Vol.37 (1) 2015 — 2 articles found.

Counting Toric Actions on Symplectic Four-Manifolds

C. R. Math. Rep. Acad. Sci. Canada Vol. 37 (1) 2015, pp. 33-40
Vol.37 (1) 2015
Y. Karshon; L. Kessler; M. Pinsonnault Details
(Received: 2014-09-15 , Revised: 2014-10-07 )
(Received: 2014-09-15 , Revised: 2014-10-07 )

Y. Karshon,Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5S 3G3; e-mail: karshon@math.toronto.edu

L. Kessler,Department of Mathematics, Physics, and Computer Science, University of Haifa, at Oranim, Tivon 36006, Israel; e-mail: liatke.math@gmail.com

M. Pinsonnault,Department of Mathematics, Middlesex College The University of Western Ontario London, Ontario N6A 5B7 Canada; e-mail: mpinson@uwo.ca

Abstract/Résumé:

Given a symplectic manifold, we ask in how many different ways can a torus act on it. Classification theorems in equivariant symplectic geometry can sometimes tell that two Hamiltonian torus actions are inequivalent, but often they do not tell whether the underlying symplectic manifolds are (non-equivariantly) symplectomorphic. For two dimensional torus actions on closed symplectic four-manifolds, we reduce the counting question to combinatorics, by expressing the manifold as a symplectic blowup in a way that is compatible with all the torus actions simultaneously.

Nous nous intéressons aux différentes actions d’un tore sur une variété symplectique donnée. En géométrie symplectique équivariante, les théorèmes de classification permettent parfois de distinguer des actions hamiltoniennes de tores géométriquement inéquivalentes. Par contre, ces théorèmes ne permettent habituellement pas de déterminer si les variétés symplectiques sous-jaçentes sont symplectomorphes. Dans le cas des variétés symplectiques de dimension \(4\), nous réduisons le problème d’énumération des actions toriques inéquivalentes à un problème combinatoire en exprimant la variété considérée comme un éclatement symplectique qui est compatible simultanément avec toutes les actions toriques. Ce résultat est obtenu en employant des techniques pseudo-holomorphes.

Keywords:

AMS Subject Classification: Momentum maps; symplectic reduction 53D20

PDF(click to download): Counting Toric Actions on Symplectic Four-Manifolds

A Classification of Tracially Approximate Splitting Interval Algebras. II. Existence Theorem

C. R. Math. Rep. Acad. Sci. Canada Vol. 37 (1) 2015, pp. 1–32
Vol.37 (1) 2015
Zhuang Niu Details
(Received: 2012-01-26 , Revised: 2013-03-26 )
(Received: 2012-01-26 , Revised: 2013-03-26 )

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

Abstract/Résumé:

Motivated by Huaxin Lin’s axiomatization of AH-algebras, the class of TASI-algebras is introduced as the class of unital C*-algebras which can be tracially approximated by splitting interval algebras—certain sub-C*-algebras of interval algebras. It is shown that the class of simple separable nuclear TASI-algebras satisfying the UCT is classified by the Elliott invariant. As a consequence, any such TASI-algebra is then isomorphic to an inductive limit of splitting interval algebras together with certain homogeneous C*-algebras—so it is an ASH-algebra.

Une classe de C*-algèbres qui généralisent la classe bien connue TAI de Lin est considérée—basées sur, au lieu de l’intervalle, ce qui pourrait s’appeler l’intervalle fendu ("splitting interval"), de sorte que l’on les appelle la classe TASI. On montre que la classe de C*-algèbres TASI qui sont simples, nucléaires, et à élément unité, qui vérifient le théorème à coefficients universels (UCT), peuvent se classifier d’après l’invariant d’Elliott.

Keywords: Classification of simple C*-algebras, inductive limits of sub-homogeneous C*- algebras, tracially approximate splitting interval algebras

AMS Subject Classification: Classifications of $C^*$-algebras; factors 46L35

PDF(click to download): A Classification of Tracially Approximate Splitting Interval Algebras. II. Existence Theorem

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