The aim of the conference is to exchange ideas developed and being developed in three fields: geometric nonlinear control theory, geometrical mechanics, and geometry of quantum systems. The three topics use similar tools from differential geometry and may offer to each other new methods and ideas. Besides current research presentations, there are planned survey lectures, introducing new language and results to all domains described above. The main emphasis will be put on systems of finite dimension.
Quantum systems appear in the Quantum Information Theory (QIT) which is now rapidly developing field. Its experimental branch aims at construction of new quantum devices capable of storing, transforming, and transmitting information with efficiency and speed surpassing their classical counterparts. QIT renewed the interest in foundations of Quantum Mechanics and the geometry of quantum states. It is desirable to develop methods for effective production of particular quantum states of real physical systems like atoms, ions or molecules manipulated with the help of resonant, coherent radiation. The problem of construction of desirable states from available ones can be conveniently formulated as a control theory problem in the Hilbert space (or its projectivization) of the quantum system in question. The possibility of such a construction can be reduced to an appropriate accessibility or controllability question, whereas various demands summarized as ‘effectiveness’ (e.g. the speed of the evolution, the energetic cost etc.) can be treated as optimal control problem.
Controllability and optimal control problems have been successfully dealt with methods and tools of geometric control theory. This theory, developed over last 30 years, treats a control system as a collection of vector fields on a manifold, often exhibiting a special structure. Controllability criteria can be expressed in terms of the Lie algebra generated by the vector fields defining the system. In optimal control theory, the fundamental tool is the Pontryagin Maximum Principle and its geometric versions. Its recent applications in simple quantum systems give hopes of solving effectively the mathematical problems appearing in this field. The geometric control theory seems a perfectly suited tool for studying the above outlined issues of quantum engineering. Its connections to geometric and Hamiltonian mechanics giving rise to various problems of integrability and nonintegrabilty, can shed a new light on possibilities of effective construction of useful quantum states.