The goal of the trimester is: to provide a broad, accessible and updated presentation of Microlocal Analysis and of some of its most recent applications to Partial Differential Equations (PDE), both linear and nonlinear, and to related fields of analysis and geometry; to introduce open problems and exciting directions of new research ; to offer a wide outline of advanced topics to the interested young mathematicians; to provide specialists from various Universities the opportunity to meet and exchange ideas.
The last half century has seen impressive developments in the analytic tools and in the ensuing deepening of our understanding of PDE theory. Such progress has been made possible in particular by the greatly expanded reach of the Fourier transform in suitably adapted versions (e.g., beyond groups to manifolds as Fourier integrals, or to the analytic category as the FBI transform), leading to:
pseudodifferential calculus (in its Weyl formulation) and its refinements based on Littlewood-Paley decompositions; paradifferential operators, extending pseudodifferential operators to nonlinear equations and allowing a deep analysis of the singularities of their solutions; Fourier integral operators and the WKB methods of linear, as well as nonlinear, geometrical optics and their connections with symplectic geometry.
Intimately linked to Fourier analysis are the Strichartz inequalities which have been shown, in the last decades, to play a crucial role in:
the iterative methods leading to the solution of nonlinear PDE from that of related linear PDE; the study of the local Lp regularity of solutions - mainly of hyperbolic and Schrödinger type equations.
Lectures, conferences and seminars will be devoted to all these aspects of modern analysis. Prominence will also be given to some of the most important applications: compressible and incompressible flows, Euler and Navier-Stokes equations, dispersive and bilinear estimates for equations of relativity theory and quantum mechanics, etc.
Pseudodifferential operators and oscillatory integrals (Fourier integral operators) are also closely related to harmonic analysis, in particular to problems on nilpotent groups and Fourier methods in complex and CR geometry. In this direction, the last month of this trimester will overlap with the trimester in Harmonic Analysis. This will allow both audiences and lecturers to venture in areas of common interest.
In conclusion, the title chosen for this scientific project, Phase Space Analysis of PDE, is a good summary of a simple fact: the modern approach to all fields of PDE requires a good understanding of many areas of mathematics, including symplecting geometry and the developments of microlocal analysis. The phase space vision of PDE has led to many new results and to wider perspectives. This trimester is intended to provide an accessible description to a large segment of this type of results.