• Matthias KAWSKI (Arizona)
  Combinatorial Aspects in Control
  Abstract: A key feature that distinguishes control systems from classical dynamical systems is that one may choose from, or switch between different flows. Much of the richness of control and many challenges arise from the common lack of noncommutativity of the flows. This motivates ever deeper investigations of the underlying geometric, algebraic, and combinatorial structures. The lectures will focus on combinatorial aspects that underlie controllability and optimality, with further applications to approximating systems and state-space realizations. From exponential Lie series and Hall sets, this leads to the chronological calculus and combinatorial Hopf algebras.


• Dominique MANCHON (CNRS) and Ander MURUA (San Sebastián)
  Combinatorial Hopf Algebras and Applications
  Abstract: This series of lectures aims to present pre-Lie algebras and connected Hopf algebras naturally associated to them, with emphasis on applications related to differential equations following the pioneering work of J. Butcher. We shall mostly focus on Hopf algebras related to pre-Lie algebras of rooted trees (i.e. free pre-Lie algebras), which are central objects for studying differential equations and their numerical approximation through Runge-Kutta methods. We also will make a small incursion in the realm of Quantum Field Theory, with an introduction to Birkhoff-Connes-Kreimer renormalization in connected Hopf algebras. The pre-requisites are basic linear algebra (including tensor product) and elementary differential calculus of several variables.


• Jean-Yves THIBON (Marne-la-Vallée)
  Noncommutative Symmetric Functions Theory
  Abstract: These lectures will focus on the simplest combinatorial Hopf algebras, Noncommutative symmetric functions and quasi-symmetric functions. The connection with Lie series, in particular the Hausdorff series, the continuous BCH, Magnus and Fer-Zassenhaus expansions will be explained in terms of Lie idempotents in Solomon's descent algebras. Motivated by these examples, other combinatorial Hopf algebras will be introduced, with their realization in terms of noncommutative polynomials.

Each lecture is accompanied by problem sessions where the students can test their understanding of the lectures in concrete problems. We trust that such an occasion will inspire especially young researchers, providing them with new horizons and exposing them to the leading scientists view of future developments in the field. It addresses physicists and mathematicians alike sharing a common interest in participating in the development of a dynamic research interaction between applied and pure mathematics centered around new methods steaming from the recent progress in the understanding of algebro-combinatorial structures in control theory.

Concerning the application procedure, please use the online APPLICATION FORM.
The conference fee is 100 euro, and 75 euro for students. Partial financial support will be available. Payment should be made by bank order to the following bank account:
Bank: Banco Santander
Account: 0049 0932 47 2090960099
IBAN: ES65 0049 0932 4720 9096 0099
BIC/SWIFT: BSCHESMM
Please, add explicitly in the reference text of your transfer order:
'COCO2010 School + surname of the participant'
Please, send a confirmation (including your name and transfer details) by e-mail to [email protected], no later than March 15, 2010.
However, to complete your application we ask you kindly to send an e-mail to [email protected] including a letter of motivation, your CV with the list of publications and the name and email address of one senior scientist willing to write a letter of recommendation for you.

The Center provides many facilities and an agreeable atmosphere to allow the participants to carry their work during the meeting.

Benasque is a beautiful mountain village located 1140 m above sea level, in a valley at the heart of the Pyrenees, about 140 km north of the city of Zaragoza. The beautiful landscapes, scenic hikes and the proximity to the highest peaks in the Pyrenees make Benasque one of the main tourist sites in the region.

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Further Information.

• List of participants.
• Scientific program.
• Pictures.