Showing papers in "Annals of Mathematics in 2013"
TL;DR: In this article, the authors introduce a new solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution, providing a pathwise notion of a solution, together with a very detailed approximation theory.
Abstract: We introduce a new concept of solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution. This notion provides a factorisation of the Cole-Hopf solution map into a \universal" measurable map from the probability space into an explicitly described auxiliary metric space, composed with a new solution map that has very good continuity properties. The advantage of such a formulation is that it essentially provides a pathwise notion of a solution, together with a very detailed approximation theory. In particular, our construction completely bypasses the Cole-Hopf transform, thus laying the groundwork for proving that the KPZ equation describes the uctuations of systems in the KPZ universality class. As a corollary of our construction, we obtain very detailed new regularity results about the solution, as well as its derivative with respect to the initial condition. Other byproducts of the proof include an explicit approximation to the stationary solution of the KPZ equation, a well-posedness result for the Fokker-Planck equation associated to a particle diusing in a rough space-time dependent potential, and a new periodic homogenisation result for the heat equation with a space-time periodic potential. One ingredient in our construction is an example of a non-Gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge.
633 citations
TL;DR: In this paper, a family of moduli spaces, a virtual cycle, and a corresponding cohomological eld theory associated to the singularity are described for any nondegenerate, quasi-homogeneous hypersurface singularity.
Abstract: For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological eld theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity Ar 1. We also resolve two outstanding conjectures of Witten. The rst conjecture is that ADE-singularities are self-dual, and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.
372 citations
TL;DR: In this article, it was shown that topological full groups of minimal systems are amenable to piecewise-translations of the integers, and that these groups are obtained by constructing a suitable family of densities on the classical Bernoulli space.
Abstract: We provide the rst examples of nitely generated simple groups that are amenable (and innite). To this end, we prove that topological full groups of minimal systems are amenable. This follows from a general existence result on invariant states for piecewise-translations of the integers. The states are obtained by constructing a suitable family of densities on the classical Bernoulli space.
204 citations
TL;DR: In this paper, it was shown that the support of a random measure on the unit ball of a separable Hilbert space that satises the Ghirlanda-Guerra identities must be ultrametric with probability one.
Abstract: In this paper we prove that the support of a random measure on the unit ball of a separable Hilbert space that satises the Ghirlanda-Guerra identities must be ultrametric with probability one. This implies the Parisi ultrametricity conjecture in mean-eld spin glass models, such as the Sherrington-Kirkpatrick and mixed p-spin models, for which Gibbs measures are known to satisfy the Ghirlanda-Guerra identities in the thermodynamic limit.
173 citations
TL;DR: In this paper, the authors complete the parametrisation of all p-blocks of nite quasi-simple groups by nding the so-called quasi-isolated blocks of exceptional groups of Lie type for bad primes.
Abstract: This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of nite quasi-simple groups by nding the so-called quasi-isolated blocks of exceptional groups of Lie type for bad primes. This relies on the explicit decomposition of Lusztig induction from suitable Levi subgroups. Our second major result is the proof of one direction of Brauer’s long-standing height zero conjecture on blocks of nite groups, using the
137 citations
TL;DR: In this article, it was shown that isoparametric hypersurfaces with (g,m) = (6, 2) are homogeneous, which answers Dorfmeister-Neher's conjecture armatively and solves Yau's problem.
Abstract: We prove that isoparametric hypersurfaces with (g;m) = (6; 2) are homogeneous, which answers Dorfmeister-Neher’s conjecture armatively and solves Yau’s problem in the case g = 6.
133 citations
TL;DR: In this paper, the Ruelle and Selberg zeta functions for C r Anosov ows, r > 2, on a compact smooth manifold were studied and the spectral properties of a transfer operator acting on suitable Banach spaces of anisotropic currents were analyzed.
Abstract: We study the Ruelle and Selberg zeta functions for C r Anosov ows, r > 2, on a compact smooth manifold. We prove several results, the most remarkable being (a) for C 1 ows the zeta function is meromorphic on the entire complex plane; (b) for contact ows satisfying a bunching condition (e.g., geodesic ows on manifolds of negative curvature better than 1 9 -pinched), the zeta function has a pole at the topological entropy and is analytic in a strip to its left; (c) under the same hypotheses as in (b) we obtain sharp results on the number of periodic orbits. Our arguments are based on the study of the spectral properties of a transfer operator acting on suitable Banach spaces of anisotropic currents.
124 citations
TL;DR: In this article, the authors prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem) for which the smoothness of the interface breaks down in nite time into a splash singularity or a splat singularity.
Abstract: In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem) for which the smoothness of the interface breaks down in nite time into a splash singularity or a splat singularity.
124 citations
TL;DR: In this article, the authors describe quantum enveloping algebras of symmetric Kac-Moody Lie algesbras via a finite eld Hall algebra construction involving Z2-graded complexes of quiver representations.
Abstract: We describe quantum enveloping algebras of symmetric Kac-Moody Lie algebras via a nite eld Hall algebra construction involving Z2-graded complexes of quiver representations.
119 citations
TL;DR: In this article, it was shown that various GIT semistabilities of polarized varieties imply semi-log-canonicity, i.e., the semistability of polarized GITs implies semidefinite Canonicity.
Abstract: We prove that various GIT semistabilities of polarized varieties imply semi-log-canonicity.
117 citations
TL;DR: In this article, the conjecture of Korevaar and Meyers that for each N cdt d, there exists a spherical t-design in the sphere S d consisting of n points, where cd is a constant depending only on d.
Abstract: In this paper we prove the conjecture of Korevaar and Meyers: for each N cdt d , there exists a spherical t-design in the sphere S d consisting of N points, where cd is a constant depending only on d.
TL;DR: In this article, it was shown that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra LG completely "remembers" the group G.
Abstract: We prove that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra LG completely \remembers" the group G. More precisely, if LG is isomorphic to the von Neumann algebra L of an arbitrary countable group , then must be isomorphic to G. This represents the rst superrigidity result pertaining to group von Neumann algebras.
TL;DR: In this article, the Moser-Trudinger inequalities on the CR sphere were derived in the Adams form, for powers of the sublaplacian and for general spectrally dened operators on the space of CR-pluriharmonic functions.
Abstract: We derive sharp Moser-Trudinger inequalities on the CR sphere. The rst type is in the Adams form, for powers of the sublaplacian and for general spectrally dened operators on the space of CR-pluriharmonic functions. We will then obtain the sharp Beckner-Onofri inequality for CRpluriharmonic functions on the sphere and, as a consequence, a sharp logarithmic Hardy-Littlewood-Sobolev inequality in the form given by Carlen and Loss.
TL;DR: Angel, Benjamini, and Schramm as discussed by the authors showed that any distributional limit of planar graphs with an exponential tail is almost surely recurrent, and showed that the probability that the simple random walk started at a uniform random vertex avoids its initial location for T steps is at most C logT.
Abstract: We prove that any distributional limit of nite planar graphs in which the degree of the root has an exponential tail is almost surely recurrent. As a corollary, we obtain that the uniform innite planar triangulation and quadrangulation (UIPT and UIPQ) are almost surely recurrent, resolving a conjecture of Angel, Benjamini and Schramm. We also settle another related problem of Benjamini and Schramm. We show that in any bounded degree, nite planar graph the probability that the simple random walk started at a uniform random vertex avoids its initial location for T steps is at most C logT .
TL;DR: In this article, a noncommutative resolution of the nilpotent cone which is derived equivalent to the Springer resolution has been derived, which is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and Rumynin.
Abstract: We prove most of Lusztig’s conjectures on the canonical basis in homology of a Springer ber. The conjectures predict that this basis controls numerics of representations of the Lie algebra of a semisimple algebraic group over an algebraically closed eld of positive characteristic. We check this for almost all characteristics. To this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the Springer resolution. On the one hand, this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and Rumynin. On the other hand, it is compatible with the t-structure arising from an equivalence with the derived category of perverse sheaves on the ane ag variety of the Langlands dual group. This equivalence established by Arkhipov and the rst author ts the framework of local geometric Langlands duality. The latter compatibility allows one to apply Frobenius purity theorem to deduce the desired properties of the basis. We expect the noncommutative counterpart of the Springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric Langlands duality.
TL;DR: In this article, the authors studied the distribution of the nodal length of random eigenfunctions for large eigenvalues, and their primary result is that the asymptotics for the variance is nonuniversal.
Abstract: Using the spectral multiplicities of the standard torus, we endow the Laplace eigenspaces with Gaussian probability measures. This induces a notion of random Gaussian Laplace eigenfunctions on the torus (\arithmetic random waves"). We study the distribution of the nodal length of random eigenfunctions for large eigenvalues, and our primary result is that the asymptotics for the variance is nonuniversal. Our result is intimately related to the arithmetic of lattice points lying on a circle with radius corresponding to the energy.
TL;DR: In this paper, it was shown that the number of birational automorphisms of a variety of general type X is bounded by c vol(X;KX), where c is a constant that only depends on the dimension of X.
Abstract: We show that the number of birational automorphisms of a variety of general type X is bounded by c vol(X;KX), where c is a constant that only depends on the dimension of X.
TL;DR: In this paper, the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky cluster algebras and their categorization via triangulated categories was proved.
Abstract: We prove the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky’s cluster algebras and their (additive) categorication via triangulated categories.
TL;DR: In this article, the authors determine the lens spaces that arise by integer Dehn surgery along a knot in the three-sphere, using tools from Floer homology and lattice theory.
Abstract: We determine the lens spaces that arise by integer Dehn surgery along a knot in the three-sphere. Specically, if surgery along a knot produces a lens space, then there exists an equivalent surgery along a Berge knot with the same knot Floer homology groups. This leads to sharp information about the genus of such a knot. The arguments rely on tools from Floer homology and lattice theory. They are primarily combinatorial in nature.
TL;DR: In this paper, an explicit family of automorphic forms of Kloosterman sums for general reductive groups is presented. But the automorphism is not used to describe the local and global monodromy of these sheaves.
Abstract: Deligne constructed a remarkable local system on P 1 f 0;1g attached to a family of Kloosterman sums. Katz calculated its monodromy and asked whether there are Kloosterman sheaves for general reductive groups and which automorphic forms should be attached to these local systems under the Langlands correspondence. Motivated by work of Gross and Frenkel-Gross we nd an explicit family of such automorphic forms and even a simple family of automorphic sheaves in the framework of the geometric Langlands program. We use these automorphic sheaves to construct ‘-adic Kloosterman sheaves for any reductive group in a uniform way, and describe the local and global monodromy of these Kloosterman sheaves. In particular, they give motivic Galois representations with exceptional monodromy groups G2;F4;E7 and E8. This also gives an example of the geometric Langlands correspondence with wild ramication for any reductive group.
TL;DR: In this article, the authors donnent une preuve de la representabilite de la K-theorie invariante par homotopie dans la categorie homotopy stable des schemas (resultat annonce par Voevodsky).
Abstract: Ces notes donnent une preuve de la representabilite de la K-theorie invariante par homotopie dans la categorie homotopique stable des schemas (resultat annonce par Voevodsky). On en deduit, grâce au theoreme de changement de base propre en theorie de l'homotopie stable des schemas, un theoreme de descente par eclatements en K-theorie invariante par homotopie.
TL;DR: In this paper, it was shown that every G -orbit closure in G/? is a finite volume homogeneous space, and related equidistribution properties were established for real Lie groups.
Abstract: Let G be a real Lie group, ? be a lattice in G and G be a compactly generated closed subgroup of G . If the Zariski closure of the group Ad(G) is semisimple with no compact factor, we prove that every G -orbit closure in G/? is a finite volume homogeneous space. We also establish related equidistribution properties.
TL;DR: In this paper, it was shown that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpinski carpet S3 is a Euclidean isometry.
Abstract: We prove that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpinski carpet S3 is a Euclidean isometry. For carpets in a more general family, the standard 1/p-Sierpinski carpets Sp, p ≥ 3 odd, we show that the groups of quasisymmetric self-maps are finite dihedral. We also establish that Sp and Sq are quasisymmetrically equivalent only if p = q. The main tool in the proof for these facts is a new invariant—a certain discrete modulus of a path family—that is preserved under quasisymmetric
TL;DR: The structure of sets of solutions to systems of equations in free and hyperbolic groups, projections of such sets (Diophantine sets), and the structure of denable sets over free groups were studied in this paper, where a modication of the sieve procedure, which was used in proving quantier elimination in the theory of a free group, was used to prove that free and torsion-free (Gromov) groups are stable.
Abstract: This paper is the eighth in a sequence on the structure of sets of solutions to systems of equations in free and hyperbolic groups, projections of such sets (Diophantine sets), and the structure of denable sets over free and hyperbolic groups. In this eighth paper we use a modication of the sieve procedure, which was used in proving quantier elimination in the theory of a free group, to prove that free and torsion-free (Gromov) hyperbolic groups are stable. In the
TL;DR: The canonical dimension of a coadmissible representation of a semisimple padic Lie group in a p-adic Banach space is either zero or at least half the dimension of the nonzero coadjoint orbit as mentioned in this paper.
Abstract: We prove that the canonical dimension of a coadmissible representation of a semisimple p-adic Lie group in a p-adic Banach space is either zero or at least half the dimension of a nonzero coadjoint orbit. To do this we establish analogues for p-adically completed enveloping algebras of Bernstein’s inequality for modules over Weyl algebras, the Beilinson-Bernstein localisation theorem and Quillen’s Lemma about the endomorphism ring of a simple module over an enveloping algebra.
TL;DR: In this paper, a rigorous proof of the KPZ relation between the uctuation exponent and the wandering exponent on integer lattices has been given assuming that the exponents exist in a certain sense.
Abstract: It has been conjectured in numerous physics papers that in ordinary rst-passage percolation on integer lattices, the uctuation exponent and the wandering exponent are related through the universal relation = 2 1, irrespective of the dimension. This is sometimes called the KPZ relation between the two exponents. This article gives a rigorous proof of this conjecture assuming that the exponents exist in a certain sense.
TL;DR: In this paper, the authors studied the algebraic structure of the n-dimensional Cremona group and showed that it is not an algebraic group of innite dimension (ind-group) if n 2.
Abstract: We study the algebraic structure of the n-dimensional Cremona group and show that it is not an algebraic group of innite dimension (ind-group) if n 2. We describe the obstruction to this, which is of a topological nature. By contrast, we show the existence of a Euclidean topology on the Cremona group which extends that of its classical subgroups and makes it a topological group.
TL;DR: In this paper, the authors obtained an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus and gave a limit theorem for these flows.
Abstract: The aim of this paper is to obtain an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus (Theorem 1) and to give a limit theorem for these flows (Theorem 2).
TL;DR: In this paper, a geometric invariant theory (GIT) construction of the log canonical model M ¯ g (a) of the pairs (M ¯ g,ad) for a?(7/10�?, 7/10] for small??Q +.
Abstract: We give a geometric invariant theory (GIT) construction of the log canonical model M ¯ g (a) of the pairs (M ¯ g ,ad) for a?(7/10�?,7/10] for small ??Q + . We show that M ¯ g (7/10) is isomorphic to the GIT quotient of the Chow variety of bicanonical curves; M ¯ g (7/10-?) is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions; e.g., they do not contain elliptic chains. We show that there is a small contraction ?:M ¯ g (7/10+?)?M ¯ g (7/10) that contracts the locus of elliptic bridges. Moreover, by using the GIT interpretation of the log canonical models, we construct a small contraction ? + :M ¯ g (7/10-?)?M ¯ g (7/10) that is the Mori flip of ? .
TL;DR: In this paper, it was shown that the Sylow p -subgroups of a finite group G/Z are abelian if p does not divide the integers for all lying over the group.
Abstract: Let Z be a normal subgroup of a finite group G , let ??Irr(Z) be an irreducible complex character of Z , and let p be a prime number. If p does not divide the integers ?(1)/?(1) for all ??Irr(G) lying over ? , then we prove that the Sylow p -subgroups of G/Z are abelian. This theorem, which generalizes the Gluck-Wolf Theorem to arbitrary finite groups, is one of the principal obstacles to proving the celebrated Brauer Height Zero Conjecture