Showing papers in "Transactions of the American Mathematical Society in 2015"

Hypocoercivity for linear kinetic equations conserving mass

Abstract: We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted $L^2$ norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models like the linear Boltzmann equation or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed.

Acylindrically hyperbolic groups

Abstract: We say that a group $G$ is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that the class of acylindrically hyperbolic groups coincides with many other classes studied in the literature, e.g., the class $C_{geom}$ introduced by Hamenstadt, the class of groups admitting a non-elementary weakly properly discontinuous action on a hyperbolic space in the sense of Bestvina and Fujiwara, and the class of groups with hyperbolically embedded subgroups studied by Dahmani, Guirardel, and the author. We also record some basic results about acylindrically hyperbolic groups for future use.

Riemannian Ricci curvature lower bounds in metric measure spaces with -finite measure

Abstract: In prior work (4) of the first two authors with Savare, a new Riemannian notion of lower bound for Ricci curvature in the class of metric measure spaces (X,d,m) was introduced, and the corresponding class of spaces denoted by RCD(K,∞). This notion relates the CD(K,N) theory of Sturm and Lott-Villani, in the case N = ∞, to the Bakry-Emery approach. In (4) the RCD(K,∞) property is defined in three equivalent ways and several properties of RCD(K,∞) spaces, including the regularization properties of the heat flow, the connections with the theory of Dirichlet forms and the stability under tensor products, are provided. In (4) only finite reference measures m have been considered. The goal of this paper is twofold: on one side we extend these results to general σ-finite spaces, on the other we remove a technical assumption appeared in (4) concerning a strengthening of the CD(K,∞) condition. This more general class of spaces includes Euclidean spaces endowed with Lebesgue measure, complete noncompact Riemannian manifolds with bounded geometry and the pointed metric measure limits of manifolds with lower Ricci curvature bounds.

GL-equivariant modules over polynomial rings in infinitely many variables

Abstract: Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group G. We study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible G-action. We define and prove finiteness properties for analogues of Hilbert series, systems of parameters, depth, local cohomology, Koszul duality, and regularity. We also show that this category is built out of a simpler, more combinatorial, quiver category which we describe explicitly. Our work is motivated by recent papers in the literature which study finiteness properties of infinite polynomial rings equipped with group actions. (For example, the paper by Church, Ellen- berg and Farb on the category of FI-modules, which is equivalent to our category.) Along the way, we see several connections with the character polynomials from the representation theory of the symmetric groups. Several examples are given to illustrate that the invariants we introduce are explicit and computable.

On uniform relationships between combinatorial problems

Abstract: The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic, with one of the most common frameworks for doing so being reverse mathematics. In this setting, one investigates which theorems provably imply which others in a weak formal theory roughly corresponding to computable mathematics. Since the proofs of such implications take place in classical logic, they may in principle involve appeals to multiple applications of a particular theorem, or to nonuniform decisions about how to proceed in a given construction. In practice, however, if a theorem Q implies a theorem P, it is usually because there is a direct uniform translation of the problems represented by P into the problems represented by Q, in a precise sense formalized by Weihrauch reducibility. We study this notion of uniform reducibility in the context of several natural combinatorial problems, and compare and contrast it with the traditional notion of implication in reverse mathematics. We show, for instance, that for all n; j; k 1, if j < k then Ramsey's theorem for n-tuples and k many colors is not uniformly, or Weihrauch, reducible to Ramsey's theorem for n-tuples and j many colors. The two theorems are classically equivalent, so our analysis gives a genuinely ner metric by which to gauge the relative strength of mathematical propositions. We also study Weak K�onig's Lemma, the Thin Set Theorem, and the Rainbow Ramsey's Theorem, along with a number of their variants investigated in the literature. Weihrauch reducibility turns out to be connected with sequential forms of mathematical principles, where one wishes to solve in nitely many instances of a particular problem simultaneously. We exploit this connection to uncover new points of di erence between combinatorial problems previously thought to be more closely related.

Linear-quadratic optimal control problems for mean-field stochastic differential equations — time-consistent solutions

Abstract: Linear-quadratic optimal control problems are considered for mean-field stochastic differential equations with deterministic coefficients. Time-inconsistency feature of the problems is carefully investigated. Both open-loop and closed-loop equilibrium solutions are presented for such kind of problems. Open-loop solutions are presented by means of variational method with decoupling of forward-backward stochastic differential equations, which lead to a Riccati equation system lack of symmetry. Closed-loop solutions are presented by means of multi-person differential games, the limit of which leads to a Riccati equation system with a symmetric structure.

Properties of squeezing functions and global transformations of bounded domains

Abstract: The central purpose of the present paper is to study boundary behavior of squeezing functions on bounded domains. We prove that the squeezing function of a strongly pseudoconvex domain tends to 1 near the boundary. In fact, such an estimate is proved for the squeezing function on any domain near its globally strongly convex boundary points. We also study the stability of squeezing functions on a sequence of bounded domains, and give comparisons of intrinsic measures and metrics on bounded domains in terms of squeezing functions. As applications, we give new and simple proofs of several well known results about geometry of strongly pseudoconvex do- mains, and prove that all Cartan-Hartogs domains are homogenous regular. Finally, some related problems that ask for further study are proposed.

The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases

Abstract: We study the ground state of a trapped Bose gas, starting from the full many-body Schrodinger Hamiltonian, and derive the nonlinear Schrodinger energy functional in the limit of large particle number, when the interaction potential converges slowly to a Dirac delta function. Our method is based on quantitative estimates on the discrepancy between the full many-body energy and its mean-field approximation using Hartree states. These are proved using finite dimensional localization and a quantitative version of the quantum de Finetti theorem. Our approach covers the case of attractive interactions in the regime of stability. In particular, our main new result is a derivation of the 2D attractive nonlinear Schrodinger ground state.

Stability and compactness for complete f -minimal surfaces

Abstract: Let $(M,\bar{g}, e^{-f}d\mu)$ be a complete metric measure space with Bakry-\'Emery Ricci curvature bounded below by a positive constant. We prove that, in $M$, there is no complete two-sided $L_f$-stable immersed $f$-minimal hypersurface with finite weighted volume. Further, if $M$ is a 3-manifold, we prove a smooth compactness theorem for the space of complete embedded $f$-minimal surfaces in $M$ with the uniform upper bounds of genus and weighted volume, which generalizes the compactness theorem for complete self-shrinkers in $\mathbb{R}^3$ by Colding-Minicozzi.

Transitive 2-representations of finitary 2-categories

Abstract: In this article, we define and study the class of simple transitive $2$-representations of finitary $2$-categories. We prove a weak version of the classical Jordan-H{\"o}lder Theorem where the weak composition subquotients are given by simple transitive $2$-representations. For a large class of finitary $2$-categories we prove that simple transitive $2$-representations are exhausted by cell $2$-representations. Finally, we show that this large class contains finitary quotients of $2$-Kac-Moody algebras.

Vapnik-Chervonenkis Density in some Theories without the Independence Property, II

Abstract: We study the Vapnik-Chervonenkis (VC) density of denable families in certain stable rst-order theories. In particular we obtain uniform bounds on VC density of denable families in nite U-rank theories without the nite cover property, and we characterize those abelian groups for which there exist uniform bounds on the VC density of denable families.

Gradient estimates of harmonic functions and transition densities for Lévy processes

Abstract: We prove gradient estimates for harmonic functions with respect to a $d$-dimensional unimodal pure-jump Levy process under some mild assumptions on the density of its Levy measure. These assumptions allow for a construction of an unimodal Levy process in $\R^{d+2}$ with the same characteristic exponent as the original process. The relationship between the two processes provides a fruitful source of gradient estimates of transition densities. We also construct another process called a difference process which is very useful in the analysis of differential properties of harmonic functions.

Positroids and non-crossing partitions

Abstract: We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then freely placing the structure of a connected positroid on each of the blocks of the partition. This structural result yields several combinatorial facts about positroids. We show that the face poset of a positroid polytope embeds in a poset of weighted non-crossing partitions. We enumerate connected positroids, and show how they arise naturally in free probability. Finally, we prove that the probability that a positroid on [n] is connected equals 1/e^2 asymptotically.

Physical Brownian motion in a magnetic field as a rough path

Abstract: The indefinite integral of the homogenized Ornstein-Uhlenbeck process is a well-known model for physical Brownian motion, modelling the behaviour of an object subject to random impulses [L. S. Ornstein, G. E. Uhlenbeck: On the theory of Brownian Motion. In: Physical Review. 36, 1930, 823-841]. One can scale these models by changing the mass of the particle and in the small mass limit one has almost sure uniform convergence in distribution to the standard idealized model of mathematical Brownian motion. This provides one well known way of realising the Wiener process. However, this result is less robust than it would appear and important generic functionals of the trajectories of the physical Brownian motion do not necessarily converge to the same functionals of Brownian motion when one takes the small mass limit. In presence of a magnetic field the area process associated to the physical process converges - but not to L\\'evy's stochastic area. As this area is felt generically in settings where the particle interacts through force fields in a nonlinear way, the remark is physically significant and indicates that classical Brownian motion, with its usual stochastic calculus, is not an appropriate model for the limiting behaviour. We compute explicitly the area correction term and establish convergence, in the small mass limit, of the physical Brownian motion in the rough path sense. The small mass limit for the motion of a charged particle in the presence of a magnetic field is, in distribution, an easily calculable, but \"non-canonical\" rough path lift of Brownian motion. Viewing the trajectory of a charged Brownian particle with small mass as a rough path is informative and allows one to retain information that would be lost if one only considered it as a classical trajectory. We comment on the importance of this point of view.

On the maximum orders of elements of finite almost simple groups and primitive permutation groups

Abstract: We determine upper bounds for the maximum order of an element of a finite almost simple group with socle T in terms of the minimum index m(T) of a maximal subgroup of T: for T not an alternating group we prove that, with finitely many excep- tions, the maximum element order is at most m(T) Moreover, apart from an explicit list of groups, the bound can be reduced to m(T)/4 These results are applied to determine all primitive permutation groups on a set of size n that contain permutations of order greater than or equal to n/4 We note again that this result gives upper bounds for meo(Aut(T)) in terms of m(T), and for meo(G) in terms of m(G) (since m(T) ≤ m(G)) Moreover equality in the up- per bound meo(Aut(T)) ≤ m(T) holds when T = PSLd(q) for all but two pairs (d,q), see Table 3 and Theorem 216 (Theorem 216 and Table 3 provide good estimates for

Two approaches to sidorenko’s conjecture

Abstract: Sidorenko’s conjecture states that for every bipartite graphH on {1, · · · , k} ∫ ∏ (i,j)∈E(H) h(xi, yj)dμ |V (H)| ≥ (∫ h(x, y) dμ )|E(H)| holds, where μ is the Lebesgue measure on [0, 1] and h is a bounded, nonnegative, symmetric, measurable function on [0, 1]2. An equivalent discrete form of the conjecture is that the number of homomorphisms from a bipartite graph H to a graph G is asymptotically at least the expected number of homomorphisms from H to the Erdős-Renyi random graph with the same expected edge density as G. In this paper, we present two approaches to the conjecture. First, we introduce the notion of tree-arrangeability, where a bipartite graph H with bipartition A∪B is tree-arrangeable if neighborhoods of vertices in A have a certain tree-like structure. We show that Sidorenko’s conjecture holds for all tree-arrangeable bipartite graphs. In particular, this implies that Sidorenko’s conjecture holds if there are two vertices a1, a2 in A such that each vertex a ∈ A satisfies N(a) ⊆ N(a1) or N(a) ⊆ N(a2), and also implies a recent result of Conlon, Fox, and Sudakov [3]. Second, if T is a tree and H is a bipartite graph satisfying Sidorenko’s conjecture, then it is shown that the Cartesian product T H of T and H also satisfies Sidorenko’s conjecture. This result implies that, for all d ≥ 2, the d-dimensional grid with arbitrary side lengths satisfies Sidorenko’s conjecture.

The metaplectic Casselman-Shalika formula

Abstract: This paper studies spherical Whittaker functions for central extensions of reductive groups over local fields. We follow the development of Chinta and Offen to produce a metaplectic Casselman-Shalika formula for tame covers of all unramified groups.

On accumulated spectrograms

Abstract: We study the eigenvalues and eigenfunctions of the time-frequency localization operator HΩ on a domain Ω of the time-frequency plane. The eigenfunctions are the appropriate prolate spheroidal functions for an arbitrary domain Ω ⊆ R. Indeed, in analogy to the classical theory of Landau-SlepianPollak, the number of eigenvalues of HΩ in [1− δ, 1] is equal to the measure of Ω up to an error term depending on the perimeter of the boundary of Ω. Our main results show that the spectrograms of the eigenfunctions corresponding to the large eigenvalues (which we call the accumulated spectrogram) form an approximate partition of unity of the given domain Ω. We derive asymptotic, non-asymptotic, and weak-L error estimates for the accumulated spectrogram. As a consequence the domain Ω can be approximated solely from the spectrograms of eigenfunctions without information about their phase.

The Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B

Abstract: We study a Bishop-Phelps-Bollobas version of Lindenstrauss properties A and B. For domain spaces, we study Banach spaces X such that (X,Y ) has the BishopPhelps-Bollobas property (BPBp) for every Banach space Y . We show that in this case, there exists a universal function ηX(e) such that for every Y , the pair (X,Y ) has the BPBp with this function. This allows us to prove some necessary isometric conditions for X to have the property. We also prove that if X has this property in every equivalent norm, then X is one-dimensional. For range spaces, we study Banach spaces Y such that (X,Y ) has the Bishop-Phelps-Bollobas property for every Banach space X. In this case, we show that there is a universal function ηY (e) such that for every X, the pair (X,Y ) has the BPBp with this function. This implies that this property of Y is strictly stronger than Lindenstrauss property B. The main tool to get these results is the study of the Bishop-Phelps-Bollobas property for c0-, `1and `∞-sums of Banach spaces.

Ice quivers with potential associated with triangulations and Cohen-Macaulay modules over orders

Abstract: Given a triangulation of a polygon P with n vertices, we associate an ice quiver with potential such that the associated Jacobian algebra has the structure of a Gorenstein tiled K[x]-order L. Then we show that the stable category of the category of Cohen-Macaulay L-modules is equivalent to the cluster category C of Dynkin type A(n-3). It gives a natural interpretation of the usual indexation of cluster tilting objects of C by triangulations of P. Moreover, it extends naturally the triangulated categorification by C of the cluster algebra of type A(n-3) to an exact categorification by adding coefficients corresponding to the sides of P. Finally, we lift the previous equivalence of categories to an equivalence between the stable category of graded Cohen-Macaulay L-modules and the bounded derived category of modules over a quiver of type A(n-3).

The quantum Gromov-Hausdorff propinquity

Abstract: We introduce the quantum Gromov-Hausdorff propinquity, a new distance between quantum compact metric spaces, which extends the GromovHausdorff distance to noncommutative geometry and strengthens Rieffel’s quantum Gromov-Hausdorff distance and Rieffel’s proximity by making *-isomorphism a necessary condition for distance zero, while being well adapted to Leibniz seminorms. This work offers a natural solution to the long-standing problem of finding a framework for the development of a theory of Leibniz Lip-norms over C*algebras.

Absolute continuity of self-similar measures, their projections and convolutions

Abstract: We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small - of co-dimension at least one in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continuity of the convolution of two measures. As a concrete application, we obtain a very strong version of Marstrand's projection theorem for planar self-similar sets.

Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation

Abstract: We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the “elastic” operator. In the homogeneous case, we investigate the phase spaces in which the initial value problem gives rise to a semigroup, and the further regularity of solutions. In the nonhomogeneous case, we study how the regularity of solutions depends on the regularity of forcing terms, and we characterize the spaces where a bounded forcing term yields a bounded solution. What we discover is a variety of different regimes, with completely different behaviors, depending on the exponent in the friction term. We also provide counterexamples in order to show the optimality of our results. Mathematics Subject Classification 2010 (MSC2010): 35L10, 35L15, 35L20.

Almost global existence for 2-D incompressible isotropic elastodynamics

Abstract: Author(s): Lei, Z; Sideris, TC; Zhou, Y | Abstract: We consider the Cauchy problem for 2-D incompressible isotropic elastodynamics. Standard energy methods yield local solutions on a time interval [0, T/∈] for initial data of the form ∈U0, where T depends only on some Sobolev norm of U0. We show that for such data there exists a unique solution on a time interval [0, exp T/∈], provided that ∈ is sufficiently small. This is achieved by careful consideration of the structure of the nonlinearity. The incompressible elasticity equation is inherently linearly degenerate in the isotropic case; in other words, the equation satisfies a null condition. This is essential for time decay estimates. The pressure, which arises as a Lagrange multiplier to enforce the incompressibility constraint, is estimated in a novel way as a nonlocal nonlinear term with null structure. The proof employs the generalized energy method of Klainerman, enhanced by weighted L2 estimates and the ghost weight introduced by Alinhac.

Observables of Macdonald processes

Abstract: We present a framework for computing averages of various observables of Macdonald processes. This leads to new contour--integral formulas for averages of a large class of multilevel observables, as well as Fredholm determinants for averages of two different single level observables.

On twisted higher-rank graph C*-algebras

Abstract: We define the categorical cohomology of a k-graphand show that the first three terms in this cohomology are isomorphic to the corresponding terms in the cohomology defined in our previous paper. This leads to an alternative characterisation of the twisted k-graph C � -algebras introduced there. We prove a gauge-invariant uniqueness theorem and use it to show that every twisted k-graph C � -algebra is isomorphic to a twisted groupoid C � -algebra. We deduce criteria for simplicity, prove a Cuntz-Krieger uniqueness theorem and establish that all twisted k-graph C � -algebras are nuclear and belong to the bootstrap class.

Noncommutative mirror symmetry for punctured surfaces

Abstract: In 2013, Abouzaid, Auroux, Efimov, Katzarkov and Orlov showed that the wrapped Fukaya categories of punctured spheres and finite unbranched covers of punctured spheres are derived equivalent to the categories of singularities of a superpotential on certain crepant resolutions of toric 3 dimensional singularities. We generalize this result to other punctured Riemann surfaces and reformulate it in terms of certain noncommutative algebras coming from dimer models. In particular, given any consistent dimer model we can look at a subcategory of noncommutative matrix factorizations and show that this category is $ \mathtt {A}_\infty $-isomorphic to a subcategory of the wrapped Fukaya category of a punctured Riemann surface. The connection between the dimer model and the punctured Riemann surface then has a nice interpretation in terms of a duality on dimer models.

The homotopy types of moment-angle complexes for flag complexes

Abstract: We study the homotopy types of moment-angle complexes, or equivalently, of complements of coordinate subspace arrangements. The overall aim is to identify the simplicial complexes K for which the corresponding moment-angle complex Z_K has the homotopy type of a wedge of spheres or a connected sum of sphere products. When K is flag, we identify in algebraic and combinatorial terms those K for which Z_K is homotopy equivalent to a wedge of spheres, and give a combinatorial formula for the number of spheres in the wedge. This extends results of Berglund and Jollenbeck on Golod rings and homotopy theoretical results of the first and third authors. We also establish a connection between minimally non-Golod rings and moment-angle complexes Z_K which are homotopy equivalent to a connected sum of sphere products. We go on to show that for any flag complex K the loop space of Z_K is homotopy equivalent to a product of spheres and loops on spheres when localised rationally or at any prime not equal to 2.