April-June 2010

Trimester in Combinatorics and Control

 

Control theory was born as an interdisciplinary branch of engineering and mathematics, concerned with the behavior of dynamical systems. It has become a major field in applied mathematics. Applications range from celestial mechanics in astronomy (including Earth and Solar System astronomy), to chemical physics, to biophysics, to the new, promising field of quantum control. Also, it is related to recently developed branches of mathematics, e.g. like rough path theory. The use of algebras of formal power series in control was first advocated in 1981 by M. Fliess, who developed a functional expansion, generalizing Lie series, today known as Fliess or Chen-Fliess series. In fact, since then calculus in free Lie algebras and formal power series played a major role in control theory.

However, Fliess' work made the introduction of Hopf-algebraic methods in control theory both indispensable and unavoidable. And the discovery of advanced combinatorial and algebraic structures lead to greater transparency eventually allowing to obtain profound insights and more information. From this perspective the explosive developments, especially during the last decade, of the theory of combinatorial Hopf algebras marks a pivotal point, right at the interface between modern mathematics (algebraic geometry, theory of noncommutative symmetric functions, theory of operads etc.) and applied fields.

The trimester Combinatorics and Control 2010 aims for nothing less than a further acceleration in innovation in the field. There is surely much more to come and there are reasons to believe we are nearing a tipping-point in basic and applied research at the interface of combinatorics and control, where the pace of discoveries will, at long last, match that of the questions.

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