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Symplectic Actions of $2$-Tori on $4$-Manifolds

Symplectic Actions of $2$-Tori on $4$-Manifolds
Author: Alvaro Pelayo
Publisher: American Mathematical Soc.
Total Pages: 96
Release: 2010-02-22
Genre: Mathematics
ISBN: 0821847139

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In this paper the author classifies symplectic actions of $2$-tori on compact connected symplectic $4$-manifolds, up to equivariant symplectomorphisms. This extends results of Atiyah, Guillemin-Sternberg, Delzant and Benoist. The classification is in terms of a collection of invariants of the topology of the manifold, of the torus action and of the symplectic form. The author constructs explicit models of such symplectic manifolds with torus actions, defined in terms of these invariants.


The Topology of Torus Actions on Symplectic Manifolds

The Topology of Torus Actions on Symplectic Manifolds
Author: Michèle Audin
Publisher: Birkhäuser
Total Pages: 181
Release: 2012-12-06
Genre: Mathematics
ISBN: 3034872216

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The material and references in this extended second edition of "The Topology of Torus Actions on Symplectic Manifolds", published as Volume 93 in this series in 1991, have been updated. Symplectic manifolds and torus actions are investigated, with numerous examples of torus actions, for instance on some moduli spaces. Although the book is still centered on convexity results, it contains much more material, in particular lots of new examples and exercises.


Torus Actions on Symplectic Manifolds

Torus Actions on Symplectic Manifolds
Author: Michèle Audin
Publisher: Birkhäuser
Total Pages: 331
Release: 2012-12-06
Genre: Mathematics
ISBN: 3034879601

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The material and references in this extended second edition of "The Topology of Torus Actions on Symplectic Manifolds", published as Volume 93 in this series in 1991, have been updated. Symplectic manifolds and torus actions are investigated, with numerous examples of torus actions, for instance on some moduli spaces. Although the book is still centered on convexity results, it contains much more material, in particular lots of new examples and exercises.


$C^*$-Algebras of Homoclinic and Heteroclinic Structure in Expansive Dynamics

$C^*$-Algebras of Homoclinic and Heteroclinic Structure in Expansive Dynamics
Author: Klaus Thomsen
Publisher: American Mathematical Soc.
Total Pages: 138
Release: 2010-06-11
Genre: Mathematics
ISBN: 0821846922

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The author unifies various constructions of $C^*$-algebras from dynamical systems, specifically, the dimension group construction of Krieger for shift spaces, the corresponding constructions of Wagoner and Boyle, Fiebig and Fiebig for countable state Markov shifts and one-sided shift spaces, respectively, and the constructions of Ruelle and Putnam for Smale spaces. The general setup is used to analyze the structure of the $C^*$-algebras arising from the homoclinic and heteroclinic equivalence relations in expansive dynamical systems, in particular, expansive group endomorphisms and automorphisms and generalized 1-solenoids. For these dynamical systems it is shown that the $C^*$-algebras are inductive limits of homogeneous or sub-homogeneous algebras with one-dimensional spectra.


Small Modifications of Quadrature Domains

Small Modifications of Quadrature Domains
Author: Makoto Sakai
Publisher: American Mathematical Soc.
Total Pages: 282
Release: 2010
Genre: Mathematics
ISBN: 0821848100

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For a given plane domain, the author adds a constant multiple of the Dirac measure at a point in the domain and makes a new domain called a quadrature domain. The quadrature domain is characterized as a domain such that the integral of a harmonic and integrable function over the domain equals the integral of the function over the given domain plus the integral of the function with respect to the added measure. The family of quadrature domains can be modeled as the Hele-Shaw flow with a free-boundary problem. The given domain is regarded as the initial domain and the support point of the Dirac measure as the injection point of the flow.


The Moduli Space of Cubic Threefolds as a Ball Quotient

The Moduli Space of Cubic Threefolds as a Ball Quotient
Author: Daniel Allcock
Publisher: American Mathematical Soc.
Total Pages: 89
Release: 2011
Genre: Mathematics
ISBN: 0821847511

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"Volume 209, number 985 (fourth of 5 numbers)."


Metrics of Positive Scalar Curvature and Generalised Morse Functions, Part I

Metrics of Positive Scalar Curvature and Generalised Morse Functions, Part I
Author: Mark P. Walsh
Publisher: American Mathematical Soc.
Total Pages: 105
Release: 2011
Genre: Mathematics
ISBN: 082185304X

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It is well known that isotopic metrics of positive scalar curvature are concordant. Whether or not the converse holds is an open question, at least in dimensions greater than four. The author shows that for a particular type of concordance, constructed using the surgery techniques of Gromov and Lawson, this converse holds in the case of closed simply connected manifolds of dimension at least five.


Centres of Centralizers of Unipotent Elements in Simple Algebraic Groups

Centres of Centralizers of Unipotent Elements in Simple Algebraic Groups
Author: Ross Lawther
Publisher: American Mathematical Soc.
Total Pages: 201
Release: 2011
Genre: Mathematics
ISBN: 0821847694

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Let G be a simple algebraic group defined over an algebraically closed field k whose characteristic is either 0 or a good prime for G, and let uEG be unipotent. The authors study the centralizer CG(u), especially its centre Z(CG(u)). They calculate the Lie algebra of Z(CG(u)), in particular determining its dimension; they prove a succession of theorems of increasing generality, the last of which provides a formula for dim Z(CG(u)) in terms of the labelled diagram associated to the conjugacy class containing u.