Showing papers in "Transactions of the American Mathematical Society in 2011"

Global heat kernel estimates for symmetric jump processes

Abstract: In this paper, we study sharp heat kernel estimates for a large class of symmetric jump-type processes in R d for all t > 0. A prototype of the processes under consideration are symmetric jump processes on R d with jumping intensity

Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces

Abstract: The action of the generalized fractional integral operators and the generalized fractional maximal operators is investigated in the framework of Morrey spaces. A typical property of the functions which belongs to Morrey spaces under pointwise multiplication by the generalized fractional integral operators and the generalized fractional maximal operators is established. The boundedness property of the fractional integral operators on the predual of Morrey spaces is shown as well. A counterexample concerning the FeffermanPhong inequality is given by the use of the characteristic function of the Cantor set.

Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^{3}$

Abstract: The existence of a random attractor in H 1 (ℝ 3 ) x L 2 (ℝ 3 ) is proved for the damped semilinear stochastic wave equation defined on the entire space ℝ 3 . The nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. The uniform pullback estimates on the tails of solutions for large space variables are established. The pullback asymptotic compactness of the random dynamical system is proved by using these tail estimates and the energy equation method.

Convenient categories of smooth spaces

Abstract: A 'Chen space' is a set X equipped with a collection of 'plots', i.e., maps from convex sets to X, satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace or quotient space of a Chen space is a Chen space, and the space of smooth maps between Chen spaces is again a Chen space. Souriau's 'diffeological spaces' share these convenient properties. Here we give a unified treatment of both formalisms. Following ideas of Penon and Dubuc, we show that Chen spaces, diffeological spaces, and even simplicial complexes are examples of 'concrete sheaves on a concrete site'. As a result, the categories of such spaces are locally Cartesian closed, with all limits, all colimits, and a weak subobject classifier. For the benefit of differential geometers, our treatment explains most of the category theory we use. © 2011 John C. Baez and Alexander E. Hoffnung.

A Chebyshev criterion for Abelian integrals

Abstract: We present a criterion that provides an easy sufficient condition in order for a collection of Abelian integrals to have the Chebyshev property. This condition involves the functions in the integrand of the Abelian integrals and can be checked, in many cases, in a purely algebraic way. By using this criterion, several known results are obtained in a shorter way and some new results, which could not be tackled by the known standard methods, can also be deduced.

Small ball probability estimates for log-concave measures

Abstract: We establish a small ball probability inequality for isotropic log-concave probability measures: there exist absolute constants c1, c2 > 0 such that if X is an isotropic log-concave random vector in R with ψ2 constant bounded by b and if A is a non-zero n × n matrix, then for every e ∈ (0, c1) and y ∈ R, P (‖Ax− y‖2 6 e‖A‖HS) 6 e ( c2 b ‖A‖HS ‖A‖op )2 , where c1, c2 > 0 are absolute constants.

Monomial ideals, almost complete intersections and the Weak Lefschetz property

Abstract: Many algebras are expected to have the Weak Lefschetz property though this is often very difficult to establish. We illustrate the subtlety of the problem by studying monomial and some closely related ideals. Our results exemplify the intriguing dependence of the property on the characteristic of the ground field, and on arithmetic properties of the exponent vectors of the monomials.

The defocusing energy-supercritical nonlinear wave equation in three space dimensions

Abstract: We consider the defocusing nonlinear wave equation u tt ― Δu + |u| p u = 0 in the energy-supercritical regime p > 4. For even values of the power p, we show that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that solutions with bounded critical Sobolev norm are global and scatter. The impetus to consider this problem comes from recent work of Kenig and Merle who treated the case of spherically-symmetric solutions.

Sharp results in the integral-form John-Nirenberg inequality

Abstract: We consider the strong form of the John-Nirenberg inequality for the L 2 -based BMO. We construct explicit Bellman functions for the inequality in the continuous and dyadic settings and obtain the sharp constant, as well as the precise bound on the inequality's range of validity, both previously unknown. The results for the two cases are substantially different. The paper not only gives another instance in the short list of such explicit calculations, but also presents the Bellman function method as a sequence of clear steps, adaptable to a wide variety of applications.

Newton polygons of higher order in algebraic number theory

Abstract: We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a p-adic eld, together with relevant arithmetic information about the elds generated by the irreducible factors. This carries out a program suggested by . Ore. As an application, we obtain fast algorithms to compute discriminants, prime ideal decomposition and integral bases of number elds.

Cubulating random groups at density less than 1/6

Abstract: We prove that random groups at density less than 1/6 act freely and cocompactly on CAT(0) cube complexes, and that random groups at density less than 1/5 have codimension-1 subgroups. In particular, Property (T ) fails to hold at density less than 1/5.

Calabi–Yau Threefolds and Moduli of Abelian Surfaces I

Abstract: We describe birational models and decide the rationality/unirationality of moduli spaces $\cal A$d (and $\cal A$levd) of (1, d)-polarized Abelian surfaces (with canonical level structure, respectively) for small values of d. The projective lines identified in the rational/unirational moduli spaces correspond to pencils of Abelian surfaces traced on nodal threefolds living naturally in the corresponding ambient projective spaces, and whose small resolutions are new Calabi–Yau threefolds with Euler characteristic zero.

$n$-representation-finite algebras and $n$-APR tilting

Abstract: We introduce the notion of n-representation-finiteness, generalizing representationfinite hereditary algebras. We establish the procedure of n-APR tilting, and show that it preserves n-representation-finiteness. We give some combinatorial description of this procedure, and use this to completely describe a class of n-representation-finite algebras called “type A”.

Isomorphism and Morita equivalence of graph algebras

Abstract: For any countable graph E, we investigate the relationship between the Leavitt path algebra L ℂ (E) and the graph C * -algebra C * (E). For graphs E and F, we examine ring homomorphisms, ring *-homomorphisms, algebra homomorphisms, and algebra *-homomorphisms between L ℂ (E) and L ℂ (F). We prove that in certain situations isomorphisms between L ℂ (E) and L ℂ (F) yield *-isomorphisms between the corresponding C * -algebras C * (E) and C * (F). Conversely, we show that *-isomorphisms between C * (E) and C * (F) produce isomorphisms between L ℂ (E) and L ℂ (F) in specific cases. The relationship between Leavitt path algebras and graph C * -algebras is also explored in the context of Morita equivalence.

On Floer homology and the Berge conjecture on knots admitting lens space surgeries

Abstract: We complete the first step in a two-part program proposed by Baker, Grigsby, and the author to prove that Berge’s construction of knots in the three-sphere which admit lens space surgeries is complete. The first step, which we prove here, is to show that a knot in a lens space with a threesphere surgery has simple (in the sense of rank) knot Floer homology. The second (conjectured) step involves showing that, for a fixed lens space, the only knots with simple Floer homology belong to a simple finite family. Using results of Baker, we provide evidence for the conjectural part of the program by showing that it holds for a certain family of knots. Coupled with work of Ni, these knots provide the first infinite family of non-trivial knots which are characterized by their knot Floer homology. As another application, we provide a Floer homology proof of a theorem of Berge.

Kolmogorov complexity and the Recursion Theorem

Abstract: Several classes of diagonally nonrecursive (DNR) functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers A can wtt-compute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of the initial segments of A. Furthermore, A can Turing compute a DNR function iff there is a nontrivial A-recursive lower bound on the Kolmogorov complexity of the initial segments of A. A is PA-complete, that is, A can compute a {0, 1}-valued DNR function, iff A can compute a function F such that F(n) is a string of length n and maximal C-complexity among the strings of length n. A ≥ T K iff A can compute a function F such that F(n) is a string of length n and maximal H-complexity among the strings of length n. Further characterizations for these classes are given. The existence of a DNR function in a Turing degree is equivalent to the failure of the Recursion Theorem for this degree; thus the provided results characterize those Turing degrees in terms of Kolmogorov complexity which no longer permit the usage of the Recursion Theorem.

Local rigidity of inversive distance circle packing

Abstract: A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle.

The Dirichlet problem for the convex envelope

Abstract: The Convex Envelope of a given function was recently characterized as the solution of a fully nonlinear Partial Differential Equation (PDE). In this article we study a modified problem: the Dirichlet problem for the underlying PDE. The main result is an optimal regularity result. Differentiability (C1,α regularity) of the boundary data implies the corresponding result for the solution in the interior, despite the fact that the solution need not be continuous up to the boundary. Secondary results are the characterization of the convex envelope as: (i) the value function of a stochastic control problem, and (ii) the optimal underestimator for a class of nonlinear elliptic PDEs.

ON THE SINGULARITIES OF THE CURVED n-BODY PROBLEM

Abstract: We study singularities of the n-body problem in spaces of constant curvature and generalize certain results due to Painleve, Weierstrass, and Sundman. For positive curvature, some of our proofs use the correspondence between total collision solutions of the original system and their orthogonal projection—a property that offers a new method of approaching the problem in this par- ticular case.

Einstein and conformally flat critical metrics of the volume functional

Abstract: Let R be a constant. Let M R γ be the space of smooth metrics g on a given compact manifold Ω n (n > 3) with smooth boundary Σ such that g has constant scalar curvature R and g|Σ is a fixed metric γ on Σ. Let V(g) be the volume of g ∈ M R γ . In this work, we classify all Einstein or conformally flat metrics which are critical points of V(·) in M R γ .

On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation

Abstract: Let QC(K, g) be a family of K-quasiconformal mappings of the open unit disk onto itself satisfying the PDE Δw = g, g ∈ C(U), w(0) = 0. It is proved that QC(K,g) is a uniformly Lipschitz family. Moreover, if |g| ∞ is small enough, then the family is uniformly bi-Lipschitz. The estimations are asymptotically sharp as K → 1 and |g| ∞ → 0, so w ∈ QC(K, g) behaves almost like a rotation for sufficiently small K and |g| ∞ .

Slow escaping points of meromorphic functions

Abstract: We show that for any transcendental meromorphic function f there is a point z in the Julia set of f such that the iterates f n (z) escape, that is, tend to ∞ , arbitrarily slowly. The proof uses new covering results for analytic functions. We also introduce several slow escaping sets, in each of which f n (z) tends to ∞ at a bounded rate, and establish the connections between these sets and the Julia set of f . To do this, we show that the iterates of f satisfy a strong distortion estimate in all types of escaping Fatou components except one, which we call a quasi-nested wandering domain. We give examples to show how varied the structures of these slow escaping sets can be.

Ground state and non-ground state solutions of some strongly coupled elliptic systems

Abstract: We study an elliptic system of the form Lu = ⌊v⌋p&1 v and Lv = ⌊u⌋ q&1 u in Ω with homogeneous Dirichlet boundary condition, where Lu:= &Δu in the case of a bounded domain and Lu:= &Δu + u in the cases of an exterior domain or the whole space RN. We analyze the existence, uniqueness, sign and radial symmetry of ground state solutions and also look for sign changing solutions of the system. More general non-linearities are also considered. © 2011 American Mathematical Society.

Einstein solvmanifolds and the pre-Einstein derivation

Abstract: An Einstein nilradical is a nilpotent Lie algebra, which can be the nilradical of a metric Einstein solvable Lie algebra. The classification of Riemannian Einstein solvmanifolds (possibly, of all noncompact homogeneous Einstein spaces) can be reduced to determining, which nilpotent Lie algebras are Einstein nilradicals and to finding, for every Einstein nilradical, its Einstein metric solvable extension. For every nilpotent Lie algebra, we construct an (essentially unique) derivation, the pre- Einstein derivation, the solvable extension by which may carry an Einstein inner product. Using the pre-Einstein derivation, we then give a variational characterization of Einstein nilradicals. As an application, we prove an easy-to-check convex geometry condition for a nilpotent Lie algebra with a nice basis to be an Einstein nilradical and also show that a typical two-step nilpotent Lie algebra is an Einstein nilradical. The theory of Riemannian homogeneous spaces with an Einstein metric splits into three very different cases depending on the sign of the Einstein constant, the scalar curvature. Among them, the picture is complete only in the Ricci-flat case: by the result of (AK), every Ricci-flat homogeneous space is flat. The major open conjecture in the case of negative scalar curvature is the Alekseevski Conjecture (Al1) asserting that a noncompact Einstein homogeneous space admits a simply transitive solvable isometry group. This is equivalent to saying that any such space is a solvmanifold, a solvable Lie group with a left-invariant Riemannian metric satisfying the Einstein condition. By a deep result of J.Lauret (La5), any Einstein solvmanifold is standard. This means that the metric solvable Lie algebra s of such a solvmanifold has the following property: the orthogonal complement to the derived algebra of s is abelian. The systematic study of standard Einstein solvmanifolds (and the term "standard") originated from the paper of J.Heber (Heb). On the Lie algebra level, all the metric Einstein solvable Lie algebras can be obtained as the result of the following construction (Heb, La1, La5, LW). One starts with the three pieces of data: a nilpotent Lie algebra n, a semisimple derivationof n, and an inner product h� , �i n on n, with respect to which � is symmetric. An extension of n byis a solvable Lie algebra s = RH ⊕ n (as a linear space) with (adH)|n := �. The inner product on s is defined by h H, ni = 0, k Hk 2 = Tr � (and coincides with the existing one on n). The resulting metric solvable Lie algebra (s, h� , �i ) is Einstein provided n is "nice" and the derivationand the inner product h� , �i n are chosen "in the correct way" (note, however, that these conditions are expressed by a system of algebraic equations, which could hardly be analyzed directly, see Section 2). Metric Einstein solvable Lie algebras of higher rank (with the codimension of the nilradical greater than one) having the same nilradical n can be obtained from s via a known procedure, by further adjoining to n semisimple derivation commuting with �. It turns out that the structure of an Einstein metric solvable Lie algebra is completely encoded in its nilradical in the following sense: given a nilpotent Lie algebra n, there is no more than one (possibly none) choice ofand of h� , �i n, up to conjugation by Aut(n) and scaling, which may result in an Einstein metric solvable Lie algebra (s, h� , �i ). Definition 1. A nilpotent Lie algebra is called an Einstein nilradical, if it is the nilradical of an Einstein metric solvable Lie algebra. A derivationof an Einstein nilradical n and an inner product h� , �i n, for which the metric solvable Lie algebra (s, h� , �i ) is Einstein are called an Einstein derivation and a nilsoliton inner product respectively.

Quantum Monodromy and nonconcentration near a closed semi-hyperbolic orbit

Abstract: For a large class of semiclassical operators P(h) ― z which includes Schrodinger operators on manifolds with boundary, we construct the Quantum Monodromy operator M(z) associated to a periodic orbit γ of the classical flow. Using estimates relating M(z) and P(h) — z, we prove semiclassical estimates for small complex perturbations of P(h) — z in the case γ is semi-hyperbolic. As our main application, we give logarithmic lower bounds on the mass of eigenfunctions away from semi-hyperbolic orbits of the associated classical flow. As a second application of the Monodromy Operator construction, we prove if γ is an elliptic orbit, then P(h) admits quasimodes which are well-localized near γ.

Quantum isometry groups of 0-dimensional manifolds

Abstract: Quantum isometry groups of spectral triples associated with ap- proximately finite-dimensional C � -algebras are shown to arise as inductive limits of quantum symmetry groups of corresponding truncated Bratteli dia- grams. This is used to determine explicitly the quantum isometry group of the natural spectral triple on the algebra of continuous functions on the middle- third Cantor set. It is also shown that the quantum symmetry groups of finite graphs or metric spaces coincide with the quantum isometry groups of the corresponding classical objects equipped with natural Laplacians.

1D symmetry for solutions of semilinear and quasilinear elliptic equations

Abstract: Several new $ 1$D results for solutions of possibly singular or degenerate elliptic equations, inspired by a conjecture of De Giorgi, are provided. In particular, $ 1$D symmetry is proven under the assumption that either the profiles at infinity are $ 2$D, or that one level set is a complete graph, or that the solution is minimal or, more generally, $ Q$-minimal.

On adapted coordinate systems

Abstract: The notion of an adapted coordinate system for a given real-analytic function, introduced by V. I. Arnol'd, plays an important role, for instance, in the study of asymptotic expansions of oscillatory integrals. In two dimensions, A. N. Varchenko gave sufficient conditions for the adaptness of a given coordinate system and proved the existence of an adapted coordinate system for analytic functions without multiple components. Varchenko's proof is based on a two-dimensional resolution of singularities result. In this article, we present a more elementary approach to these results, which is based on the Puiseux series expansion of roots of the given function. This approach is inspired by the work of D. H. Phong and E. M. Stein on the Newton polyhedron and oscillatory integral operators. It applies to arbitrary real-analytic functions, and even to arbitrary smooth functions of finite type. In particular, we show that Varchenko's conditions are in fact necessary and sufficient for the adaptedness of a given coordinate system and that adapted coordinates always exist in two dimensions, even in the smooth, finite type setting. For analytic functions, a construction of adapted coordinates by means of Puiseux series expansions of roots has already been carried out in work by D. H. Phong, E. M. Stein and J. A. Sturm on the growth and stability of real-analytic function, as we learned after the completion of this paper. In contrast to their work, however, our proof more closely follows Varchenko's algorithm for the construction of an adapted coordinate system, which turns out to be useful for the extension to the smooth setting.