new approach, which could have general applicability.
68 citations
TL;DR: In this paper, a tropicalization functor is defined to send closed subschemes of a toric variety over a ring R with non-Archimedean valuation to closed tropical toric varieties.
Abstract: We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as T=(R∪{−∞},max,+) by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-Archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of T-points this reduces to Kajiwara–Payne’s extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of T-schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.
TL;DR: In this article, a cycle-theoretic avatar of a refined Artin conductor in ramification theory was constructed for higher-dimensional class field theory over finite fields, which describes the abelian fundamental group of U by Chow groups of moduli.
Abstract: One of the main results of this article is a proof of the rank-one case of an existence conjecture on lisse (Q) over bar (l)-sheaves on a smooth variety U over a finite field due to Deligne and Drinfeld. The problem is translated into the language of higher-dimensional class field theory over finite fields, which describes the abelian fundamental group of U by Chow groups of 0-cycles with moduli. A key ingredient is the construction of a cycle-theoretic avatar of a refined Artin conductor in ramification theory originally studied by Kazuya Kato.
TL;DR: In this article, the authors acknowledge 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.
Abstract: The first author was supported by NSF grant DMS-1361701. The second 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 third author was supported by the Alfred P. Sloan
Fellowship, the NSF CAREER Award DMS 1056004, the NSF INSPIRE Award DMS 1344235, and
the NSF Materials Research Science and Engineering Center Seed Grant DMR 0212302.
TL;DR: In this paper, a desingularization theorem for Hitchin's equations was proved for elements in the ends of the Higgs bundle moduli space and identified a dense open subset of the boundary of the compactification of this modulus space.
Abstract: We associate to each stable Higgs pair (A(0), Phi(0)) on a compact Riemann surface X a singular limiting configuration (A(infinity), Phi(infinity)), assuming that det Phi has only simple zeroes. We then prove. a desingularization theorem by constructing a family of solutions (A(t), t Phi(t) to Hitchin's equations, which converge to this limiting configuration as t -> infinity. This provides a new proof via gluing methods, for elements in the ends of the Higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space.
TL;DR: In this paper, the existence of K-polystable Q-Gorenstein smoothable Q-Fano varieties was proved in the Gromov-Hausdorff sense.
Abstract: In this article we prove the existence of Kahler–Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties, and we show how these metrics behave, in the Gromov–Hausdorff sense, under Q-Gorenstein smoothings.
TL;DR: In this paper, the functor of noncommutative deformations of irreducible rational curves in a projective 33-fold is shown to be representable, and every such curve is associated with AconAcon.
Abstract: We prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 33-fold is representable, and hence, we associate to every such curve a noncommutative deformation algebra AconAcon. This new invariant extends and unifies known invariants for flopping curves in 33-folds, such as the width of Reid and the bidegree of the normal bundle. It also applies in the settings of flips and singular schemes. We show that the noncommutative deformation algebra AconAcon is finite-dimensional, and give a new way of obtaining the commutative deformations of the curve, allowing us to make explicit calculations of these deformations for certain (−3,1)(−3,1)-curves.
We then show how our new invariant AconAcon also controls the homological algebra of flops. For any flopping curve in a projective 33-fold with only Gorenstein terminal singularities, we construct an autoequivalence of the derived category of the 33-fold by twisting around a universal family over the noncommutative deformation algebra AconAcon, and prove that this autoequivalence is an inverse of Bridgeland’s flop-flop functor. This demonstrates that it is strictly necessary to consider noncommutative deformations of curves in order to understand the derived autoequivalences of a 33-fold and, thus, the Bridgeland stability manifold.
TL;DR: In this paper, the authors give an explicit uniform bound on the number of geometric torsion points of a Jacobian lying on the image of a curve of genus g ≥ 2 over a number field F of degree d=[F:Q] under an Abel-Jacobi map.
Abstract: Let X be a curve of genus g≥2 over a number field F of degree d=[F:Q]. The conjectural existence of a uniform bound N(g,d) on the number #X(F) of F-rational points of X is an outstanding open problem in arithmetic geometry, known by the work of Caporaso, Harris, and Mazur to follow from the Bombieri–Lang conjecture. A related conjecture posits the existence of a uniform bound Ntors,†(g,d) on the number of geometric torsion points of the Jacobian J of X which lie on the image of X under an Abel–Jacobi map. For fixed X, the finiteness of this quantity is the Manin–Mumford conjecture, which was proved by Raynaud. We give an explicit uniform bound on #X(F) when X has Mordell–Weil rank r≤g−3. This generalizes recent work of Stoll on uniform bounds for hyperelliptic curves of small rank to arbitrary curves. Using the same techniques, we give an explicit, unconditional uniform bound on the number of F-rational torsion points of J lying on the image of X under an Abel–Jacobi map. We also give an explicit uniform bound on the number of geometric torsion points of J lying on X when the reduction type of X is highly degenerate. Our methods combine Chabauty–Coleman’s p-adic integration, non-Archimedean potential theory on Berkovich curves, and the theory of linear systems and divisors on metric graphs.
TL;DR: In this paper, it was shown that the set of singular vectors in Rd, d≥2, has Hausdorff dimension d2d+1 plus a power of e between d2 and d.
Abstract: We prove that the set of singular vectors in Rd, d≥2, has Hausdorff dimension d2d+1 and that the Hausdorff dimension of the set of e-Dirichlet improvable vectors in Rd is roughly d2d+1 plus a power of e between d2 and d. As a corollary, the set of divergent trajectories of the flow by diag(et,…,et,e−dt) acting on SLd+1(R)/SLd+1(Z) has Hausdorff codimension dd+1. These results extend the work of the first author.
TL;DR: In this article, the p-adic eigencurve is smooth at classical weight 1 points which are regular at p and give a precise criterion for etaleness over the weight space at those points.
Abstract: We show that the p-adic eigencurve is smooth at classical weight 1 points which are regular at p and give a precise criterion for etaleness over the weight space at those points. Our approach uses deformations of Galois representations.
TL;DR: In this article, the authors generalize Bogomolov's inequality for Higgs sheaves to the logarithmic case and show some examples of smooth nonconnected curves on smooth rational surfaces that cannot be lifted modulo p2.
Abstract: We generalize Bogomolov’s inequality for Higgs sheaves and the Bogomolov– Miyaoka–Yau inequality in positive characteristic to the logarithmic case. We also generalize Shepherd-Barron’s results on Bogomolov’s inequality on surfaces of special type from rank 2 to the higher-rank case. We use these results to show some examples of smooth nonconnected curves on smooth rational surfaces that cannot be lifted modulo p2. These examples contradict some claims by Xie.
TL;DR: In this paper, it was shown that F vanishes on at most O(n(11/6) points of the Cartesian product A X B X C unless F has a special group-related form.
Abstract: Let F 2 C[x; y; z] be a constant-degree polynomial, and let A; B; C subset of C be finite sets of size n. We show that F vanishes on at most O(n(11/6))points of the Cartesian product A X B X C, unless F has a special group-related form. This improves a theorem of Elekes and Szab and generalizes a result of Raz, Sharir, and Solymosi. The same statement holds over R, and a similar statement holds when A; B; C have different sizes (with a more involved bound replacing O(n(11/6)). This result provides a unified tool for improving bounds in various Erdos-type problems in combinatorial geometry, and we discuss several applications of this kind.
TL;DR: In this article, it was shown that the scenario of formation of singularities discovered by Castro, Cordoba, Fefferman, Gancedo, and Gomez-Serrano in the case of the water waves system, in which the interface remains locally smooth but self-intersects in finite time, is completely prevented for two-fluid interfaces with positive densities.
Abstract: We show that so-called splash singularities cannot develop in the case of locally smooth solutions of the two-fluid interfaces in two dimensions. More precisely, we show that the scenario of formation of singularities discovered by Castro, Cordoba, Fefferman, Gancedo, and Gomez-Serrano in the case of the water waves system, in which the interface remains locally smooth but self-intersects in finite time, is completely prevented in the case of two-fluid interfaces with positive densities.
TL;DR: In this article, the Ax-Schanuel theorem for the exponential function has been proved for the j-function, and it is shown that atypical algebraic relations among functions and their compositions with the j function are governed by modular relations.
Abstract: In this paper we prove a functional transcendence statement for the j-function which is an analogue of the Ax–Schanuel theorem for the exponential function. It asserts, roughly, that atypical algebraic relations among functions and their compositions with the j-function are governed by modular relations.
TL;DR: In this paper, it was shown that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three-circle theorem and two sharp monotonicity formulae were derived as corollaries.
Abstract: The classical Hadamard three-circle theorem is generalized to complete Kahler manifolds. More precisely, we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three-circle theorem. Two sharp monotonicity formulae are derived as corollaries. Among applications, we obtain sharp dimension estimates (with rigidity) of holomorphic functions with polynomial growth when the holomorphic sectional curvature is nonnegative. When the bisectional curvature is nonnegative, the sharp dimension estimate was due to Ni.
TL;DR: In this article, the vanishing of the geometric Bloch-Kato Selmer group for the adjoint representation of a Galois representation associated to regular algebraic polarized cuspidal automorphic representations under an assumption on the residual image was proved.
Abstract: We prove the vanishing of the geometric Bloch–Kato Selmer group for the adjoint representation of a Galois representation associated to regular algebraic polarized cuspidal automorphic representations under an assumption on the residual image. Using this, we deduce that the localization and completion of a certain universal deformation ring for the residual representation at the characteristic zero point induced from the automorphic representation is formally smooth of the correct dimension. We do this by employing the Taylor–Wiles–Kisin patching method together with Kisin’s technique of analyzing the generic fiber of universal deformation rings. Along the way we give a characterization of smooth closed points on the generic fiber of Kisin’s potentially semistable local deformation rings in terms of their Weil–Deligne representations.
TL;DR: In this paper, a formalism of mixed modular perverse sheaves for varieties equipped with a stratification by affine spaces is developed, based on the theory of parity sheaves due to Juteau, Mautner and Williamson.
Abstract: Building on the theory of parity sheaves due to Juteau, Mautner, and Williamson, we develop a formalism of “mixed modular perverse sheaves” for varieties equipped with a stratification by affine spaces. We then give two applications: (1) a “Koszul-type” derived equivalence relating a given flag variety to the Langlands dual flag variety and (2) a formality theorem for the modular derived category of a flag variety (extending a previous result of Riche, Soergel, and Williamson).
TL;DR: In this article, it was shown that if a train track map has a word-hyperbolic fundamental group, then it can act freely and cocompactly on a CAT(0) cube complex.
Abstract: Let V be a finite graph, and let ϕ:V→V be an irreducible train track map whose mapping torus has word-hyperbolic fundamental group G. Then G acts freely and cocompactly on a CAT(0) cube complex. Hence, if F is a finite-rank free group and if Φ:F→F is an irreducible monomorphism so that G=F∗Φ is word-hyperbolic, then G acts freely and cocompactly on a CAT(0) cube complex. This holds, in particular, if Φ is an irreducible automorphism with G=F⋊ΦZ word-hyperbolic.
TL;DR: In this paper, the authors conjecture that there is a basis for the second homology of a Bianchi manifold, where each basis element is represented by a surface of low genus.
Abstract: Let M be an arithmetic hyperbolic 3-manifold, such as a Bianchi manifold. We conjecture that there is a basis for the second homology of M, where each basis element is represented by a surface of “low” genus, and we give evidence for this. We explain the relationship between this conjecture and the study of torsion homology growth.
TL;DR: In this article, a deformation of Plancherel measure linked to polynomials is considered and the first and second-order asymptotics of the bulk of a random Young diagram under this distribution are described.
Abstract: In this paper, we consider a deformation of Plancherel measure linked to Jack polynomials. Our main result is the description of the first- and second-order asymptotics of the bulk of a random Young diagram under this distribution, which extends celebrated results of Vershik, Kerov, Logan, and Shepp (for the first-order asymptotics) and Kerov (for the second-order asymptotics). This gives more evidence for the connection with the Gaussian $\beta$-ensemble, already suggested by a work of Matsumoto.
Our main tool is a polynomiality result for the structure constants of some quantities that we call Jack characters, recently introduced by Lassalle. We believe that this result is also interesting in itself and we give several other applications of it.
TL;DR: In this paper it was shown that the Grothendieck-Teichm\"uller group can be seen as a properad governing Lie bialgebras by automorphisms, and that there is an associated homotopy Batalin-Vilkovisky algebra structure on the associated Chevalley-Eilenberg complex.
Abstract: We show Koszulness of the prop governing involutive Lie bialgebras and also of the props governing non-unital and unital-counital Frobenius algebras, solving a long-standing problem. This gives us minimal models for their deformation complexes, and for deformation complexes of their algebras which are discussed in detail. Using an operad of graph complexes we prove, with the help of an earlier result of one of the authors, that there is a highly non-trivial action of the Grothendieck-Teichm\"uller group $GRT_1$ on (completed versions of) the minimal models of the properads governing Lie bialgebras and involutive Lie bialgebras by automorphisms. As a corollary one obtains a large class of universal deformations of any (involutive) Lie bialgebra and any Frobenius algebra, parameterized by elements of the Grothendieck-Teichm\"uller Lie algebra.We also prove that, for any given homotopy involutive Lie bialgebra structure in a vector space, there is an associated homotopy Batalin-Vilkovisky algebra structure on the associated Chevalley-Eilenberg complex.