Showing papers in "Inventiones Mathematicae in 2009"
TL;DR: Theorem 4.1 and 5.1 of Khare and Wintenberger as mentioned in this paper have been shown to be true and they provide proofs of the theorem 4.2.
Abstract: We provide proofs of Theorems 4.1 and 5.1 of Khare and Wintenberger (Invent. Math., doi:10.1007/s00222-009-0205-7, 2009).
367 citations
TL;DR: For a nonsingular projective 3-fold X, the authors define integer invariants virtually enumerating pairs (C,D) where C is an embedded curve and D is a divisor.
Abstract: For a nonsingular projective 3-fold X, we define integer invariants virtually enumerating pairs (C,D) where C⊂X is an embedded curve and D⊂C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of X. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category.
364 citations
TL;DR: In this paper, an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) cyclotome Khovanov-Lauda algesbras in type A was constructed.
Abstract: We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) cyclotomic Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki’s categorification theorem. The Khovanov-Lauda algebras are naturally graded, which allows us to exhibit a non-trivial ℤ-grading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic.
307 citations
TL;DR: In this article, the authors considered the anisotropic Calderon problem and related inverse problems, and characterized those Riemannian manifolds which admit limiting Carleman weights, and gave a complex geometrical optics construction for a class of such manifolds.
Abstract: In this article we consider the anisotropic Calderon problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig et al. (Ann. Math. 165:567–591, 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via the attenuated geodesic ray transform. Earlier results in dimension n≥3 were restricted to real-analytic metrics.
297 citations
TL;DR: In this paper, the authors consider the problem of global in time existence and uniqueness of solutions of the 2-D infinite depth full water wave equation and show that for such data there exists a unique solution for a time period [0,T/e] for initial data of the form e − Ψ, where T depends only on Ψ.
Abstract: We consider the problem of global in time existence and uniqueness of solutions of the 2-D infinite depth full water wave equation. It is known that this equation has a solution for a time period [0,T/e] for initial data of the form e
Ψ, where T depends only on Ψ. In this paper, we show that for such data there exists a unique solution for a time period [0,e
T/e
]. This is achieved by better understandings of the nature of the nonlinearity of the full water wave equation.
254 citations
TL;DR: In this article, a series of *-representations of quantized cluster varieties were constructed using quantum dilogarithm kernels, which can be viewed as analogs of the Weil representation.
Abstract: We construct, using the quantum dilogarithm, a series of *-representations of quantized cluster varieties. This includes a construction of infinite dimensional unitary projective representations of their discrete symmetry groups - the cluster modular groups. The examples of the latter include the classical mapping class groups of punctured surfaces.
One of applications is quantization of higher Teichmuller spaces.
The constructed unitary representations can be viewed as analogs of the Weil representation. In both cases representations are given by integral operators. Their kernels in our case are the quantum dilogarithms.
We introduce the symplectic/quantum double of cluster varieties and related them to the representations.
250 citations
TL;DR: The authors showed that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general β siblings converge to Sine β, a translation invariant point process.
Abstract: We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general β siblings converge to Sine β, a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane.
223 citations
TL;DR: For convex domains in Euclidean space, Cheeger's isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori weakest requirement that Lipschnitz functions have arbitrarily slow uniform tail-decay, are all quantitatively equivalent as mentioned in this paper.
Abstract: We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori weakest requirement that Lipschitz functions have arbitrarily slow uniform tail-decay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz’ya, Cheeger, Gromov–Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are “on-average” Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the “worst” subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne–Weinberger, Li–Yau, Kannan–Lovasz–Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semi-group following Bakry–Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0,∞) curvature-dimension condition of Bakry-Emery.
220 citations
TL;DR: In this paper, the authors define a double point cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities, and prove the equivalence of this theory to the theory of algebraic cobordisms previously defined by Levine and Morel.
Abstract: We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations.
200 citations
TL;DR: In this article, a modularity lifting theorem for two-dimensional, 2-adic, potentially Barsotti-Tate representations was proved, and a classification of connected finite flat group schemes over rings of integers of finite extensions of ℚ2 was given.
Abstract: We prove a modularity lifting theorem for two dimensional, 2-adic, potentially Barsotti-Tate representations. This proves hypothesis (H) of Khare-Wintenberger, and completes the proof of Serre’s conjecture. The main new ingredient is a classification of connected finite flat group schemes over rings of integers of finite extensions of ℚ2.
192 citations
TL;DR: In this paper, the Ricci flow with surgeries was introduced to define a flow after singularities by a new approach based on a surgery procedure, which can be applied to classify all geometries that are possible for the initial manifold.
Abstract: We consider a closed smooth hypersurface immersed in euclidean space evolving by mean curvature flow. It is well known that the solution exists up to a finite singular time at which the curvature becomes unbounded. The purpose of this paper is to define a flow after singularities by a new approach based on a surgery procedure. Compared with the notions of weak solutions existing in the literature, the flow with surgeries has the advantage that it keeps track of the
changes of topology of the evolving surface and thus can be applied to classify all geometries that are possible for the initial manifold.
Our construction is inspired by the procedure originally introduced by Hamilton for the Ricci flow, and then employed by Perelman in the proof of Thurston's geometrization conjecture.
In this paper we consider initial hypersurfaces which have dimension at least three and are two-convex, that is, such that the sum of the two smallest principal curvatures is nonnegative everywhere. Under these assumptions, we construct a flow with surgeries which has uniformly bounded curvature until the evolving manifold is split in finitely many components with known topology. As a corollary, we obtain a classification up to diffeomorphism of the hypersurfaces under consideration.
TL;DR: The notion of L 2-rigidity for von Neumann algebras was introduced in this article, which can be viewed as an analogue for the vanishing of 1-cohomology into the left regular representation of a group.
Abstract: We introduce the notion of L
2-rigidity for von Neumann algebras, a generalization of property (T) which can be viewed as an analogue for the vanishing of 1-cohomology into the left regular representation of a group. We show that L
2-rigidity passes to normalizers and is satisfied by nonamenable II1 factors which are non-prime, have property Γ, or are weakly rigid. As a consequence we obtain that if M is a free product of diffuse von Neumann algebras, or if M=LΓ where Γ is a finitely generated group with β1
(2)(Γ)>0, then any nonamenable regular subfactor of M is prime and does not have properties Γ or (T). In particular this gives a new approach for showing solidity for a free group factor thus recovering a well known recent result of N. Ozawa.
TL;DR: The main result is that, to varying degrees, this recursive-theoretic condition is also sufficient for a general effective system to be realized, modulo a small extension, as the subaction of a shift of finite type or as the action of a cellular automaton on its limit set.
Abstract: We study the (sub)dynamics of multidimensional shifts of finite type and sofic shifts, and the action of cellular automata on their limit sets. Such a subaction is always an effective dynamical system: i.e. it is isomorphic to a subshift over the Cantor set the complement of which can be written as the union of a recursive sequence of basic sets. Our main result is that, to varying degrees, this recursive-theoretic condition is also sufficient. We show that the class of expansive subactions of multidimensional sofic shifts is the same as the class of expansive effective systems, and that a general effective system can be realized, modulo a small extension, as the subaction of a shift of finite type or as the action of a cellular automaton on its limit set (after removing a dynamically trivial set). As applications, we characterize, in terms of their computational properties, the numbers which can occur as the entropy of cellular automata, and construct SFTs and CAs with various interesting properties.
TL;DR: In this article, a Gysin-type exact sequence was established in which the symplectic homology SH(W) of W maps to HC(M), which in turn maps to H(M) by a map of degree -2, which then maps to W(W), and a description of the degree 2 map in terms of rational holomorphic curves with constrained asymptotic markers was given.
Abstract: A symplectic manifold W with contact type boundary M=∂W induces a linearization of the contact homology of M with corresponding linearized contact homology HC(M). We establish a Gysin-type exact sequence in which the symplectic homology SH(W) of W maps to HC(M), which in turn maps to HC(M), by a map of degree -2, which then maps to SH(W). Furthermore, we give a description of the degree -2 map in terms of rational holomorphic curves with constrained asymptotic markers, in the symplectization of M.
TL;DR: In this article, two new one-parameter families of monotonicity formulas were constructed to study the free boundary points in the lower dimensional obstacle problem and proved the uniqueness and continuous dependence of the blowups at singular points of given homogeneity.
Abstract: We construct two new one-parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem. The first one is a family of Weiss type formulas geared for points of any given homogeneity and the second one is a family of Monneau type formulas suited for the study of singular points. We show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity. This allows to prove a structural theorem for the singular set. Our approach works both for zero and smooth non-zero lower dimensional obstacles. The study in the latter case is based on a generalization of Almgren’s frequency formula, first established by Caffarelli, Salsa, and Silvestre.
TL;DR: In this paper, the authors provide applications of patching to quadratic forms and central simple algebras over function fields of curves over Henselian valued fields, and use a patching approach to reprove and generalize a recent result of Parimala and Suresh (in Preprint arXiv:0708.3128, 2007).
Abstract: This paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over Henselian valued fields. In particular, we use a patching approach to reprove and generalize a recent result of Parimala and Suresh (in Preprint arXiv:0708.3128, 2007) on the u-invariant of p-adic function fields, p≠2. The strategy relies on a local-global principle for homogeneous spaces for rational algebraic groups, combined with local computations.
TL;DR: The n-th Milnor K-group of an essentially smooth local ring over an infinite field coincides with the (n,n)-motivic cohomology of the ring as discussed by the authors.
Abstract: We prove that the n-th Milnor K-group of an essentially smooth local ring over an infinite field coincides with the (n,n)-motivic cohomology of the ring. This implies Levine’s generalized Bloch–Kato conjecture.
TL;DR: In this article, the authors examined the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out.
Abstract: We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension $d_{s}=\frac{4}{3}$
, that is, p
t
(x,x)=t
−2/3+o(1). This establishes a conjecture of Alexander and Orbach (J. Phys. Lett. (Paris) 43:625–631, 1982). En route we calculate the one-arm exponent with respect to the intrinsic distance.
TL;DR: In this paper, an invariant of a contact 3-manifold with convex boundary is described as an element of Juhasz's sutured Floer homology.
Abstract: We describe an invariant of a contact 3-manifold with convex boundary as an element of Juhasz’s sutured Floer homology. Our invariant generalizes the contact invariant in Heegaard Floer homology in the closed case, due to Ozsvath and Szabo.
TL;DR: In this article, the authors give a positive answer to von Neumann's problem of knowing whether a non-amenable countable discrete group contains a noncyclic free subgroup.
Abstract: We give a positive answer, in the measurable-group-theory context, to von Neumann’s problem of knowing whether a non-amenable countable discrete group contains a non-cyclic free subgroup. We also get an embedding result of the free-group von Neumann factor into restricted wreath product factors.
TL;DR: In this paper, the authors show that the period map of cubic fourfolds is an open embedding in the representation space Sym(3(ℂ6) with a certain algebra of meromorphic automorphic forms on a symmetric domain of orthogonal type of dimension 20.
Abstract: The period map for cubic fourfolds takes values in a locally symmetric variety of orthogonal type of dimension 20 We determine the image of this period map (thus confirming a conjecture of Hassett) and give at the same time a new proof of the theorem of Voisin that asserts that this period map is an open embedding An algebraic version of our main result is an identification of the algebra of SL (6,ℂ)-invariant polynomials on the representation space Sym 3(ℂ6)* with a certain algebra of meromorphic automorphic forms on a symmetric domain of orthogonal type of dimension 20 We also describe the stratification of the moduli space of semistable cubic fourfolds in terms of a Vinberg-Dynkin diagram
TL;DR: In this article, it was shown that a non-positively curved twin building lattice is not simple if the corresponding buildings are irreducible and not of affine type (i.e., they are not Bruhat-Tits buildings).
Abstract: Kac-Moody groups over finite fields are finitely generated groups. Most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of non-positively curved twinned buildings: those are the most important (but not the only) examples of twin building lattices. We prove that these lattices are simple if the corresponding buildings are irreducible and not of affine type (i.e. they are not Bruhat-Tits buildings). Many of them are finitely presented and enjoy property (T). Our arguments explain geometrically why simplicity fails to hold only for affine Kac-Moody groups. Moreover we prove that a nontrivial continuous homomorphism from a completed Kac-Moody group is always proper. We also show that Kac-Moody lattices fulfill conditions implying strong superrigidity properties for isometric actions on non-positively curved metric spaces. Most results apply to the general class of twin building lattices.
TL;DR: In this article, the authors characterize all linear operators on finite or infinite-dimensional spaces of multivariate polynomials preserving the property of being non-vanishing whenever the variables are in prescribed open circular domains.
Abstract: In 1952 Lee and Yang proposed the program of analyzing phase transitions in terms of zeros of partition functions. Linear operators preserving non-vanishing properties are essential in this program and various contexts in complex analysis, probability theory, combinatorics, and matrix theory. We characterize all linear operators on finite or infinite-dimensional spaces of multivariate polynomials preserving the property of being non-vanishing whenever the variables are in prescribed open circular domains. In particular, this solves the higher dimensional counterpart of a long-standing classification problem originating from classical works of Hermite, Laguerre, Hurwitz and Polya-Schur on univariate polynomials with such properties.
TL;DR: In this paper, the authors studied the microscopic convexity property of fully nonlinear elliptic and parabolic partial differential equations and established that the rank of Hessian ∇cffff 2 istg u is of constant rank for any convex solution u of equation F(∇¯¯¯¯ 2¯¯ u,∇ u, u,u,x)=0.
Abstract: We study microscopic convexity property of fully nonlinear elliptic and parabolic partial differential equations. Under certain general structure condition, we establish that the rank of Hessian ∇
2
u is of constant rank for any convex solution u of equation F(∇
2
u,∇
u,u,x)=0. The similar result is also proved for parabolic equations. Some of geometric applications are also discussed.
TL;DR: In this paper, the Faltings height pairing of arithmetic special divisors and CM cycles on Shimura varieties associated to orthogonal groups was studied and a conjecture relating the total pairing to the central derivative of a Rankin L-function was derived.
Abstract: We study the Faltings height pairing of arithmetic special divisors and CM cycles on Shimura varieties associated to orthogonal groups. We compute the Archimedean contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a Rankin L-function. We prove the conjecture in certain cases where the Shimura variety has dimension 0, 1, or 2. In particular, we obtain a new proof of the Gross-Zagier formula.
TL;DR: In this paper, the Pieri rules for the classical cohomology and small quantum cohoms of Grassmannian varieties were established, with integer coefficients, in terms of special Schubert class generators and relations.
Abstract: We study the three point genus zero Gromov-Witten invariants on the Grassmannians which parametrize non-maximal isotropic subspaces in a vector space equipped with a nondegenerate symmetric or skew-symmetric form We establish Pieri rules for the classical cohomology and the small quantum cohomology ring of these varieties, which give a combinatorial formula for the product of any Schubert class with certain special Schubert classes We also give presentations of these rings, with integer coefficients, in terms of special Schubert class generators and relations
TL;DR: In this article, the existence of a semi-conjugacy to an irrational rotation for conservative toral homeomorphisms of the two-torus has been shown to be true for all toral homomorphisms with all points non-wandering.
Abstract: We give an equivalent condition for the existence of a semi-conjugacy to an irrational rotation for conservative homeomorphisms of the two-torus. This leads to an analogue of Poincare’s classification of circle homeomorphisms for conservative toral homeomorphisms with unique rotation vector and a certain bounded mean motion property. For minimal toral homeomorphisms, the result extends to arbitrary dimensions. Further, we provide a basic classification for the dynamics of toral homeomorphisms with all points non-wandering.
TL;DR: In this article, a functorial normal crossing compactification of the moduli space of smooth cubic surfaces was proposed, analogous to the Grothendieck-Knudsen compactification.
Abstract: We give a functorial normal crossing compactification of the moduli space of smooth cubic surfaces entirely analogous to the Grothendieck-Knudsen compactification \(M_{0,n}\subset\overline{M}_{0,n}\) .
TL;DR: In this paper, it was shown that the domain of outer communication of a smooth, regular, stationary, four dimensional, vacuum black hole solution is locally isometric to the outer communication domain of a Kerr spacetime.
Abstract: A fundamental conjecture in general relativity asserts that the domain of outer communication of a regular, stationary, four dimensional, vacuum black hole solution is isometrically diffeomorphic to the domain of outer communication of a Kerr black hole. So far the conjecture has been resolved, by combining results of Hawking [17], Carter [4] and Robinson [28], under the additional hypothesis of non-degenerate horizons and real analyticity of the space-time. We develop a new strategy to bypass analyticity based on a tensorial characterization of the Kerr solutions, due to Mars [24], and new geometric Carleman estimates. We prove, under a technical assumption (an identity relating the Ernst potential and the Killing scalar) on the bifurcate sphere of the event horizon, that the domain of outer communication of a smooth, regular, stationary Einstein vacuum spacetime of dimension 4 is locally isometric to the domain of outer communication of a Kerr spacetime.