Noncommutative Curves of Genus Zero
Noncommutative Curves of Genus Zero

Author: Dirk Kussin

Publisher: American Mathematical Soc.

Published: 2009-08-07

Total Pages: 146

ISBN-13: 0821844008

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In these notes the author investigates noncommutative smooth projective curves of genus zero, also called exceptional curves. As a main result he shows that each such curve $\mathbb{X}$ admits, up to some weighting, a projective coordinate algebra which is a not necessarily commutative graded factorial domain $R$ in the sense of Chatters and Jordan. Moreover, there is a natural bijection between the points of $\mathbb{X}$ and the homogeneous prime ideals of height one in $R$, and these prime ideals are principal in a strong sense.

Symplectic, Poisson, and Noncommutative Geometry
Symplectic, Poisson, and Noncommutative Geometry

Author: Tohru Eguchi

Publisher: Cambridge University Press

Published: 2014-08-25

Total Pages: 303

ISBN-13: 1107056411

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This volume contains seven chapters based on lectures given by invited speakers at two May 2010 workshops held at the Mathematical Sciences Research Institute.

Noncommutative Differential Geometry and Its Applications to Physics
Noncommutative Differential Geometry and Its Applications to Physics

Author: Yoshiaki Maeda

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 310

ISBN-13: 9401007047

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Noncommutative differential geometry is a new approach to classical geometry. It was originally used by Fields Medalist A. Connes in the theory of foliations, where it led to striking extensions of Atiyah-Singer index theory. It also may be applicable to hitherto unsolved geometric phenomena and physical experiments. However, noncommutative differential geometry was not well understood even among mathematicians. Therefore, an international symposium on commutative differential geometry and its applications to physics was held in Japan, in July 1999. Topics covered included: deformation problems, Poisson groupoids, operad theory, quantization problems, and D-branes. The meeting was attended by both mathematicians and physicists, which resulted in interesting discussions. This volume contains the refereed proceedings of this symposium. Providing a state of the art overview of research in these topics, this book is suitable as a source book for a seminar in noncommutative geometry and physics.

Towards Non-Abelian P-adic Hodge Theory in the Good Reduction Case
Towards Non-Abelian P-adic Hodge Theory in the Good Reduction Case

Author: Martin C. Olsson

Publisher: American Mathematical Soc.

Published: 2011-02-07

Total Pages: 170

ISBN-13: 082185240X

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The author develops a non-abelian version of $p$-adic Hodge Theory for varieties (possibly open with ``nice compactification'') with good reduction. This theory yields in particular a comparison between smooth $p$-adic sheaves and $F$-isocrystals on the level of certain Tannakian categories, $p$-adic Hodge theory for relative Malcev completions of fundamental groups and their Lie algebras, and gives information about the action of Galois on fundamental groups.

Points and Curves in the Monster Tower
Points and Curves in the Monster Tower

Author: Richard Montgomery

Publisher: American Mathematical Soc.

Published: 2010-01-15

Total Pages: 154

ISBN-13: 0821848186

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Cartan introduced the method of prolongation which can be applied either to manifolds with distributions (Pfaffian systems) or integral curves to these distributions. Repeated application of prolongation to the plane endowed with its tangent bundle yields the Monster tower, a sequence of manifolds, each a circle bundle over the previous one, each endowed with a rank $2$ distribution. In an earlier paper (2001), the authors proved that the problem of classifying points in the Monster tower up to symmetry is the same as the problem of classifying Goursat distribution flags up to local diffeomorphism. The first level of the Monster tower is a three-dimensional contact manifold and its integral curves are Legendrian curves. The philosophy driving the current work is that all questions regarding the Monster tower (and hence regarding Goursat distribution germs) can be reduced to problems regarding Legendrian curve singularities.

The Creation of Strange Non-Chaotic Attractors in Non-Smooth Saddle-Node Bifurcations
The Creation of Strange Non-Chaotic Attractors in Non-Smooth Saddle-Node Bifurcations

Author: Tobias H. JŠger

Publisher: American Mathematical Soc.

Published: 2009-08-07

Total Pages: 120

ISBN-13: 082184427X

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The author proposes a general mechanism by which strange non-chaotic attractors (SNA) are created during the collision of invariant curves in quasiperiodically forced systems. This mechanism, and its implementation in different models, is first discussed on an heuristic level and by means of simulations. In the considered examples, a stable and an unstable invariant circle undergo a saddle-node bifurcation, but instead of a neutral invariant curve there exists a strange non-chaotic attractor-repeller pair at the bifurcation point. This process is accompanied by a very characteristic behaviour of the invariant curves prior to their collision, which the author calls `exponential evolution of peaks'.

Center Manifolds for Semilinear Equations with Non-Dense Domain and Applications to Hopf Bifurcation in Age Structured Models
Center Manifolds for Semilinear Equations with Non-Dense Domain and Applications to Hopf Bifurcation in Age Structured Models

Author: Pierre Magal

Publisher: American Mathematical Soc.

Published: 2009

Total Pages: 84

ISBN-13: 0821846531

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Several types of differential equations, such as delay differential equations, age-structure models in population dynamics, evolution equations with boundary conditions, can be written as semilinear Cauchy problems with an operator which is not densely defined in its domain. The goal of this paper is to develop a center manifold theory for semilinear Cauchy problems with non-dense domain. Using Liapunov-Perron method and following the techniques of Vanderbauwhede et al. in treating infinite dimensional systems, the authors study the existence and smoothness of center manifolds for semilinear Cauchy problems with non-dense domain. As an application, they use the center manifold theorem to establish a Hopf bifurcation theorem for age structured models.