XXth century appears sometimes as mathematically non linear when one thinks to several achievements inherited from XIXth century like chaotic dynamics, non linear partial differential equations, non linear algebra to quote just a few. At the same time last century invented quantum mechanics as a linear necessary replacement of classical Newtonian dynamics, and linear logic  as a refinement of classical and intuitionist logic.

Will XXIth century be linear or non linear? ArpiLYSM opens the debate.

The school will provide, among several seminars, three minicourses devoted to  the different themes involved, in the title of the school, and will not  necessitate any prerequisit other than elementary mathematics and physics knowledge . Wednesday will be concerned by the opposition linear versus non linear in other fields of mathematics and more, such as economy, art, and more general processes of thinking.

        Thierry Paul, Stefano Rossi, Lorenzo Tortora di Falco, Lionel Vaux-Auclair, org.

Courses

• Logic, by Thomas Erhard and Laurent Regnier

"Title and abstract TBA"

• Control, by Emmanuel Trélat

"Control theory: linear vs nonlinear aspects."

In this short course I will give an introduction to control theory, in finite and in infinite dimension, with a particular focus on linear and nonlinear aspects. 

For linear systems, the theory is well established and controllability properties can be derived, by duality, by establishing observability inequalities. 

For nonlinear systems, linearization (around some equilibrium or along a reference trajectory) is a powerful tool to establish local controllability properties. I will introduce several other approaches, such as the Coron return method, or the use of Lie brackets. 

• Quantum, by Thierry Paul  (et al TBC)

"The unreasonnable linearity of Quantum Mechanics and the beauty of the square root"

Quantum Mechanics is by essence linear. This unintuitive fact is essential for the understanding of the stability of the world, but also induces many surprising facts when comparing quantum structures with their classical counterpart, by essence non linear.

How can a non linear paradigm fit into a  linear one without loosing its deep meaning?  Square root, square root, square root.

In this short course we will show and link very different situations illustrating this pheomenon, like dynamical systems, optimal transport, Legendre transform and this fantastic transformer from non linear to linear which is the process of quantization and its relation with calculus of variation.