Showing papers in "Transactions of the American Mathematical Society in 2007"

Strongly self-absorbing C*-algebras

Abstract: Say that a separable, unital C*-algebra V ? C is strongly self absorbing if there exists an isomorphism V such that lx> ai>e approximately unitarily equivalent *-homomorphisms. We study this class of algebras, which includes the Cuntz algebras ?2, Ooo, the UHF algebras of infinite type, the Jiang-Su algebra Z and tensor products of ?00 with UHF algebras of infinite type. Given a strongly self-absorbing C*-algebra V we characterise when a separable C*-algebra absorbs V tensorially (i.e., is P-stable), and prove closure properties for the class of separable Testable C* algebras. Finally, we compute the possible If-groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing C*-algebras.

Supercharacters and superclasses for algebra groups

Abstract: We study certain sums of irreducible characters and compatible unions of conjugacy classes in finite algebra groups. These groups generalize the unimodular upper triangular groups over a finite field, and the supercharacter theory we develop extends results of Carlos Andre and Ning Yan that were originally proved in the upper triangular case. This theory sometimes allows explicit computations in situations where it would be impractical to work with the full character table. We discuss connections with the Kirillov orbit method and with Gelfand pairs, and we give conditions for a supercharacter or a superclass to be an ordinary irreducible character or conjugacy class, respectively. We also show that products of supercharacters are positive integer combinations of supercharacters.

A Koszul duality for props

Abstract: The notion of prop models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the Koszul duality theory of associative algebras and operads to props.

Erdös distance problem in vector spaces over finite fields

Abstract: We study the Erdos/Falconer distance problem in vector spaces over finite fields. Let be a finite field with elements and take , . We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in to provide estimates for minimum cardinality of the distance set in terms of the cardinality of . Bounds for Gauss and Kloosterman sums play an important role in the proof.

Clusters, Coxeter-sortable elements and noncrossing partitions

Abstract: We introduce Coxeter-sortable elements of a Coxeter group For finite we give bijective proofs that Coxeter-sortable elements are equinumerous with clusters and with noncrossing partitions. We characterize Coxeter-sortable elements in terms of their inversion sets and, in the classical cases, in terms of permutations

Legendrian contact homology in $P \times \mathbb{R}$

Abstract: A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form P x R, where P is an exact symplectic manifold, is established The class of such contact manifolds includes 1-jet spaces of smooth manifolds As an application, contact homology is used to provide (smooth) isotopy invariants of submanifolds ofR" and, more generally, invariants of self transverse immersions into R n up to restricted regular homotopies When n = 3, this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of Ng

Mass concentration phenomena for the $L^2$-critical nonlinear Schrödinger equation

Abstract: In this paper, we show that any solution of the nonlinear Schr{o}dinger equation $iu_t+\Delta u\pm|u|^\frac{4}{N}u=0,$ which blows up in finite time, satisfies a mass concentration phenomena near the blow-up time. Our proof is essentially based on the Bourgain's one~\cite{MR99f:35184}, which has established this result in the bidimensional spatial case, and on a generalization of Strichartz's inequality, where the bidimensional spatial case was proved by Moyua, Vargas and Vega~\cite{MR1671214}. We also generalize to higher dimensions the results in Keraani~\cite{MR2216444} and Merle and Vega~\cite{MR1628235}.

Generalized Killing spinors in dimension 5

Abstract: We study the intrinsic geometry of hypersurfaces in Calabi-Yau manifolds of real dimension 6 and, more generally, -structures on 5-manifolds defined by a generalized Killing spinor. We prove that in the real analytic case, such a 5-manifold can be isometrically embedded as a hypersurface in a Calabi-Yau manifold in a natural way. We classify nilmanifolds carrying invariant structures of this type, and present examples of the associated metrics with holonomy .

Riemannian flag manifolds with homogeneous geodesics

Abstract: A geodesic in a Riemannian homogeneous manifold (M = G/K, g) is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of the Lie group G. We investigate G-invariant metrics with homogeneous geodesics (i.e., such that all geodesics are homogeneous) when M = G/K is a flag manifold, that is, an adjoint orbit of a compact semisimple Lie group G. We use an important invariant of a flag manifold M = G/K, its T-root system, to give a simple necessary condition that M admits a non-standard G-invariant metric with homogeneous geodesics. Hence, the problem reduces substantially to the study of a short list of prospective flag manifolds. A common feature of these spaces is that their isotropy representation has two irreducible components. We prove that among all flag manifolds M = G/K of a simple Lie group G, only the manifold Com(R 2l+2 ) = SO(2l +1)/U(l) of complex structures in R 2l+2 , and the complex projective space CP 2l-1 = Sp(l)/U(1) " Sp(l- 1) admit a non-naturally reductive invariant metric with homogeneous geodesics. In all other cases the only G-invariant metric with homogeneous geodesics is the metric which is homothetic to the standard metric (i.e., the metric associated to the negative of the Killing form of the Lie algebra g of G). According to F. Podesta and G.Thorbergsson (2003), these manifolds are the only non-Hermitian symmetric flag manifolds with coisotropic action of the stabilizer.

Extension d'une valuation

Abstract: We want to determine all the extensions of a valuation v of a field K to a cyclic extension L of K, i.e. L = K(x) is the field of rational functions of x or L = K(θ) is the finite separable extension generated by a root 0 of an irreducible polynomial G(x). In two articles from 1936, Saunders MacLane has introduced the notions of key polynomial and of augmented valuation for a given valuation μ of K[x], and has shown how we can recover any extension to L of a discrete rank one valuation v of K by a countable sequence of augmented valuations (μ ι ) ι ∈ with I ⊂ N. The valuation μ i is defined by induction from the valuation μ i-1 , from a key polynomial o i and from the value γ, = μ(oi). In this article we study some properties of the augmented valuations and we generalize the results of MacLane to the case of any valuation v of K. For this we need to introduce simple admissible families of augmented valuations A (μα) α ∈A where A is not necessarily a countable set, and to define a limit key polynomial and limit augmented valuation for such families. Then. any extension μ to L of a valuation ν on K is again a limit of a family of augmented valuations. We also get a "factorization" theorem which gives a description of the values (μ α (f)) for any polynomial f in K [x] .

Nonsmooth analysis on smooth manifolds

Abstract: We study infinitesimal properties of nonsmooth (nondifferentiable) functions on smooth manifolds. The eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function. A subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems, nonclassical problems of calculus of variations, and generalized solutions of first-order partial differential equations on manifolds. We also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions.

Allee effect and bistability in a spatially heterogeneous predator-prey model

Abstract: A spatially heterogeneous reaction-diffusion system modelling predator-prey interaction is studied, where the interaction is governed by a Holling type II functional response. Existence of multiple positive steady states and global bifurcation branches are examined as well as related dynamical behavior. It is found that while the predator population is not far from a constant level, the prey population could be extinguished, persist or blow up depending on the initial population distributions, the various parameters in the system, and the heterogeneous environment. In particular, our results show that when the prey growth is strong, the spatial heterogeneity of the environment can play a dominant role for the presence of the Allee effect. Our mathematical analysis relies on bifurcation theory, topological methods, various comparison principles and elliptic estimates. We combine these methods with monotonicity arguments to the system through the use of some new auxiliary scalar equations, though the system itself does not keep an order structure as the competition system does. Among other things, this allows us to obtain partial descriptions of the dynamical behavior of the system.

On symplectic fillings of lens spaces

Abstract: Le ξ st be the contact structure naturally induced on the lens space L(p, q) = S 3 /Z/pZ by the standard contact structure ξ st on the three-sphere S 3 . We obtain a complete classification of the symplectic fillings of (L(p, q),(ξ st ) up to orientation-preserving diffeomorphisms. In view of our results, we formulate a conjecture on the diffeomorphism types of the smoothings of complex two-dimensional cyclic quotient singularities.

Projectivity and freeness over comodule algebras

Abstract: Let H be a Hopf algebra and A an H-simple right H-comodule algebra It is shown that under certain hypotheses every (H, A)-Hopf module is either projective or free as an A-module and A is either a quasi-Frobenius or a semisimple ring As an application it is proved that every weakly finite (in particular, every finite dimensional) Hopf algebra is free both as a left and a right module over its finite dimensional right coideal subalgebras, and the latter are Frobenius algebras Similar results are obtained for H-simple H-module algebras

General logical metatheorems for functional analysis

Abstract: In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, Holder-Lipschitz, uniformly continuous, bounded and weakly quasi-nonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.

Asymptotic estimates of multi-dimensional stable densities and their applications

Abstract: The relation between the upper and lower asymptotic estimates of the density and the fractal dimensions on the sphere of the spectral measure for a multivariate stable distribution is discussed. In particular, the problem and the conjecture on the asymptotic estimates of multivariate stable densities in the work of Pruitt and Taylor in 1969 are solved. The proper asymptotic orders of the stable densities in the case where the spectral measure is absolutely continuous on the sphere, or discrete with the support being a finite set, or a mixture of such cases are obtained. Those results are applied to the moment of the last exit time from a ball and the Spitzer type limit theorem involving capacity for a multi-dimensional transient stable process.

Finite generation of symmetric ideals

Abstract: Let be a commutative Noetherian ring, and let be the polynomial ring in an infinite collection of indeterminates over . Let be the group of permutations of . The group acts on in a natural way, and this in turn gives the structure of a left module over the group ring . We prove that all ideals of invariant under the action of are finitely generated as -modules. The proof involves introducing a certain well-quasi-ordering on monomials and developing a theory of Grobner bases and reduction in this setting. We also consider the concept of an invariant chain of ideals for finite-dimensional polynomial rings and relate it to the finite generation result mentioned above. Finally, a motivating question from chemistry is presented, with the above framework providing a suitable context in which to study it.

The Auslander-Reiten translation in submodule categories

Abstract: Let A be an artin algebra or, more generally, a locally bounded associative algebra, and S(Λ) the category of all embeddings (A C B) where B is a finitely generated A-module and A is a submodule of B. Then S(Λ) is an exact Krull-Schmidt category which has Auslander-Reiten sequences. In this manuscript we show that the Auslander-Reiten translation in S(Λ) can be computed within mod A by using our construction of minimal monomorphisms. If in addition A is uniserial, then any indecomposable nonprojective object in S(Λ) is invariant under the sixth power of the Auslander-Reiten translation.

A sharp form of the Moser-Trudinger inequality on a compact Riemannian surface

Abstract: In this paper, a sharp form of the Moser-Trudinger inequality is established on a compact Riemannian surface via the method of blow-up analysis, and the existence of an extremal function for such an inequality is proved

Cohomogeneity one actions on noncompact symmetric spaces of rank one

Abstract: We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces CH", n ≥ 3. For the quaternionic hyperbolic spaces MH n , n > 3, we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classification problem was essentially solved by Elie Cartan.

Convolutions and multiplier transformations of convex bodies

Abstract: Rotation intertwining maps from the set of convex bodies in into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even Blaschke-Minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even Blaschke-Minkowski homomorphism maps a convex body to a polytope, then it is a constant multiple of the projection body operator

QUANTUM COHOMOLOGY AND THE k-SCHUR BASIS

Abstract: We prove that structure constants related to Hecke algebras at roots of unity are special cases of k-Littlewood-Richardson coefficients associated to a product of k-Schur functions. As a consequence, both the 3-point Gromov-Witten invariants appearing in the quantum cohomology of the Grassmannian, and the fusion coefficients for the WZW conformal field theories associated to su(l) are shown to be k-Littlewood-Richardson coefficients. From this, Mark Shimozono conjectured that the k-Schur functions form the Schubert basis for the homology of the loop Grassmannian, whereas k-Schur coproducts correspond to the integral cohomology of the loop Grassmannian. We introduce dual k-Schur functions defined on weights of k-tableaux that, given Shimozono's conjecture, form the Schubert basis for the cohomology of the loop Grassmannian. We derive several properties of these functions that extend those of skew Schur functions.

Recursive subhomogeneous algebras

Abstract: We introduce and characterize a particularly tractable class of unital type 1 C*-algebras with bounded dimension of irreducible representations. Algebras in this class are called recursive subhomogeneous algebras, and they have an inductive description (through iterated pullbacks) which allows one to carry over from algebras of the form many of the constructions relevant in the study of the stable rank and K-theory of simple direct limits of homogeneous C*-algebras. Our characterization implies, in particular, that if is a separable C*-algebra whose irreducible representations all have dimension at most and if for each the space of -dimensional irreducible representations has finite covering dimension, then is a recursive subhomogeneous algebra. We demonstrate the good properties of this class by proving subprojection and cancellation theorems in it. Consequences for simple direct limits of recursive subhomogeneous algebras, with applications to the transformation group C*-algebras of minimal homeomorphisms, will be given in separate papers.

Rational irreducible characters and rational conjugacy classes in finite groups

Abstract: We prove that a finite group G has two rational-valued irreducible characters if and only if it has two rational conjugacy classes, and determine the structure of any such group. Along the way we also prove a conjecture of Gow stating that any finite group of even order has a non-trivial rational-valued irreducible character of odd degree.

Extensions of $p$-local finite groups

Abstract: A p-local finite group consists of a finite p-group S, together with a pair of categories which encode "conjugacy" relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we study and classify extensions of p-local finite groups, and also compute the fundamental group of the classifying space of a p-local finite group.

Wiener’s lemma for infinite matrices

Abstract: In this paper, we introduce a class of infinite matrices related to the Beurling algebra of periodic functions, and we show that it is an inverse-closed subalgebra of \({\mathcal{B}}(\ell^{q}_{w})\), the algebra of all bounded linear operators on the weighted sequence space \(\ell^{q}_{w}\), for any 1≤q<∞ and any discrete Muckenhoupt Aq-weight w.

Spectral zeta functions of fractals and the complex dynamics of polynomials

Abstract: We obtain formulas for the spectral zeta function of the Laplacian on symmetric finitely ramified fractals, such as the Sierpinski gasket, and a fractal Laplacian on the interval. These formulas contain a new type of zeta function associated with a polynomial (rational functions also can appear in this context). It is proved that this zeta function has a meromorphic continuation to a half-plane with poles contained in an arithmetic progression. It is shown as an example that the Riemann zeta function is the zeta function of a quadratic polynomial, which is associated with the Laplacian on an interval. The spectral zeta function of the Sierpinski gasket is a product of the zeta function of a polynomial and a geometric part; the poles of the former are canceled by the zeros of the latter. A similar product structure was discovered by M.L. Lapidus for self-similar fractal strings.

Theory of valuations on manifolds, III. Multiplicative structure in the general case

Abstract: This is the third part of a series of articles where the theory of valuations on manifolds is constructed. In the second part of this series the notion of a smooth valuation on a manifold was introduced. The goal of this article is to put a canonical multiplicative structure on the space of smooth valuations on general manifolds, thus extending some of the affine constructions from the first author's 2004 paper and, from the first part of this series.

Central invariants and higher indicators for semisimple quasi-Hopf algebras

Abstract: In this paper, we define the higher Frobenius-Schur (FS-)indicators for finite-dimensional modules of a semisimple quasi-Hopf algebra H via the categorical counterpart developed in a 2005 preprint. When H is an ordinary Hopf algebra, we show that our definition coincides with that introduced by Kashina, Sommerhauser, and Zhu. We find a sequence of gauge invariant central elements of H such that the higher FS-indicators of a module V are obtained by applying its character to these elements. As an application, we show that FS-indicators are sufficient to distinguish the four gauge equivalence classes of semisimple quasi-Hopf algebras of dimension eight corresponding to the four fusion categories with certain fusion rules classified by Tambara and Yamagami. Three of these categories correspond to well-known Hopf algebras, and we explicitly construct a quasi-Hopf algebra corresponding to the fourth one using the Kac algebra. We also derive explicit formulae for FS-indicators for some quasi-Hopf algebras associated to group cocycles.

On operator-valued Fourier multiplier theorems

Abstract: The classical Fourier multiplier theorems of Marcinkiewicz and Mikhlin are extended to vector-valued functions and operator-valued multiplier functions on Z d or R d which satisfy certain R-boundedness conditions.