George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Mohammad Rouzbehani, School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran; e-mail: rouzbehani.m.math@gmail.com

Abstract/Résumé:

We show that a C*-algebra with topological dimension zero is purely infinite if it is weakly purely infinite (a question of Kirchberg and Rørdam). We give an application of this result.

On démontre qu’une C*-algèbre de dimension topologique égale à zéro est purement infinie si elle est faiblement purement infinie (une question de Kirchberg et de Rørdam). On donne une application de ce résultat.

Keywords: (weak) pure infiniteness, C*-algebra, topological dimension zero

AMS Subject Classification: General theory of $C^*$-algebras 46L05

PDF(click to download): Weakly Purely Infinite C*-algebras with Topological Dimension Zero are Purely Infinite

Gabriela Estevez, Instituto de Matematica e Estatıstica, Universidade Federal Fluminense, Rua Prof. Marcos Waldemar de Freitas Reis, S/N, 24.210-201, Bloco H, Campus do Gragoata, Niteroi, Rio de Janeiro RJ, Brasil; e-mail: gestevez@id.uff.br

Michael Yampolsky, Department of Mathematics, University of Toronto, 40 St George Street, Toronto, Ontario, Canada; e-mail: yampol@math.toronto.edu

Abstract/Résumé:

We construct a hyperbolic attractor of renormalization of bi-cubic circle maps with bounded combinatorics, with a codimension-two stable foliation.

On construit un attracteur hyperbolique de renormalisation pour une application de cercle bicubique à combinatoire bornée, dont la foliation est de codimension deux.

Keywords: Bi-cubic circle maps, bounded type rotation number, hyperbolicity of renormalization, renormalization operator

AMS Subject Classification: Maps of the circle, Universality; renormalization 37E10, 37E20

PDF(click to download): Hyperbolicity of Renormalization for Bi-cubic Circle Maps with Bounded Combinatorics

Igor Kukavica, Department of Mathematics, University of Southern California, Los Angeles, CA 90089; e-mail: kukavica@usc.edu

Linfeng Li, Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095; e-mail: lli265@math.ucla.edu

Amjad Tuffaha, Department of Mathematics and Statistics, American University of Sharjah, Sharjah, UAE; e-mail: atufaha@aus.edu

Abstract/Résumé:

We provide a maximal regularity theorem for the linear Stokes equation with a non-homogeneous divergence condition in a bounded domain \(\Omega \subseteq \mathbb{R}^3\) and with the Neumann boundary conditions. We prove the existence and uniqueness of solutions such that the velocity belongs to the space \(H^{(s+1)/2,s+1}((0,T) \times \Omega)\), where \(s\in [1, 1.5 )\cup (1.5, 2)\).

Nous fournissons un théorème de régularité maximale pour l’équation linéaire de Stokes avec une condition de divergence non homogène dans un domaine borné \(\Omega \subseteq \mathbb{R}^3\) et avec les conditions aux limites de Neumann. On prouve l’existence et l’unicité de solutions telles que la vitesse appartient à l’espace \(H^{(s+1)/2,s+1}((0,T) \times \Omega)\), où \(s\in [1, 1.5 )\cup (1.5, 2)\).

Keywords: Local existence, Navier-Stokes equations, maximal regularity, trace regularity

AMS Subject Classification: , , Smoothness and regularity of solutions of PDE 35-11, 35A01, 35B65

PDF(click to download): Maximal regularity for the Neumann-Stokes problem in $H^{r/2,r}$ spaces

Yoshiki Aibara, Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan; e-mail: y.aibara.math95@gmail.com

Yoshimichi Ueda, Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan; e-mail: ueda@math.nagoya-u.ac.jp

Abstract/Résumé:

We explain how Pusz–Woronowicz’s notion of functional calculus fits the theory of Lebesgue decomposition for positive operators on Hilbert spaces initially developed by Ando. In this way, we reconstruct the essential and fundamental part of the theory.

On montre comment la notion de calcul fonctionnel de Pusz–Woronomicz s’adapte à la théorie de décomposition de Lebesgue pour les opérateurs positifs sur un espace de Hilbert, initialement dévelopée par Ando. De cette façon on reconstruit la partie essentielle et fondamentale de cette théorie.

Keywords: Binary operation, Functional calculus, Lebesgue decomposition, Positive operator

AMS Subject Classification: None of the above; but in this section, Functional calculus, Operator means; shorted operators; etc., 28E99, 47A60, 47A64, 47B02

PDF(click to download): Lebesgue Decomposition for Positive Operators Revisited

George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Qingzhai Fan , Department of Mathematics, Shanghai Maritime University, Shangha, China 201306; e-mail: fanqingzhai@fudan.edu.cn,qzfan@shmtu.edu.cn

Xiaochun Fang, Department of Mathematics, Tongji University, Shanghai, China 200092; e-mail: xfang@tongji.edu.cn

Abstract/Résumé:

In this paper, we introduce some classes of generalized tracial approximation C*-algebras. Consider the class of unital C*-algebras which are tracially 𝒵-absorbing (or have tracial nuclear dimension at most n, or have the property SP, or are m-almost divisible). Then A is tracially 𝒵-absorbing (respectively, has tracial nuclear dimension at most n, has the property SP, is weakly (n, m)-almost divisible) for any simple unital C*-algebra A in the corresponding class of generalized tracial approximation C*-algebras. As an application, let A be an infinite-dimensional unital simple C*-algebra, and let B be a centrally large subalgebra of A. If B is tracially 𝒵-absorbing, then A is tracially 𝒵-absorbing. This result was obtained by Archey, Buck, and Phillips in Archey et al. (2018).

On introduit la notion d’approximation traciale généralisée d’une C*-algèbre par des C*-algèbres dans une class donnée. Cette notion généralise la notion de Lin d’approximation triviale simple, et aussi la notion d’Archey et de Phillips de centralement grande sousalgèbre, deux notions qui se sont démontrées très importantes.

Keywords: Cuntz Semigroup, C∗-algebras, tracial approximation

AMS Subject Classification: General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 46L05, 46L35, 46L80

PDF(click to download): Generalized Tracially Approximated C*-algebras

Kam Hang Cheng, Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong; e-mail: henry.cheng@family.ust.hk

Yik Man Chiang, Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong; e-mail: machiang@ust.hk

Avery Ching, Department of Statistics, The University of Warwick, Coventry, CV4 7AL, United Kingdom. (Current address: Department of Mathematics, University of Dundee, Nethergate, Dundee DD1 4HN, Scotland, United Kingdom. e-mail: AChing001@dundee.ac.uk); e-mail: avery.ching@warwick.ac.uk

Abstract/Résumé:

This is a research announcement on a unifying study of generating functions of various sequences of special functions, using Bernstein’s theory of holonomic \(D\)-modules. Both new and well-known generating functions have been obtained in a systematic and algebraic way. New difference analogues of some special functions are also discovered. This announcement focuses on particular results about Hermite functions, Bessel functions and polynomials, Laguerre polynomials, and Gegenbauer polynomials.

Il s’agit d’une annonce de recherche sur une étude unificatrice des fonctions génératrices de diverses séquences de fonctions spéciales, en utilisant la théorie de Bernstein des \(D\)-modules holonomes. Des fonctions génératrices nouvelles et bien connues ont été obtenues de manière systématique et algébrique. De nouveaux analogues discrets de certaines fonctions spéciales sont également découverts. Cette annonce se concentre sur des résultats particuliers concernant les fonctions d’Hermite, les fonctions et polynômes de Bessel, les polynômes de Laguerre et les polynômes de Gegenbauer.

Keywords: D-modules, generating functions, holonomic systems of PDEs, special functions, transmutation formulae

AMS Subject Classification: Representations of entire functions by series and integrals, Monodromy; relations with differential equations and $D$-modules, Orthogonal polynomials and functions of hypergeometric type (Jacobi; Laguerre; Hermite; Askey scheme; etc.), None of the above; but in this section, Weyl theory and its generalizations, Algebraic aspects (differential-algebraic; hypertranscendence; group-theoretical), Commutators; derivations; elementary operators; etc. 30D10, 32S40, 33C45, 33E99, 34B20, 34M15, 47B47

PDF(click to download): $D$-module Approach to Special Functions and Generating Functions

Charles Starling, Carleton University, School of Mathematics and Statistics. 4302 Herzberg Laboratories, Ottawa ON, K1S 5B6; e-mail: cstar@math.carleton.ca

Abstract/Résumé:

We prove a new uniqueness theorem for the tight C*-algebras of an inverse semigroup by generalising the uniqueness theorem given for étale groupoid C*-algebras by Brown, Nagy, Reznikoff, Sims, and Williams. We use this to show that in the nuclear and Hausdorff case, a *-homomorphism from the boundary quotient C*-algebra of a right LCM monoid is injective if and only if it is injective on the subalgebra generated by the core submonoid. We also use our result to clarify the identity of the tight C*-algebra of an inverse semigroup we previously associated to a subshift and erroneously identified as the Carlsen-Matsumoto algebra.

Nous prouvons un nouveau thèoréme d’unicité pour les C*-algèbres serrées d’un semi-groupe inverse en généralisant le théorème d’unicité donné pour les C*-algèbres groupoides étales par Brown, Nagy, Reznikoff, Sims et Williams. Nous utilisons ceci pour montrer que dans le cas nucléaire et de Hausdorff, un *-homomorphisme de l’algèbre C* du quotient aux limites d’un monoïde LCM droit est injectif si et seulement s’il est injectif sur la sous-algèbre générée par le sous-monoide de noyau. Nous utilisons également notre résultat pour clarifier l’identité de l’algèbre C* serrée d’un semi-groupe inverse que nous avons précédemment associé à un sous-décalage et identifié à tort comme l’algèbre de Carlsen-Matsumoto.

Keywords: C*-algebra, LCM semigroup, groupoid, inverse semigroup, subshift, tight representation, uniqueness

AMS Subject Classification: Groupoids; semigroupoids; semigroups; groups (viewed as categories), Inverse semigroups, General theory of $C^*$-algebras 18B40, 20M18, 46L05

PDF(click to download): A New Uniqueness Theorem for the Tight C*-algebra of an Inverse Semigroup

Juncheng Wei, FRSC, Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada; e-mail: jcwei@math.ubc.ca

Qidi Zhang, Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada; e-mail: qidi@math.ubc.ca

Yifu Zhou, Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA; e-mail: yzhou173@jhu.edu

Abstract/Résumé:

Singularity formation for evolution equations has attracted much
attention in recent years. In this survey article, we will introduce
some recent progress on the parabolic gluing
method and its applications in investigating the mechanism
of singularity formation for parabolic flows. Two model problems will be
revisited to illustrate the ideas, and recent developments and
techniques will be presented.

La formation de singularités pour les équations d’évolution a attiré
beaucoup d’attention ces dernières années. Dans cet article d’enquête,
nous présenterons quelques progrès récents sur la méthode de
collage parabolique et ses applications dans l’étude du
mécanisme de formation de singularités pour les écoulements
paraboliques. Deux problèmes modèles seront revisités pour illustrer les
idées, et les développements et techniques récents seront présentés.

Keywords: Blow-up, Fujita equation, Sobolev critical exponent, parabolic gluing method, slow decay

AMS Subject Classification: Asymptotic behavior of solutions, 35B40, 35K58

PDF(click to download): On the Parabolic Gluing Method and Singularity Formation

Vesna Iršič, Faculty of Mathematics and Physics, University of Ljubljana and Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia; e-mail: vesna.irsic@fmf.uni-lj.si

Bojan Mohar, FRSC, Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada; e-mail: mohar@sfu.ca

Alexandra Wesolek, Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada; e-mail: agwesole@sfu.ca

Abstract/Résumé:

A recently introduced variation of the game of cops and robber is played on geodesic spaces. In this paper we establish some general strategies for the players, in particular the generalized radial strategy and the covering space strategy. Those strategies are then applied to the game on the \(n\)-dimensional ball, the sphere, and the torus.

Le jeu aux gendarmes et aux voleurs sur les espaces géodésiques est analysé. On établit quelques stratégies générales pour les joueurs, en particulier la stratégie radiale généralisée et la stratégie d’espaces de couverture universels. Ces stratégies sont ensuite appliquées aux jeux sur la boule de dimension \(n\), sur la sphère, et sur le tore.

Keywords: Game of cops and robber, geodesic space, radial strategy

AMS Subject Classification: Metric spaces; metrizability, Differential games, Discrete-time games 54E35, 91A23, 91A50

PDF(click to download): Cops and Robber Game in Higher-dimensional Manifolds with Spherical and Euclidean Metric

Askold Khovanskii, FRSC, University of Toronto, Toronto, Canada; e-mail: askold@math.toronto.edu

Sushil Singla, Department of Mathematics, Shiv Nadar University, Greater Noida, India 201314; e-mail: ss774@snu.edu.in

Aaron Tronsgard, University of Toronto, Toronto, Canada; e-mail: tronsgar@math.utoronto.ca

Abstract/Résumé:

We reconsider the theory of Lagrange interpolation polynomials with multiple interpolation points and apply it to linear algebra. In particular, we show that one can evaluate a meromorphic function at a matrix, using only an interpolation polynomial.

On reconsidère la thèorie des polynômes d’interpolation de Lagrange et l’applique à l’algèbre linéaire. En particulier, on peut évaluer une fonction méromorphe à une matrice seulement avec un polynôme d’interpolation.

Keywords: Cayley Hamilton theorem, Interpolation polynomials, meromorphic function at a matrix

AMS Subject Classification: Instructional exposition (textbooks; tutorial papers; etc.), , Canonical forms; reductions; classification, Interpolation 15-01, 15A16, 15A21, 41A05

PDF(click to download): Interpolation Polynomials and Linear Algebra

Nathan Grieve, Department of Mathematics & Computer Science, Royal Military College of Canada, P.O. Box 17000, Station Forces, Kingston, ON, K7K 7B4, Canada; School of Mathematics and Statistics, 4302 Herzberg Laboratories, Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6, Canada; Departement de mathematiques, Universite du Quebec a Montreal, Local PK-5151, 201 Avenue du President-Kennedy, Montreal, QC, H2X 3Y7, Canada; e-mail: nathan.m.grieve@gmail.com

Abstract/Résumé:

We revisit work of S. Boucksom, C. Favre, and M. Jonsson (J. Algebraic Geom. 18 (2009), no. 2, 279–308); Boucksom and H. Chen (Compos. Math. 147 (2011), no. 4, 1205–1229); and S. Boucksom, A. Küronya, C. Maclean, and T. Szemberg (Math. Ann. 361 (2015), no. 3–4, 811–834). The key point is to associate a Duistermaat-Heckman measure to a filtered big linear series on a given projective variety. The expectation of the measure admits a description via the theory of Newton-Okounkov bodies. Such considerations have origins in symplectic geometry. They have applications for \(\mathrm{K}\)-stability and Diophantine arithmetic geometry of projective varieties.

Nous revisitons les travaux de S. Boucksom, C. Favre, and M. Jonsson (J. Algebraic Geom. 18 (2009), no. 2, 279–308); Boucksom et H. Chen (Compos. Math. 147 (2011), no. 4, 1205–1229); et S. Boucksom, A. Küronya, C. Maclean, et T. Szemberg (Math. Ann. 361 (2015), no. 3–4, 811–834). Nous étudions deux résultats, qui sont à l’intersection de la \(\mathrm{K}\)-stabilité, de la géométrie arithmétique Diophantienne, et de la théorie des corps convexe de Newton-Okounkov. Ils se rapportent à des filtrations de grands systèmes linéaires sur des variétés projectives et à l’existence de une mesure de Duistermaat-Heckman. La mesure de Duistermaat-Heckman vient de la géométrie symplectique. L’espérance de la mesure peut être calculée par la théorie de Newton-Okounkov.

Keywords: Duistermaat-Heckman measure, Newton-Okounkov body, restricted volume functions

AMS Subject Classification: Valuations and their generalizations, Divisors; linear systems; invertible sheaves 13A18, 14C20

PDF(click to download): On Duistermaat-Heckman Measure for Filtered Linear Series

George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Abstract/Résumé:

It is shown that for a unital C*-algebra, what is sometimes referred to as the Elliott invariant—loosely speaking, K-theory and traces— i.e., the order-unit K\(_0\)-group, the K\(_1\)-group, and the trace simplex, paired in the natural way with K\(_0\), can be expressed purely in terms of K-theory, with the trace simplex and its pairing with K\(_0\) recoverable in a simple way (using polar decomposition) from algebraic K\(_1\), defined as in the purely algebraic context using invertible elements rather than just unitaries.

L’invariant naïf d’Elliott, qui est à la base de la classification complète récente d’une énorme classe de C*-algèbres simples (celles qui sont de dimension nucléaire finie, qui sont séparables, et qui satisfont à l’UCT), peut s’exprimer entièrement dans le cadre de K-théorie algébrique.

Keywords: Algebraic K1-group of a C*-algebra encodes bounded traces

AMS Subject Classification: None of the above; but in this section, $K_0$ as an ordered group; traces, General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 19B99, 19K14, 46L05, 46L35, 46L80

PDF(click to download): K-Theory and Traces

George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Klaus Thomsen, Department of Mathematics, Aarhus University, Ny Munkegade, 8000 Aarhus C, Denmark; e-mail: matkt@math.au.dk

Abstract/Résumé:

It is shown that, for any unital simple infinite-dimensional AF algebra, the KMS-state bundle for a one-parameter automorphism group is isomorphic to an arbitrary proper simplex bundle over the real line with (as is necessary) fibre at (inverse temperature) zero isomorphic to the trace simplex.

On démontre que, pour toute C*-algèbre AF simple à élément unité et à dimension infinie, le faisceau d’états KMS pour un grouped’automorphismes à un paramètre est isomorphe à un faisceau de simplices propre arbitraire sur la ligne réelle tel que (nécessairement) le fibre sur la température inverse zéro est isomorphe au simplex tracial.

Keywords: One-parameter automorphism group of an AF algebra, bundle of equilibrium (KMS) states arbitrary

AMS Subject Classification: General theory of $C^*$-algebras, Noncommutative dynamical systems 46L05, 46L55

PDF(click to download): The Bundle of KMS State Spaces for Flows on a Unital AF C*-algebra

Bit Na Choi, Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA; e-mail: Bitna.Choi@unh.edu

Andrew J. Dean, Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON, Canada P7B 5E1; e-mail: ajdean@lakeheadu.ca

Abstract/Résumé:

We show that a \(\mathbb{Z}_2\) action on a lattice-ordered dimension group will arise as an inductive limit of \(\mathbb{Z}_2\) actions on simplicial groups. The motivation for this study is the range of invariant problem in Elliott and Su’s classification of AF type \(\mathbb{Z}_2\) actions. We modify the proof of the Effros-Handelman-Shen theorem to include \(\mathbb{Z}_2\) actions.

Nous montrons qu’une action de \(\mathbb{Z}_2\) sur un groupe de dimension ordonné par treillis apparaît comme une limite inductive d’actions de \(\mathbb{Z}_2\) sur des groupes simpliciaux. La motivation de cette étude est le problème de la gamme de l’invariant dans la classification d’Elliott et de Su des actions de \(\mathbb{Z}_2\) de type AF. Nous modifions la preuve du théorème d’Effros-Handelman-Shen pour inclure les actions de \(\mathbb{Z}_2\).

Keywords: Dimension groups, K-theory, classification

AMS Subject Classification: Ordered abelian groups; Riesz groups; ordered linear spaces, $K_0$ as an ordered group; traces 06F20, 19K14

PDF(click to download): A Modification of the Effros-Handelman-Shen Theorem with ${Z}_2$ actions

Michael Yampolsky, Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 2E4; e-mail: yampol@math.toronto.edu

Abstract/Résumé:

In this note, we extend the renormalization horseshoe we have recently constructed with N. Goncharuk for analytic diffeomorphisms of the circle to their small two-dimensional perturbations. As one consequence, Herman rings with rotation numbers of bounded type survive on a codimension one set of parameters under small two-dimensional perturbations.

On étend le fer à cheval de renormalisation récemment construit avec N. Goncharuk pour les difféomorphismes analytiques du cercle à leurs petites perturbations à deux dimensions. Il suit que les anneaux de Herman à nombre de rotation de type borné survivent sur un ensemble de paramètres à codimension un sous petites perturbations à deux dimensions.

Keywords: Henon-like maps, Herman rings, rotation domains

AMS Subject Classification: Renormalization, Small divisors; rotation domains and linearization; Fatou and Julia sets, 37F25, 37F50, 37F80

PDF(click to download): KAM-renormalization and Herman Rings for 2D Maps

M. Manickam , Indian Institute of Science Education and Research Bhopal, Madhya Pradesh 462066 INDIA; e-mail: manickam@iiserb.ac.in, murugumanick@gmail.com

V. Kumar Murty, FRSC,Department of Mathematics, University of Toronto, Ontario, Canada, M5S 2E4; e-mail: murty@math.toronto.edu

E. M. Sandeep, Kerala School of Mathematics, Kunnamangalam, Kozhikode-673571, Kerala INDIA; e-mail: sandeep@ksom.res.in, mepeednas@gmail.com

Abstract/Résumé:

We consider a kernel function introduced by Kohnen and prove an asymptotic formula for a weighted sum of \(L\)-functions of modular forms.

On considère une fonction noyau introduite par Kohnen et démontre une formule asymptotique pour une somme pondérée de fonctions L de forms modulaires.

Keywords: Hecke eigenforms, Modular L-function, cusp forms, full modular group, integral weight, lower bound, non-vanishing.

AMS Subject Classification: Modular forms; one variable, Automorphic forms; one variable, Dirichlet series and functional equations in connection with modular forms 11F11, 11F12, 11F66

PDF(click to download): A Weighted Average of L-functions of Modular Forms

Luc Vinet, FRSC ,Centre de Recherches Mathematiques, Universite de Montreal, P.O. Box 6128, Centre-ville Station, Montreal (Quebec), H3C 3J7, Canada and Centre de Recherches Mathematiques, Universite de Montreal, P.O. Box 6128, Centre-ville Station, Montreal (Quebec), H3C 3J7, Canada; e-mail: vinet@CRM.UMontreal.CA

Alexei Zhendanov, School of Mathematics, Renmin University of China, Beijing, 100872, China; e-mail: zhedanov@yahoo.com

Abstract/Résumé:

Eigenvalue problems on irreducible \(\mathfrak{su}(2)\) modules and their adjoints are considered in the Bargmann, Barut-Girardello and finite difference models. The biorthogonality relations that arise between the corresponding generating functions of the Krawtchouk polynomials are sorted out. A link with Padé approximation is made.

Des problèmes aux valeurs propres sur les modulesirréductibles de \(\mathfrak{su}(2)\) et leurs adjoints sont examinés dans les modèles de Bargmann, Barut–Girardello et aux différences finies. Les relations de biorthogonalité qui apparaissent entre les fonctions génératrices correspondantes des polynômes de Krawtchouk sont identifiées. Un lien avec l’approximation de Padé est fait.

Keywords: Krawtchouk polynomials, Pade approximation., biorthogonality, generating functions, su(2) models

AMS Subject Classification: Representations; algebraic theory (weights), Orthogonal polynomials and functions of hypergeometric type (Jacobi; Laguerre; Hermite; Askey scheme; etc.), Pad_¸ approximation 17B10, 33C45, 41A21

PDF(click to download): A Note on su(2) Models and the Biorthogonality of Generating Functions of Krawtchouk Polynomials

George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Abstract/Résumé:

The flow of weights was introduced by Connes and Takesaki as a functor on the category of von Neumann algebras with isomorphisms as maps. While it is easy to see that this functor cannot be extended to the category of all von Neumann algebra homomorphisms, it is in fact possible to extend it to a certain extent. This can also be done, fairly extensively, for the Falcone and Takesaki non-commutative flow of weights.

Le flot des poids a été introduit par Connes et Takesaki comme foncteur sur la catégorie des algèbres de von Neumann avec isomorphismes comme flèches. On peut étendre ce foncteur jusqu’à un certain point dans les directions et covariante et contravariante. Le foncteur flot des poids non-commutatif peut aussi s’étendre, bien entendant pas aux homomorphismes arbitraires.

Keywords: Connes-Takesaki flow of weights, functoriality, non-commutative flow of weights

AMS Subject Classification: , Categories; functors 46L36, 46M15

PDF(click to download): A Remark on the Functoriality of the Connes-Takesaki Flow of Weights

Francois Lalonde, Département de mathématiques et de statistique, Université de Montréal, Montréal, Québec, Canada; e-mail: lalonde@dms.umontreal.ca

Abstract/Résumé:

In this paper, I explain the emergence of Symplectic geometry and Symplectic topology, as it occurred historically, from three sources: classical and quantum physics, complex Algebraic geometry, and String theory. Symplectic topology is nowadays one of the most fascinating subjects in mathematics, and has reached since the 1980’s a maturity that deserves the attention of all mathematicians and physicists. It combines topology, geometry, non-linear partial differential equations or relations, \(A^{\infty}\) algebras, and String theory in a powerful setting that addresses some of the most elusive questions of our times. It is made of soft \(h\)-principles coupled with hard transcendental, rigid, moduli spaces of solutions to PDE’s on manifolds. I will end the paper with some conjectures in Symplectic topology, after explaining the difference between smooth and \(C^0\) Symplectic topology.

Dans cet article, j’explique l’émergence de la géométrie symplectique et de la topologie symplectique, en suivant leur naissance et leur évolution au cours des siècles, à partir de trois sources: la physique classique et quantique, la géométrie algébrique complexe, et la théorie des cordes. La topologie symplectique est aujourd’hui l’un des domaines les plus fascinants de la recherche mathématique mondiale et a atteint, depuis les années 1980 une maturité qui mérite l’attention de tous les mathématiciens et physiciens. Elle rassemble la topologie, la géométrie, les équations aux dérivées partielles non-linéaires, les \(A^{\infty}\)-algèbres et la théorie des cordes dans une théorie puissante qui s’adresse aux problèmes les plus subtils de notre époque. Elle est faite du \(h\)-principe topologique, une théorie “soft”, couplée à un vaste ensemble d’espaces de modules d’EDP sur les variétés, que l’on peut qualifier de “hard” ou transcendental. Je conclurai cet article avec quelques conjectures en topologie symplectique après avoir expliqué la différence entre les topologies symplectiques lisse et \(C^0\).

Keywords: $C^0$-Symplectic topology, Capacities, Gromov-Witten invariants, Hamiltonian dynamics, Lagrangian submanifolds, String theory, Symplectic topology

AMS Subject Classification: Global theory of symplectic and contact manifolds, Spectral sequences and homology of fiber spaces, Symplectic and contact topology, Topological properties of groups of homeomorphisms or diffeomorphisms 53D35, 55R20, 57R17, 57S05

PDF(click to download): $C^0$ Symplectic Topology

Guihua Gong, Department of Mathematics, Hebei Normal University, Shijiazhuang, Hebei 050016, China
and Department of Mathematics, University of Puerto Rico, Rio Piedras, PR 00936, USA; e-mail: ghgong@gmail.com

Huaxin Lin, Department of Mathematics, East China Normal University, Shanghai 200062, China and
(Current) Department of Mathematics, University of Oregon, Eugene, Oregon, 97402, USA; e-mail: hlin@uoregon.edu

Zhuang Niu, Department of Mathematics, University of Wyoming, Laramie, WY, USA, 82071; e-mail: zniu@uwyo.edu,

Abstract/Résumé:

A classification theorem is obtained for a class of unital simple separable amenable \({\cal Z}\)-stable C*-algebras which exhausts all possible values of the Elliott invariant for unital stably finite simple separable amenable \({\cal Z}\)-stable C*-algebras. Moreover, it contains all unital simple separable amenable C*-algebras which satisfy the UCT and have finite rational tracial rank.

Dans cet article et le précédent on donne une classification complète, au moyen de l’invariant d’Elliott, d’une sous-classe de la classe des C*-algèbres simples, moyennables, séparables, à élément unité, absorbant l’algèbre de Jiang-Su, et satisfaisant au UCT, qui épuise l’ensemble des valeurs possibles de l’invariant pour cette class. La partie I réalise une grande partie de ce projet, et la partie II l’achève.

Keywords: Classication of Simple C*-algebras

AMS Subject Classification: General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 46L05, 46L35, 46L80

PDF(click to download): A Classification of Finite Simple Amenable Z-stable C*-algebras, II: C*-algebras with Rational Generalized Tracial Rank One