## ICIAM 07The International Council for Industrial and Applied Mathematics (ICIAM) is the worldwide organisation of societies which are dedicated primarily of significantly to applied and/or industrial mathematics. The ICIAM Congresses, held every 4 years, are run under the auspices of the Council with the aim to advance the applications of mathematics in all parts of the world. The 6th ICIAM Congress was held in Zurich, Switzerland, 16-20 July 2007, and was attended by more than 3000 scientists from 47 countries. This volume collects the invited lectures of this Congress, the appreciations of the ICIAM Prize winners' achievements and the Euler Lecture celebrating the 300th anniversary of Euler. The authors of these papers are leading researchers of their fields, rigorously selected by a distinguished international program committee. The book presents an overview of contemporary applications of mathematics, new perspectives and open problems. |

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### Contents

Gregoire Allaire Francois Jouve and Nicolas Van Goethem | 3 |

Christian Ebenbauer Tobias Raff and Frank Allgower | 23 |

Luis A Caffarelli | 43 |

Claudio Canuto | 57 |

Margaret Cheney | 89 |

Weinan | 113 |

Michel Fortin Carl Robitaille Andre Fortin and Ali Rezgui | 131 |

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algebraic algorithm Anal analysis Appl applications approximation assume asymptotic boundary conditions Cantor set Comput conservation laws consider constant convergence convex corresponding decomposition defined denote density derivative dimensions Dirichlet discretization discussed dissipation inequality domain dynamics eigenvalues elliptic equations episodic memory estimates ETH Zurich Euler exact boundary controllability example Figure finite element methods formulation fully nonlinear function Gaussian geometric given global gradient greedy algorithm grids Hamilton-Jacobi equations ICIAM initial integral inverse problems iterative kernel Lagrangian linear Markov Math Mathematics Maxwell's equations measure mesh metastable metastable sets minimization minimum phase Monge-Ampere equation multigrid multigrid methods norm numerical observability obtained optimal parameters partial differential equations Phys polynomial preconditioner properties satisfies Section SIAM simulation singular smooth solve space stability stochastic structure subspace Theorem theory tion transfer operator transition matrix triangulation unique variables vector viscosity solutions wave