We propose a new framework for the study of continuous time dynamical systems on networks. We view such dynamical systems as collections of interacting control systems. We show that a class of maps between graphs called graph fibrations give rise to maps between dynamical systems on networks. This allows us to produce conjugacy between dynamical systems out of combinatorial data. In particular we show that surjective graph fibrations lead to synchrony subspaces in networks. The injective graph fibrations, on the other hand, give rise to surjective maps from large dynamical systems to smaller ones. One can view these surjections as a kind of "fast/slow" variable decompositions or as "abstractions" in the computer science sense of the word.
In order to deform a Lagrangian submanifold in Cn, we must understand how a tubular neighbourhood looks like. We prove here that a Lagrangian submanifold has a neighbourhood which is diffeomorphic to a neighbourhood of the zero section in its cotangent bundle. To be precise and explicit, we need to define a symplectic structure on the cotangent bundles and more generally to say what a symplectic structure on a manifold is
In this paper, we use the language of operads to study open dynamical systems. More specifically, we study the algebraic nature of assembling complex dynamical systems from an interconnection of simpler ones. The syntactic architecture of such interconnections is encoded using the visual language of wiring diagrams. We define the symmetric monoidal category W, from which we may construct an operad O(W), whose objects are black boxes with input and output ports, and whose morphisms are wiring diagrams, thus prescribing the algebraic rules for interconnection. We then define two W-algebras, G and L, which associate semantic content to the structures in W. Respectively, they correspond to general and to linear systems of differential equations, in which an internal state is controlled by inputs and produces outputs. As an example, we use these algebras to formalize the classical problem of systems of tanks interconnected by pipes, and hence make explicit the algebraic relationships among systems at different levels of granularity.
We show that if a Lie group acts properly on a co-oriented contact manifold preserving the contact structure, then the contact quotient is topologically a stratified space (in the sense that a neighborhood of a point in the quotient is a product of a disk with a cone on a compact stratified space). As a corollary, we obtain that symplectic quotients for proper Hamiltonian actions are topologically stratified spaces in this strong sense thereby extending and simplifying previous work.
These are notes for a course on contact manifolds and torus actions delivered at the summer school on Symplectic Geometry of Integrable Hamiltonian Systems at Centre de Recerca Matemàtica in Barcelona in July 2001. To be published by Birkhauser.
In this paper I construct, using off the shelf components, a compact symplectic manifold with a non-trivial Hamiltonian circle action that admits no Kaehler structure.The non-triviality of the action is guaranteed by the existence of an isolated fixed point.The motivation for this work comes from the program of classification of Hamiltonian group actions.The Audin-Ahara-Hattori-Karshon classification of Hamiltonian circle actions on compact symplectic 4-manifolds showed that all of such manifolds are Kaehler.Delzant's classification of 2n-dimensional symplectic manifolds with Hamiltonian action of ndimensional tori showed that all such manifolds are projective toric varieties, hence Kaehler.An example in this paper shows that not all compact symplectic manifolds that admit non-trivial Hamiltonian torus actions are Kaehler.Similar techniques allow us to construct a compact symplectic manifold with a Hamiltonian circle action that admits no invariant complex structures, no invariant polarizations, etc.
We show that the category of vector fields on a geometric stack has the structure of a Lie 2-algebra. This proves a conjecture of R.~Hepworth. The construction uses a Lie groupoid that presents the geometric stack. We show that the category of vector fields on the Lie groupoid is equivalent to the category of vector fields on the stack. The category of vector fields on the Lie groupoid has a Lie 2-algebra structure built from known (ordinary) Lie brackets on multiplicative vector fields of Mackenzie and Xu and the global sections of the Lie algebroid of the Lie groupoid. After giving a precise formulation of Morita invariance of the construction, we verify that the Lie 2-algebra structure defined in this way is well-defined on the underlying stack.
We provide a complete and self-contained classification of (compact connected) contact toric manifolds thereby finishing the work initiated by Banyaga and Molino and by Galicki and Boyer. Our motivation comes from the conjectures of Toth and Zelditch on the uniqueness of toric integrable actions on the punctured cotangent bundles on n-toru 𝕋n and of the two-sphere S2. The conjectures are equivalent to the uniqueness, up to conjugation, of maximal tori in the contactomorphism groups of the cosphere bundles of 𝕋n and S2 respectively.
We prove that the action of a reductive complex Lie group on a Kähler manifold can be linearized in the neighbourhood of a fixed point, provided that the restriction of the action to some compact real form of the group is Hamiltonian with respect to the Kähler form.
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