Geodesics in subriemannian manifolds: Dido meets Heisenberg Chow's theorem: Getting from A to B A remarkable horizontal curve Curvature and nilpotentization Singular curves and geodesics A zoo of distributions Cartan's approach The tangent cone and Carnot groups Discrete groups tending to Carnot geometries Open problems Mechanics and geometry of bundles: Metrics on bundles Classical particles in Yang-Mills fields Quantum phases Falling, swimming, and orbiting Appendices: Geometric mechanics Bundles and the Hopf fibration The Sussmann and Ambrose-Singer theorems Calculus of the endpoint map and existence of geodesics Bibliography Index.

A Tour of Subriemannian Geometries, Their Geodesics and Applications

Introduction What are Poincare and Sobolev inequalities? Poincare inequalities, pointwise estimates, and Sobolev classes Examples and necessary conditions Sobolev type inequalities by means of Riesz potentials Trudinger inequality A version of the Sobolev embedding theorem on spheres Rellich-Kondrachov Sobolev classes in John domains Poincare inequality: examples Carnot-Caratheodory spaces Graphs Applications to PDE and nonlinear potential theory Appendix References.

Sobolev met Poincaré

Symplectic Manifolds.- Symplectic Forms.- Symplectic Form on the Cotangent Bundle.- Symplectomorphisms.- Lagrangian Submanifolds.- Generating Functions.- Recurrence.- Local Forms.- Preparation for the Local Theory.- Moser Theorems.- Darboux-Moser-Weinstein Theory.- Weinstein Tubular Neighborhood Theorem.- Contact Manifolds.- Contact Forms.- Contact Dynamics.- Compatible Almost Complex Structures.- Almost Complex Structures.- Compatible Triples.- Dolbeault Theory.- Kahler Manifolds.- Complex Manifolds.- Kahler Forms.- Compact Kahler Manifolds.- Hamiltonian Mechanics.- Hamiltonian Vector Fields.- Variational Principles.- Legendre Transform.- Moment Maps.- Actions.- Hamiltonian Actions.- Symplectic Reduction.- The Marsden-Weinstein-Meyer Theorem.- Reduction.- Moment Maps Revisited.- Moment Map in Gauge Theory.- Existence and Uniqueness of Moment Maps.- Convexity.- Symplectic Toric Manifolds.- Classification of Symplectic Toric Manifolds.- Delzant Construction.- Duistermaat-Heckman Theorems.

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Lectures on Symplectic Geometry

A reversed-field pinch is a toroidal device for magnetically confining plasmas, and a potential alternative to the tokamak for a future fusion reactor. Observations of the evolution of a reversed-field-pinch plasma towards a self-organized single-helicity state suggest that instability problems, which have previously hindered the development of these devices, could now be overcome.

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Self-organized helical equilibria as a new paradigm for ohmically heated fusion plasmas

This is a tutorial on some of the main ideas in KAM theory. The goal is to present the background and to explain and compare somewhat informally some of the main methods of proof. It is an expanded version of the lectures given by the author in the Summer Research Institute on Smooth Ergodic Theory Seattle, 1999. The style is pedagogical and expository and it only aims to be an introduction to the primary literature. It does not aim to be a systematic survey nor to present full proofs.

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A Tutorial on Kam Theory

Symplectic twist maps

We show that strictly abnormal geodesics arise in graded nilpotent Lie groups. We construct a group, with a left invariant bracket-generating distribution, for which some Carnot geodecics are strictly abnormal and, in fact, not normal in any subgroup. In the 2-step case we also prove that these geodesics are always smooth. Our main technique is based on the equations for the normal and abnormal curves, which we derive (for any Lie group) explicitly in terms of the structure constants.

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A note on Carnot geodesics in nilpotent Lie groups

This article presents new methods for the geometrical analysis of phyllotactic patterns and their comparison with patterns produced by simple, discrete dynamical systems We introduce the concept of ontogenetic graph as a parsimonious and mechanistically relevant representation of a pattern The ontogenetic graph is extracted from the local geometry of the pattern and does not impose large-scale regularity on it as for the divergence angle and other classical descriptors We exemplify our approach by analyzing the phyllotaxis of two asteraceae inflorescences in the light of a hard disk model The simulated patterns offer a very good match to the observed patterns for over 150 iterations of the model

The Possible and the Actual in Phyllotaxis: Bridging the Gap between Empirical Observations and Iterative Models

Ghost Circles for Twist Maps

Why living forms develop in a relatively robust manner, despite various sources of internal or external variability, is a fundamental question in developmental biology. Part of the answer relies on the notion of developmental constraints: at any stage of ontogenesis, morphogenetic processes are constrained to operate within the context of the current organism being built. One such universal constraint is the shape of the organism itself, which progressively channels the development of the organism toward its final shape. Here, we illustrate this notion with plants, where strikingly symmetric patterns (phyllotaxis) are formed by lateral organs. This Hypothesis article aims first to provide an accessible overview of phyllotaxis, and second to argue that the spiral patterns in plants are progressively canalized from local interactions of nascent organs. The relative uniformity of the organogenesis process across all plants then explains the prevalence of certain patterns in plants, i.e. Fibonacci phyllotaxis.

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Christophe Golé

Papers

Symplectic twist maps

A note on Carnot geodesics in nilpotent Lie groups

The Possible and the Actual in Phyllotaxis: Bridging the Gap between Empirical Observations and Iterative Models

Ghost Circles for Twist Maps

Phyllotaxis as geometric canalization during plant development