Preface The geometrica l stud y o f differentia l equation s ha s a lon g an d distinguishe d history, goin g bac k t o th e classica l investigation s o f Sophu s Lie , Gasto n Darbou x and Eli e Cartan . Thei r idea s ar e a t th e sourc e o f a number o f developments whic h currently occupy a central position in several areas of pure and applied mathematics , including th e theor y o f completel y integrabl e evolutio n equations , th e calculu s o f variations an d th e stud y o f conservation laws . Our objectiv e i n these lecture s i s to giv e an overview o f a number o f significan t ideas an d result s tha t hav e been develope d ove r the pas t decad e i n the geometrica l study of differential equations . I t is of course impossible in the course often lecture s to cove r al l th e importan t advance s tha t hav e take n plac e i n suc h a broa d field o f research. Thi s survey is therefore far from complete , and it does not succeed in doing full justic e t o al l th e idea s tha t i t aim s t o convey . W e hav e chose n t o focu s ou r attention on a number of topics which we have found t o be of particular significance , or i n whic h w e have bee n involve d throug h ou r ow n research . W e hav e als o tried , in the spiri t o f the NSF-CBM S Researc h Conferenc e Series , to kee p a good par t o f the expositio n a t a level sufficiently elementar y a s to enabl e th e non-exper t reade r to gain a reasonable understandin g o f the mai n idea s and result s of this monograp h by a n independen t study . In wha t follows , w e give a brief descriptio n o f the content s o f each chapter . Chapter 1 serves to motivate som e of the mai n idea s an d principle s underlyin g the geometrica l stud y o f differential equations . Thes e for m th e threa d unifyin g th e whole series of lectures. The y includ e th e questio n o f the solvabilit y o f the Cauch y problem fo r first-order partia l differentia l equation s an d secon d orde r hyperboli c partial differentia l equation s b y ordinar y differentia l equatio n methods , th e con - cepts o f internal , externa l an d generalize d symmetries , an d th e local , globa l an d equivariant invers e problem s i n th e calculu s o f variations . W e als o lis t som e im - portant topic s i n th e geometrica l stud y o f differentia l equation s whic h wil l no t b e covered i n thi s monograph . Chapter 2 gives an introductio n t o the theor y o f external an d generalize d sym - metries o f differential equations . Externa l symmetrie s wer e discovere d an d studie d by Lie . A n externa l symmetr y i s a vecto r field i n th e ambien t je t bundl e which , when restricte d t o th e submanifol d define d b y a differential equation , i s tangent t o that submanifold . Th e flow o f a symmetr y thu s map s solution s t o solutions . Li e proved tha t th e existenc e o f a continuou s grou p o f external symmetrie s allow s on e in man y case s t o construc t group-invarian t solution s whic h ar e governe d b y differ - ential equation s involvin g fewe r independen t variables . Thi s metho d o f symmetr y reduction ha s thus becom e a very powerfu l too l fo r constructin g exac t solution s of partial differentia l equations . I t i s fair t o sa y tha t thi s represent s on e o f th e mos t successful application s o f the geometric study o f differential equations . Generalize d ix

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