Geometry of Maslov cycles

We introduce the notion of induced Maslov cycle, which describes and unifies geometrical and topological invariants of many apparently unrelated situations, from real algebraic geometry to sub-Riemannian geometry

Quantum Control Landscapes

Numerous lines of experimental, numerical and analytical evidence indicate that it is surprisingly easy to locate optimal controls steering quantum dynamical systems to desired objectives. This has enabled the control of complex quantum systems despite the expense of solving the Schrodinger equation in simulations and the complicating effects of environmental decoherence in the laboratory. Recent work indicates that this simplicity originates in universal properties of the solution sets to quantum control problems that are fundamentally different from their classical counterparts. Here, we review studies that aim to systematically characterize these properties, enabling the classification of quantum control mechanisms and the design of globally efficient quantum control algorithms.Comment: 45 pages, 15 figures; International Reviews in Physical Chemistry, Vol. 26, Iss. 4, pp. 671-735 (2007

On the Alexandrov Topology of sub-Lorentzian Manifolds

It is commonly known that in Riemannian and sub-Riemannian Geometry, the metric tensor on a manifold defines a distance function. In Lorentzian Geometry, instead of a distance function it provides causal relations and the Lorentzian time-separation function. Both lead to the definition of the Alexandrov topology, which is linked to the property of strong causality of a space-time. We studied three possible ways to define the Alexandrov topology on sub-Lorentzian manifolds, which usually give different topologies, but agree in the Lorentzian case. We investigated their relationships to each other and the manifold's original topology and their link to causality.Comment: 20 page

On the geometry of the set of symmetric matrices with repeated eigenvalues

We investigate some geometric properties of the real algebraic variety \u394 of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart\u2013Young\u2013Mirsky-type theorem for the distance function from a generic matrix to points in \u394. We exhibit connections of our study to real algebraic geometry (computing the Euclidean distance degree of \u394) and random matrix theory

Almost exponential maps and integrability results for a class of horizontally regular vector fields

We show a higher order integrability theorem for distributions generated by a family of vector fields under a horizontal regularity assumption on their coefficients. We use as chart a class of almost exponential maps which we discuss in detailsComment: arXiv admin note: material from arXiv:1106.2410v1, now three separate articles arXiv:1106.2410v2, arXiv:1201.5228, arXiv:1201.520

Geometry of optimal control problems and Hamiltonian systems

We explain a general variational and dynamical nature of nice and powerful concepts and results mainly known in the narrow framework of Riemannian geometry. This concerns Jacobi fields, Morse's index formula, Levi Civita connection, Riemannian curvature and related topics. There is an evidence that the described results are just parts of a united beautiful subject to be discovered on the crossroads of Differential Geometry, Dynamical Systems, and Optimal Control Theory

Sub-Riemannian curvature in contact geometry

We compare different notions of curvature on contact sub-Riemannian manifolds. In particular we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the sub-Riemannian distance. We explicitly compute their expressions in terms of the standard tensors of contact geometry. As an application of these results, we prove a version of the sub-Riemannian Bonnet-Myers theorem that applies to any contact manifold, with special attention to contact Yang-Mills structures

On conjugate times of LQ optimal control problems

Motivated by the study of linear quadratic optimal control problems, we consider a dynamical system with a constant, quadratic Hamiltonian, and we characterize the number of conjugate times in terms of the spectrum of the Hamiltonian vector field H . We prove the following dichotomy: the number of conjugate times is identically zero or grows to innity. The latter case occurs if and only if H has at least one Jordan block of odd dimension corresponding to a purely imaginary eigenvalue. As a byproduct, we obtain bounds from below on the number of conjugate times contained in an interval in terms of the spectrum of H