Nonlinear Partial Differential Equations in Geometry and Physics -

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Résumé :
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Sommaire:
New Directions in 4-Manifold Theory.- Lecture 1: Donaldson and Seiberg-Witten Invariants.- Lecture 2: The Immersed Thorn Conjecture.- Lecture 3: Intersection Forms of Smooth 4-Manifolds.- References.- On the Regularity of Classical Field Theories in Minkowski Space-Time E3+1.- 1 Relativistic Field Theories.- 2 The Problem of Break-down.- 3 Energy estimates and the Problem of Optimal Local Well Posedness.- 4 Proof of the Null Estimates.- 5 The Proof of Theorem 4.- 6 Conclusions.- Static and Moving Vortices in Ginzburg-Landau Theories.- Lecture 1.- 1 Background and Models.- 2 The Work of Bethuel-Br?zis-H?lein and Others.- 3 Some Generalizations.- Lecture 2.- 1 Renormalized Energy.- 2 A Technical Result.- 3 Proof of Theorem A.- 4 Proof of Theorem B.- Lecture 3: The Dynamical Law of Ginzburg-Landau Vortices.- 1 Gor?kov-Eliashberg?s Equation.- 2 Uniqueness of Asymptotic Limit.- 3 Vortex Motion Equations.- References.- Wave Maps.- 1 Local existence. Energy method.- 1.1 The setting.- 1.2 Wave Maps.- 1.3 Examples.- 1.4 Basic questions.- 1.5 Energy estimates.- 1.6 L2-theory.- 1.7 Local existence for smooth data.- 1.8 A slight improvement.- 1.9 Global existence, the case m = 1.- 2 Blow-up and non-uniqueness.- 2.1 Overview.- 2.2 Regularity in the elliptic and parabolic cases.- 2.3 Regularity in the hyperbolic case.- 3 The conformai case m = 2.- 3.1 Overview.- 3.2 The equivariant case.- 3.3 Towards well-posedness for general targets.- 3.4 Approximation solutions.- 3.5 Convergence.- References.

All the essays are on the topmost professional level and are highly recommended to researchers and especially to young specialists in mathematical physics, PDE, differential geometry and topology, because they illustrate brilliantly the recent tendency in the theory of nonlinear PDE...namely the realization that structures originally introduced in the context of mathematical models in theoretical physics may turn out to have important applications in topology and (differential) geometry. --Mathematica Bohemica